Fundamentals of Quantum Mechanics

Electronic Engineering Materials & Nanotechnology

Chapter 4

Fundamentals of Quantum Mechanics Contents: 4-1. Introduction 4-2. Development of Quantum Mechanics 4-3. Uncertainty Principle 4-4. Wave Mechanics 4-5. Wave Packets in Space and Time 4-6. Pauli’s Exclusion Principle 4-7. Complementarity Principle 4-8. Correspondence Principle 4-9. Hidden Variables 4-10. Quantum Entanglement 4-11. Bosons and Fermions 4-12. Observables and Operators 4-13. Probability Current Density 4-14. Schrödinger's Equation 4-15. Dirac's Equation 4-16. Solution of the Schrödinger Equation 4-16.1. Case A. 4-16.2. Case B. 4-16.3. Case C. 4-16.4. Case D. 4-16.5. Case E. 4-16.6. Case F. 4-16.7. Case G.

1-Dimensional Potential Well 3-Dimensional Potential Well 1-Dimensional Potential Barrier The Hydrogen Atom 1-Dimensional Harmonic Oscillator Quantum Anharmonic Oscillator Coupled Harmonic Oscillators

-183Prof. Dr. Muhammad EL-SABA


Chapter 4

Contents of Chapter 4 (Cont.) 4-17. Approximate Methods to solve the Schrödinger Equation 4-17.1. Gundlatch Method 4-17.2. WKB. (Wentzel-Kramers-Brillouin) Approximation 4-17.3. Transmission Matrix Method (TMM) 4-17.4. QTBM

4-18. Matrix Formulation of Quantum Mechanics 4-19. Dirac Notations 4-19.1. 4-19.2. 4-19.3. 4-19.4. 4-19.5.

Hilbert Space Probability Amplitude Expectation Value of an Operator Projection Operator Density Operator

4-20. Quantum Pictures 4-21. Quantum Perturbation Theory 4-21.1. Stationary Perturbation Theory 4-21.2. Applications of the Stationary Perturbation Theory i. Zeeman Effect ii. Stark Effect 4-21.3. Dynamic Perturbation (Scattering) Theory 4-21.4. Transition Probability 4-21.5. Optical Transitions i. Absorption probability ii. Emission probability

4-22. Second Quantization 4-22.1. Creation and Annihilation (destruction) Operators 4-22.2. Fermion Operators 4-22.3. Boson Operators 4-23. Correlation of Solid-State Physics, Nuclear Physics & Cosmology 4-23.1. Matter Basic Particles 4-23.2. Physical Forces between Basic Particles 4-23.3. The Standard Model

4-24. 4-25. 4-26. 4-27. 4-28. 4-29. 4-30.

Quantum Field Theory Unified Field Theory Gauge Field Theory Final Theory and Theory of Everything (ToE) Summary Problems Bibliography

“I think it is safe to say that no one understands quantum mechanics” Physics Professor: Richard P. Feynman



Chapter 4

Fundamentals of Quantum Physics 4-1. Introduction The field of material physics includes several domains. The following figure depicts the basic theories of material physics and how they are related to size of the studied objects as well as their speed.

Fig. 4-1. Domains of physics

Classical mechanics (developed by the Islamic era scientists1, Newton, Lagrange and the Hamiltonian principles) are concerned with classical 1

In 1000-1030, Abul Rayhan al-Biruni introduces experimental scientific methods in statics and dynamics, and unified them into the science of mechanics; he also combined the fields of hydrostatics with dynamics to create the field of hydrodynamics, which he helped to mathematize and realize that acceleration is connected with non-uniform (variable speed) motion. In 1000-1030, Ibn al-Haytham (Alhazen) and Ibn Sina (Avicenna) developed the concept of inertia (known as Newton’s 1st law) and the concept momentum (p=m.v). In 1100-1138, Abū-Bakr Muhammad ibn Yahya ibn al-Sāyigh (Avempace) developed the concept of a reaction force (known as Newton’s 3rd law). In 1130-1165, Hibat Allah Abu'l-Barakat al-Baghdaadi discovered that force is proportional to acceleration rather than speed, a fundamental law in classical mechanics (known as Newton’s 2nd law). In 1121, Al-Khazini published The Book of the Balance of Wisdom, in which he developed the concepts of gravitational potential energy and gravity at-a-distance (known as Newton gravity law). In 1687 - Isaac Newton published his Philosophiae Naturalis Principia Mathematica, in which he formulated Newton's laws of motion and Newton's law of gravity, after centuries of their discovery and publishing, by Islamic era scientists! -185Prof. Dr. Muhammad EL-SABA


Chapter 4

systems whose dimensions are much greater than 1nm and their speed is much smaller than the speed of light. When the speed of a studied body is approaching the speed of light, the relativistic mechanics (developed by Einstein and others) should be used. When the size of the studied system is getting smaller and approaches 1nm (molecules and atoms) the classical mechanics is no longer valid, and should be replaced by the socalled quantum mechanics (developed by Planck, De Broglie, Heisenberg, Schrodinger and others). The so-called, quantum field theory is concerned with small particles which move with relativistic speeds (developed by Dirac and others). In solid-state materials there are about 1023 electron and ion packed into a volume of 1cm3. This high packing density implies a very small inter-particle distances (about 1Å) and high collision rate (about 1013 collision/sec). Because of the small inter-particle distance, the classical mechanics are no longer valid as the instantaneous position of the electron cannot be accurately defined, For instance, the electron motion in solids should be better analyzed by a probabilistic theory, such as the quantum theory, instead of the deterministic classical mechanics. Therefore, the explanation of material properties requires the knowledge of particles and their quantum nature. 4-2. Development of Quantum Physics Historically, the earliest versions of quantum physics were formulated in the first decade of the 20th century at around the same time as the atomic theory and the corpuscular theory of electromagnetic waves and light. Classical physics fails in explaining many material properties, such as color and specific heat. We have pointed out, so far, that the classical Dullong - Petit law for specific heat (Cv=3NkB) does not hold in solids at low temperature. Also, the classical physics failed to explain some phenomena in the absorption and emission of heat radiation. In fact, the classical physics was unable to account for some observations in blackbody radiation. It predicted that the emission or radiation of heat should continue to rise at shorter wavelengths, which is not true. Actually, the experiment shows that the intensity of heat radiation increases as wavelength decreases until a certain point, where it reaches to a certain maximum (not infinity! as predicted by classical physics; this is paradox called the ultraviolet catastrophe), and then decreases again. 4-2.1. Quantization of Energy (Max Planck) In order to explain black body radiation, Max Planck suggested (in 1900) that electromagnetic energy of radiation could only be emitted in quantized form, i.e. the energy could only be a multiple of elementary



Chapter 4

units (quanta)2 whose energy is proportional to the frequency of radiation.

E=hf = 

(4-1.a)

where the proportionality factor h is the Planck constant and = h/2 . The value of Planck’s constant can be determined from measurements of black bodies or other light sources. All such measurements coincide and yield ħ = 1.054571726x10−34 J.s. All attempts to observe physical actions values smaller than this fail. Planck’s constant is considered as the fundamental action constant in quantum physics. According to this observation, Planck postulated that the emitted radiation is composed of bunches or packets of discrete quanta whose total energy is given by:

En = n h f = n ħ n

(4-1b)

where n is an integer. Using this postulate, Planck was able to derive his famous radiation law, which expresses the intensity of radiation I from a hot body in terms of its temperature T and radiation wavelength .

1  8hc  I( T , )   5     exp( hc / k B T )  1

(4.1c)

Fig. 4-2. Radiation intensity of a hot body versus radiation wavelength, according to Planck's formula and the classical Rayleugh-Jeans law. The shown curve corresponds to the sum body, whose surface temperature is about 6000K

2

In his 1901 article in Annalen der Physik Max Planck called these packets "energy elements". The word quanta (singular quantum) was used even before 1900 to mean particles or amounts of different quantities, including electricity. -187Prof. Dr. Muhammad EL-SABA


Chapter 4

This relation coincides with the experimental observations, which indicate that the intensity of radiation decreases at short wavelengths. This would imply that higher modes would be less populated and avoid the ultraviolet catastrophe of the classical Rayleigh-Jeans law. The quantum idea was used by A. Einstein, in 1905, to interpret some experimental observation on the photoelectric effect (reported in 1887by Heinrich Hertz, who observed that shining light can eject electrons from certain materials). .Einstein proposed that the relation between the energy absorbed (KE) by the electron, which is ejected from a metal by light quanta, and the light quanta frequency (v) is given by the relation:

KE = h – e m

(4-1d)

where em is known as the metal work function, which is defined as the energy required to eject a free electron just outside the metal. This relation was verified experimentally, in 1916, by Millikan, as shown in figure 4-3. On the basis of Planck’s quantum hypothesis, Einstein postulated that light itself consists of individual quantum particles (photons). However, many researchers in the past were unconvinced that the photoelectric effect proves the existence of photons. Historically, the most important argument for the necessity of light quanta was given by Henri Poincare, in 1911 and 1912.

Fig. 4-3 Kinetic energy of ejected electron versus incident light frequency. Data from Robert Millikan, 1916.



Chapter 4

4-2.2. Bohr’s Model The idea that atoms are small solar systems was proposed in 1901, by Jean Perrin. In 1913, Niels Bohr used this idea, combining it with the Planck’s quantization idea, to formulate his model for the hydrogen atom. The Bohr model is based on the following postulates: 1- Electrons move in closed orbits of fixed radii (rn) around the nucleus. 2- Electrons may only move from one closed orbit to another by emission or absorption of quanta of energy (E = ). 3- The magnitude of angular momentum (m v rn) of an electron in a closed orbit is n  where n is an integer. Note that the first postulate is based on Rutherford's theorem (1911) and the second postulate is based on Planck's theorem (En=n). Also, the third postulate is based on Hartey's quantization theorem (1913), Wilson's rule (1915) and Sommerfeld's quantization theorem (1916), which states that the integral of each canonical momentum with respect to its coordinate over a cycle of motion must be an integer multiple of h. This means that ∮p.dl = nh. In other words we can write:

∮ p.r d = 2 rn p = 2 rn p = nh

or m v rn = n

(4-2)

Using the above postulates, Bohr derived expressions for the discrete orbital radii of the hydrogen atom rn and their energies En as well as the frequencies of absorbed or emitted light from an excited hydrogen atom. By equating the coulomb force (between one electron and one proton) to the centripetal force (due to the orbital motion of the electron around the atom center) such that e2/(4orn2 ) = m v2/rn we get:

 4  0  2  2  n  0.53 n 2 rn =  2  me 

A

(4-3a)

 m e4  1  1   En = -   13.6  2 2 2  2 n   8 o h  n

eV

(4-3b)

o

Also the frequency of absorbed or emitted light due to a transfer from one orbit to another is given by:

f=

ΔE -13.6  1 1 =  2 - 2 h h  n2 n1  -189-

Prof. Dr. Muhammad EL-SABA

(4-4)


Chapter 4

Fig. 4-4 Illustration of the Bohr atom model.

While the Bohr model was a good step towards understanding the quantum theory of atoms, it is not in fact a correct description of the nature of electron orbits. Some of the pitfalls of the Bohr model are: It fails to provide any justification of why certain spectral lines are brighter than others. There is no mechanism for the calculation of transition probabilities in this model. The Bohr model treats the electrons as if they were miniature planets, with definite radii and momenta. This is in direct violation of the uncertainty principle which dictates that position and momentum of particles cannot be simultaneously determined. Note 4-1. What’s Momentum and What’s Angular Momentum? In classical mechanics, the momentum of an object is defined as its mass multiplied by its velocity (p = mv). Momentum is a physical quantity which is closely related to forces. In both classical and quantum mechanics, the following relation stands valid: F = dp / dt Remarkably momentum is a conserved quantity. This makes momentum extremely useful in solving a great variety of real-world problems. The angular momentum is defined as follows: L = rxp In one dimension L = rp = (C/2) (h/ ) = (ℓ /2)(h/) = ℓ  -190Prof. Dr. Muhammad EL-SABA


Chapter 4

The angular momentum of a particle due to its motion through space is quantized in units of . In 3-dimension we have: L = rxp = Lx ax + Ly ay + Lz, az such that Lz = xpy − ypx,

Lx = ypz − zpy

Ly = zpx − xpz

The total angular momentum associated with a particular state is a physical quantity of interest. This is measured by the operator corresponding to the sum of squares of its components,

Note 4-3. Fine Structure Constant and Bohr’s Magneton It comes from the above discussion that the ratio of the electron velocity around a proton and the velocity of light is given by:

 = v/c = e2/4oc = 1/137 This is called the Sommerfeld fine structure constant. This means the electron orbiting around a proton with a velocity about 2200km/s. The magnetic field due to this circular motion can be calculated from Maxwell’s equations. It is given by:

B = o ( e / 2m) This is called the Bohr magneton. 4-2.3. Duality Principle (Louis de Broglie) Until 1924, it seemed that matter has a particular nature. Louis de Broglie suggested in his doctoral dissertation in 1923, that all forms of matter have wave as well as particle properties, just like light. The wavelength, of a particle, such as an electron, is related to its momentum, by the same relationship as for a photon:

= h / p -191Prof. Dr. Muhammad EL-SABA

(4-5)


Chapter 4

where p is the momentum of the particle and  is the wavelength of the associated wave. It follows form this relation that the momentum p of a particle is related to the associated wave vector k = 2 by: p=ħk

(4-6)

The de-Broglie hypothesis was confirmed experimentally three years later by Davisson and Germer in 1927 and later by G. Thomson in 1928. The Davisson-Germer experiment demonstrated the wave nature of the electron, confirming the earlier hypothesis of de-Broglie. A beam of electrons was guided through a crystal grid and the predicted interference patterns were observed. As shown in figure 4-5, Davisson and Germer built a vacuum apparatus for the purpose of measuring the energies of electrons scattered from a metal surface. Electrons from a heated filament were accelerated by a voltage and allowed to strike the surface of nickel metal. It was a great surprise to find that at certain angles there was a peak in the intensity of the scattered electron beam. These interference patterns indicated the wave behavior for electrons.

Fig, 4-5. Illustration of the Davisson-Germer experiment of electron diffraction



Chapter 4

Note 4-4. Illustration of the Davisson-Germer 1- Experimentally The path length difference for constructive interference d= 0.215 nm (the nickel lattice spacing) and the angle of peak intensity  = 50ᵒ Such that  = d sin  = 0.165 nm

2- Theoretically, we have E = 54 eV. Therefore,

= 0.167 nm

Fig, 4-6. Illustration of the Thomson experiment of electron diffraction by graphite. (1) filament heating supply, (2) cathode connection, (3) internal resistor, (4) filament. (5) cathode, (6) anode, (7) anode connection (HV), (8) focusing electrode, (9) polycrystalline graphite grating, (10) Boss, (11) fluorescent screen. -193Prof. Dr. Muhammad EL-SABA


Chapter 4

A similar experiment which demonstrates the interference and diffraction of small particles is the tow-slit interference experiment. Feynman said that this phenomenon has in it the heart of quantum mechanics. In fact, it contains the mystery of wave-particle duality. In telling you how it works, you will understand the basic peculiarities of quantum mechanics. Here a source of light shines on a screen in which have been cut two vertical slits, A and B, as shown in the following figure.

Fig, 4-7. Illustration of the two-slit interference experiment.



Chapter 4

Light which passes through the slits is detected on a distant screen. We might expect the illumination on the detector screen to consist of two paler patches where light from only one slit falls, and a brighter panel in the centre (in the region between a and b in the diagram) where the two beams overlap. In fact we see a pattern of light and dark vertical stripes in the central region. This is the 2-slit interference pattern. The interference pattern is easily understood if we think of waves from the two slits reaching the same point on the detector and interfering. But it would not be expected for streams of particles. Each particle would go through one slit or the other, and we should expect an increased number of particles to arrive in the central area of the detector. But the detection of one particle cannot be cancelled out by the arrival of another. The dark stripes - the minima - of the pattern cannot be explained in terms of particles. Since the pattern can be built up with particles going through the apparatus one at a time, the path of each individual particle must be constrained so that it avoids arriving at an interference minimum. If we close slit B, to try and establish which regions of the pattern are due to particles going through slit A, the pattern vanishes and particles begin to arrive at minima - points which they would be unable to reach if both slits were open. It seems that the path of a particle passing through one slit is affected by whether or not the other slit is open or not. To explain the two-slit experiment we need a wave to be associated with each particle. This wave determines the interference pattern. Table 4.1 shows in ascending order some of the significant theories and experiments before and during the development of the theory of quantum mechanics. Since the demonstrations of wave-like properties in photons and electrons, similar experiments have been conducted with neutrons and protons. Among these famous experiments are those of Estermann and Stern in 1929. Authors of similar recent experiments with atoms and molecules claim that such particles also act like waves. For instance, the diffraction of fullerene molecules (C60) was reported in 1999. However, it worth noting that the Planck constant h is extremely small and that explains why we don't perceive a wave-like quality of everyday objects: their wavelengths are exceedingly small. Following the initial success of the Bohr model, and the experiments, which confirmed the duality nature of matter, quantum mechanics was about to extend its frontiers.



Chapter 4

Table 4-1. Development of quantum physics.

Experiment/Theory Atomic theory Diffraction of light EM waves Discovery of electron Black-body radiation

Reference John Dalton Young J. Maxwell J.J. Thomson M. Planck

Description Matter consists of atoms Light is a wave Light is an EM wave, Electron is a particle E=nhf, Particle-like waves Photoelectric effect A. Einstein E=hf-e, Light particlelike wave Atomic structure E. Rutherford Atom = nucleus+electrons Crystal specific heat Debye Quantized values of Cv Electron absorption Frank Hearty Quantized atomic states Bohr Atom Bohr Bohr Atom model Electron spin Stern-Gerlach Quantized angular momentum (spin) Duality principle L. de Broglie =h/p Exclusion Principle W. Pauli Pauli exclusion principle Schrodinger equation E. Schrodinger E= H Uncertainty principle Heisenberg x.px > /2, E.t > /2 Davisson-Germer, Confirmation Electron diffraction of de G.P. Thomson by crystal Broglie hypothesis. Electrons are wave-like

Date 1803 1803 1869 1897 1900 1904 1911 1912 1913 1913 1922 1924 1925 1926 1927 1927 1928

4-3. Uncertainty Principle (Heisenberg) There is one important difference between classical and quantum mechanics. In quantum mechanics, we do not speak about what will happen in a certain circumstance to the electron. Rather we speak about the probability that an electron will arrive in a given circumstance. The only thing that we can predict in quantum mechanics is the probability of different events. Perhaps the electron has some kind of internal works some inner variables - that we do not yet know about. Perhaps that is why we cannot predict precisely what will happen. If we could look more closely at the electron, we could be able to tell where it would end up. As far as we know today this is impossible! In 1927, Werner Heisenberg introduced the uncertainty principle which states that: “It is impossible to specify precisely and simultaneously the value of both members of particular pairs of physical variables that describe the behavior of an atomic system”. The members of these pairs are canonically conjugated to each other in the Hamiltonian sense. For instance, the -196Prof. Dr. Muhammad EL-SABA


Chapter 4

position & momentum and the energy & time are physically conjugated to each other. According to Heisenberg statement we have3:

p x . x   / 2 , E. t   / 2

(4-7)

In his original statement, Heisenberg said that, if you make the measurement on an object, and you can determine the x-component of its momentum with an uncertainty px , you cannot , at the same time know its x-position more accurately than x = /2px , where  is a definite fixed constant given by nature (= h/2 =1.05x10-34 J.s).

Fig, 4-8. Illustration of the Heisenberg uncertainty principle. The smaller the error in momentum (p), the greater the error in position (x),, and vice versa .

The mathematic-al derivation of this principle is described in problem 5 of this chapter. The more general statement of the uncertainty principle is that one cannot design equipment in any way to determine which of two alternatives is taken without, at the same time, disturbing the accuracy of the other one. Therefore, in quantum mechanics, it is impossible to follow electrons in their motion precisely. In fact if one was able to localize exactly electrons at a certain instant ti, their momentum (and hence their velocity) should be entirely unknown. Therefore, one could not know where they would be at the next instant.

x(ti + t) = x(ti) + vg(ti) .t

(4-8)

One way in which Heisenberg originally argued for the uncertainty principle is by using an imaginary microscope as a measuring device. He imagines an experimenter trying to measure the position and momentum of an electron by shooting a photon at it. If the photon has a short This is the form of the Heisenberg inequality when the error distribution is Gaussian. This inequality was put in its modern form, as proved by Kennard in 1927, with the standard deviations replacing the errors (i.e. x p ≥ ½ ).3 -197Prof. Dr. Muhammad EL-SABA


Chapter 4

wavelength, and therefore a large momentum, the position can be measured accurately. But the photon scatters in a random direction, transferring a large and uncertain amount of momentum to the electron. If the photon has a long wavelength and low momentum, the collision does not disturb the electron's momentum very much, but the scattering will reveal its position only vaguely. The Heisenberg uncertainly principal became an important consideration in any discipline, which involves extremely small dimension. In fact, the uncertainly principal protects quantum mechanics. Heisenberg recognized that if it were possible to measure the momentum and the position simultaneously with a greater accuracy (than h) the quantum mechanics would be breakdown.

Fig, 4-9. Illustration of the Heisenberg uncertainty principle, by the virtual gammaray microscope experiment.

4-4. Wave Mechanics Since particles have a wave nature, they can be described by wave functions. Our knowledge of a wave is defined by its eigenfunctions . A wave function has its existence within defined region. Once the boundaries of that region are known, then the probability of finding the particle somewhere (between r1 and r2) can be calculated as follows: r2

P  r2  r < r1  =



 * d 3r

(4-9)

r1

It is evident that if  is normalized we shall have:

P    r <  =





 * d 3r = 1

-

The product * is called the probability density function: -198Prof. Dr. Muhammad EL-SABA

(4-10)


Pd 

Chapter 4

dP =  * 3 dr

(4-11)

The square root of the probability density is called the probability amplitude:

A =  . *

1

2

 2

1

2

(4-12)

It follows from the above relations that  is large in magnitude where the particle is likely to be, and small elsewhere. In general there are 3 conditions (Born’s conditions) that a wave function must satisfy: 123-

 and its spatial derivative d/dr are continues,  must be single valued so that   * has only one value. The limits of  must be null at infinity because the integral of the product   * and hence the probability must be finite.

If the wave function is monochromatic and progressive, it may have the following traveling waveform:

(r, t) = A. exp [ j(k.r –t)]

(4-13a)

where A is a normalization constant (the probability amplitude). Alternatively,  may have one of the following harmonic forms:

 r ,t  = A cos  k.r -  t   r ,t  = A sin  k.r -  t 

(4-13b) (4-13c)

The above traveling waves or traveling harmonics are characterized by constant amplitudes and hence their integral ∫*dV over great distances diverges unless it is artificially bounded. These waves correspond to traveling particles or photons, which are neither localized nor restrained. Moreover, there exists another class of wave functions called wave packets which are well localized and for which the integral ∫*dV converges. The wave packet corresponds to a particle, which is localized or restrained by an external force (or potential energy).



Chapter 4

4-5. Wave Packets in Space and Time A wave packet is a concentrated bunch of waves. In one-dimensional space, the wave packet (x, t) may be plotted against x at a certain time t`. The average wavelength  and approximate extension x of the packet are indicated in figure 4-10(a), together with the Fourier integral transform (wave packet representation in the reciprocal space) against the wave vector k. Alternatively,(x,t) may be plotted as a function of t at a given point x` as shown in figure 4-10(b) The Fourier integral which transforms (x`,t) from the time domain to the frequency domain is also indicated on the same figure. k = F{(x)}

(x)

k x ko = 2/

x

1 F{(x)} = 2

(a) (t)

k



 ( x ) e

jkx

dx





 = F{(t)}

 t o = 2/To

t

F{(t)} =

(b)

1 2



 (t ) e

 jt

dt



Fig. 4-10. Different representations of a wave packet in space and time and their Fourier integral transforms in the k- and -domains.



Chapter 4

It can be shown mathematically, for a wave packet of extension x and duration t that: 



xk



½ , t  ½

(4-14a)

The above relations are other forms of the Heisenberg uncertainty relations:

xp



½ , Et  ½ 

(4-14b)

The group velocity, vg, of the center of the wave packet is given by:

g 

d 1 dE dE   dk  dk dp

(4-15a)

The group velocity is the actual particle velocity and cannot exceed the speed of light (  g  c ). In contrast, the phase velocity, vph, which is defined as the velocity of a point of fixed magnitude, may exceed the speed of light. The phase velocity, vph, is given by:

vph = / k = E / p

(4-15b)

Finally it should be noted that (x) is regarded as describing the behavior of a single particle, not the statistics of a number of particles. When we wish to describe more than one particle, we must make use of a wave function that depends on the coordinates of all of them (x1, x2, ..). 4-6. Pauli’s Exclusion Principle The Pauli Exclusion Principle (1927) states that no two electrons can occupy the same energy state (including spin4). Therefore, it is not possible for two electrons to have the same momentum and spin direction. If they are at the same state of motion, the only possibility is that they must be spinning opposite to each other (s = ±1/2). There are some remarkable consequences of this fact. The variety of materials in the periodic table is a consequence of this principle. The exclusion principle is also responsible for the stability of matter on the large scale. In fact, when the two spins of electrons at a certain state are opposite, there is a much stronger total attraction in the atom. In addition, we know that the individual atoms in matter do not collapse because of the 4

The spin motion of electrons (around themselves) was first discovered by G. Uhlembeck and S. Goudsmit. So, the electron energy state is described by four quantum numbers (n, ,,m,s).



Chapter 4

uncertainly principle. Nevertheless, this does not explain why not all the protons of atoms get close together with one smear of electrons around them in a very condensed state. This is because electrons obey the exclusion rule. No more than two electrons, with opposite spin, can be in roughly the same place, and their atoms must keep away from each other. 4-7. Complimentarity Principle In order to understand the implication of uncertainty principle in more physical terms, Bohr introduced the Complementarily Principle in 1928. The complementarily principle states: the atomic phenomena cannot be described with completeness demanded by classic dynamics; some of the elements that complement each other to make up a complete classical description are mutually exclusive, and these complementary elements are all necessary for the description of various aspects of the phenomena. 4-8. Correspondence Principle The correspondence principle affirms that the expected results of quantum and classic mechanics are matched at large dimensions and high energies, i.e., when the quantum number tends to infinity. When the dimensions of the system of particles are much larger than the de Broglie wavelength =h/p of particles in question, the classic solution tends to the quantum one. In fact, the smallness of h makes the uncertainty principle - upon which quantum mechanics is based- interest only about systems of atomic size. For example a semiconductor device of dimensions in the order of 0.1m in which electrons are propagating with =100Å, one can still use classic mechanics. However, beyond this limit quantum mechanics should be utilized. We now provide a demonstration of how large quantum numbers can give rise to classical (continuous) behavior. Consider the case of onedimensional quantum harmonic oscillator (described in section 4-16-5). Quantum mechanics tells us that the total (kinetic and potential) energy of the harmonic oscillator, E, has a set of quantized values: E= (n + ½ ), n =0,1,2,3,.. where  is the angular frequency of the oscillator. However, in a classical harmonic oscillator such as a ball attached to a spring, we do not perceive any quantization. Instead, the energy of such a macroscopic system appears to vary over a continuum of values. We can verify, however, that such a macroscopic system falls within the correspondence limit. The energy of the classical harmonic oscillator with amplitude is



Chapter 4

Thus, the quantum number has the value

If we apply typical human-scale values m = 1kg,  = 1 rad/s, and A = 1m, then n ≈ 4.74×1033. This is a very large number, so the system is indeed in the correspondence limit. It is simple to see why we perceive a continuum of energy in said limit. With  = 1 rad/s, the difference between each energy level is  ≈ 10-34J, well below what we can detect. 4-9. Hidden Variables: The indeterministic nature of the quantum mechanics has been a debate between scientists for long time. So many experiments have been performed in order to answer the question if quantum mechanics is complete or if its indeterministic nature is only apparent and there exist some hidden variables that would enable a deterministic description. The majority of these experiments used spatially separated (but correlated) photons generated in an atomic decay process. These photons are passed through two polarizers (as shown in the following figure) and counted using a coincidence counter depending on the relative orientation of the two polarizers. The experiment setup is shown, below in figure 4-11(a).

Fig. 4-11. One of the proposed experiments for the illustration of the quantum entanglement. The polarization directions of photons A and B are bound by a virtual shaft. Therefore, the polarization direction of photons A and B rotate as if these photons are one particle. -203Prof. Dr. Muhammad EL-SABA


Chapter 4

According to the Bell theorem (Bell's inequality5) this should yield different results for the quantum mechanics and the hidden variables assumption. All experiments are claimed to rule out the existence of hidden variables as Bell's inequality is violated. The original inequality that Bell derived was:

1+ C(b,c) ≥ |C(a,b) – C(a,c)|

(4-16)

where C is the correlation of the particle pairs and a, b and c settings of the apparatus. This inequality is not used in practice. For one thing, it is true only for genuinely two-outcome systems, not for the three-outcome ones (with possible outcomes of zero) encountered in real experiments. For another, it applies only to a very restricted set of hidden variable theories, for which the outcomes on both sides of the experiment are exactly anti-correlated when the analyzers are parallel, in agreement with the quantum mechanical prediction. Let’s imagine that two entangled particles in a spin singlet are shot out to two distant locations, and the spins of both are measured in the direction A. The spins are 100% correlated. The same is true if both spins are measured in directions B or C. It is safe to conclude that any hidden variables which determine the A,B, and C measurements in the two particles are 100% correlated and can be used interchangeably. If A is measured on one particle and B on the other, the correlation between them is 99%. If B is measured on one and C on the other, the correlation is 99%. This allows us to conclude that the hidden variables determining A and B are 99% correlated and B and C are 99% correlated. But if A is measured in one particle and C in the other, the results are only 96% correlated, which is a contradiction. 4-10. Quantum Entanglement Quantum entanglement (or quantum nonlocal connection) is a quantum phenomenon in which the quantum states of two particles are described with reference to each other, even though these particles may be spatially separated. According to this phenomenon, it is possible to prepare two interacting particles and separate them such that when one spins-up, the other one will spin-down and vice versa. As a result, measurements performed on one system seem to be instantaneously influencing other systems entangled with it. Although the quantum entanglement phenomenon was presented in 1935, by Erwin Schrödinger, it only began to be taken seriously by the scientific community at the end of the 5

In 1964, John Bell proposed that if particles have permanent properties as Einstein claimed, there must be a limit to how similar the pairs of particles in his experiment could be, set by the finite speed of light. Bell showed that if Einstein was right, the correlation in the properties of the particles could never exceed a certain level, which became known as the Bell Inequality.



Chapter 4

twentieth century. In 1982, the French scientist Alain Aspect published a paper detailing his experiments showing that nonlocal interactions do occur. However, scientists claim that classical information cannot be transmitted through entanglement faster than the speed of light. Quantum entanglement produces some theoretically and philosophically disturbing aspects of the quantum theory, as one can show that the correlations predicted by quantum mechanics are inconsistent with the seemingly obvious principle of local realism. Local realism dictates that information about the state of a system should only be mediated by interactions in its immediate surroundings. Different views of what is actually occurring in the process of quantum entanglement give rise to different interpretations of quantum mechanics.

Fig. 4-12. Illustration of the quantum entanglement of two photons.

In 1935, Einstein, Podolsky, and Rosen formulated the EPR (A. Einstein, B. Podolsky, and N. Rosen) paradox, demonstrating that entanglement makes quantum mechanics a non-local theory. The EPR paradox is a thought experiment which demonstrates that the result of a measurement performed on one part of a quantum system can have an instantaneous effect on the result of a measurement performed on another part, regardless of the distance separating the two parts. This is paradoxal with the intuition of special relativity, which states that information cannot be transmitted faster than the speed of light. Therefore, the ERP argued that quantum mechanics is not a complete physical theory. It was Einstein’s belief that future mathematicians would discover that quantum entanglement entailed nothing more or less than an error in the calculations. However, some physicists today regard the EPR paradox as an illustration of how quantum mechanics violates classical intuitions, -205Prof. Dr. Muhammad EL-SABA


Chapter 4

and not as an indication that quantum mechanics is fundamentally flawed theory. So many theories, involving hidden variables, have been proposed in order to explain the quantum entanglement. The hidden variables would account for the spin of each particle, and would be determined when the entangled pair is created. It may appear then that the hidden variables must be in communication no matter how far apart the particles are that the hidden variable describing one particle must be able to change instantly when the other is measured. If the hidden variables stop interacting when they are far apart, the statistics of multiple measurements must obey an inequality (Bell's inequality), which is, however, violated both by quantum mechanical theory and experimental evidence. Quantum entanglement is considered as the basis of some emerging technologies such as quantum computing and quantum cryptography. There are some who believe that Quantum entanglement or quantum non-locality is the key to time travel. 4-11. Bosons and Fermions As identical particles are indistinguishable in quantum mechanics (because of the uncertainly principal), one cannot determine the wave function of each particle but only the total wave function of the system and the spin of each particle. For example if we have two particles we can know (x1, x2), x1, x2 and the spin of each particle. If the particles are exchanged each with the other, the crystal state will not change. The system wave function  should only change its phase angle  (i.e. multiplied by ej) so that:

  x2 , x1  = e j   x1 x2 

(4-17a)

If the two particles are again exchanged then:

  x1 , x2  = e2j   x1 , x2 

(4-17b)

Then, the phase angle  should satisfy the following condition:

e2j = 1 , or e j =  1

(4-17c)

Thus, we have two cases or two types of particles: Bosons and Fermions. 4-11.1. Case of Bosons : (ej = +1) In this case the wave function of the system  is symmetric when 2 particles are exchanged. Therefore, if we have N particles (Bosons), the -206Prof. Dr. Muhammad EL-SABA


Chapter 4

wave function of the system  (x1, x2,...xN) can be expressed as the product of individual particle waves:

 (x1, x2, ...xN) =  (x1) (x2) ........ (xN)

(4-18a)

The spin of Bosons is either a zero or integer number (s=0, ±1, ±2, etc). Photons (light waves), phonon (lattice vibration waves) and some nuclei such as He4 are examples of Bosons. Specifically, photons have s=±1. 4-11.2. Case of Fermions : (ej = -1) In this case the wave function of the system is asymmetric when 2 particles are exchanged so, the wave function of a system of N particle can be expressed in terms of the particles individual wave functions by mean of the so-called Slater determinant

 1 (q1 )   x1 , x2 ..xN  =

where

1 N!

.  n (q1 ) . . .  n (qn )

.

. .  1 ( qn )

.

(4-18b)

1 is a normalization factor and the coordinates q1...qn include N!

the positions x1...xN and the spin of each particle. If two particles are exchanged then the sign of  will change. The spin of Fermions is a half an odd integer number (s=  1/2,  3/2,..). Electrons, protons and odd-atomic number nuclei as He3 are examples of Fermions. Specifically, electrons have s=  1/2. (for spin-up and spindown). The probability of occupation of certain energy level (or state) by a Fermion (e.g., an electron) is defined by the Fermi-Dirac distribution:

fn  E  =

1 1+ exp



E -EF kBT



(4-19)

where Ef is the Fermi energy or the chemical potential of the system. According to equation (4-19), the Fermi energy is the energy at which the probability of occupation is 1/2. It should be noted that the probability of no occupation of a state by an electron (finding a hole) is given by:



fp(E) = 1- fn(E) or f p  E  =

Chapter 4

1 1+ exp



EF  E kBT



(4-20)

Also the probability of occupation of a certain state (certain energy) by a Boson (e.g., a phonon) is defined by the Bose-Einstein distribution (as shown in Chapter 3):

f BE  E  =

exp



1

1

E -EF k BT

(4-21)

In the case of phonons, as the number of phonons is not determinable one cannot apply the conservation rules and the relation becomes:

f BE  E  =

1

exp

 

(4-22)

1

E

kB T

4-12. Observables and Operators Observables are defined as the normal mechanical properties (variables), which can be measured such as position, momentum, energy, etc. In wave mechanics, observables are represented by corresponding operators, which are nearly mathematical instructions, as shown in table 4-2. The application of operators can be summarized in the following postulates: 1- Every classical observable has a corresponding quantum operator. 2- The possible measurable values for an observable are these for which:

[operator] n = (measured values)n where n is the wave function (or eigenfunction). The measured value is the eigenvalue corresponding to the eigenfunction n 3- The expected mean value of a sequence of many measurements of an observable, having an operator Â, is given by:

  * Â  dV

AP

d

dV

= Spacea   *  dV  P dV

 Â   Space

d

Space

Space


(4-23)


Chapter 4

where Pd = * is the probability density function of the system. Table 4-2. Observables and their corresponding Quantum operators Quantity Position Momentum Momentum x-component

Classical Observables Quantum Operators r, r x, y, z x , y , z 

dr dt dx = m dt

p = m

p   j.

px

p x   j.





Angular momentum L = r x p Hamiltonian (K.E + P.E) Total Energy

 x

H = p2/2m + V(r)

L   j.( rx ) 2 2 Ĥ = - (ħ /2m) +V(r)

E =

Ề = j ħ∂/∂t

For any operator A, we have the following time dependence: (4-24a) This is called the Ehrenfest theorem6, named after Paul Ehrenfest, the Austrian physicist. The Ehrenfest’s theorem, says that expectation values (means or averages) of observable quantities in quantum mechanics obey classical laws. It relates the time derivative of the expectation value for a quantum mechanical operator to the commutator of that operator with the Hamiltonian of the system. The commutator of two operators A and B is defined as follows: [A, B] = AB – BA. For instance, (4-24b)

6

The Ehrenfest theorem is closely related to the classical Liouville's theorem from Hamiltonian mechanics (see Chapter 9). It involves the Poisson bracket {,} instead of a commutator [,]. In fact, it is a rule of thumb that a theorem in quantum mechanics which contains a commutator can be turned into a theorem in classical mechanics by changing the commutator into a Poisson bracket and multiplying by jħ. -209Prof. Dr. Muhammad EL-SABA


Chapter 4

This is actually the quantum form of Newton second law in classical mechanics. In fact, any conservative force in classical physics can be expressed as the gradient of a potential. This is an example of the correspondence principle between quantum mechanics and classical mechanics. Also, we can calculate the instantaneous change in the position expectation value (4-24c) This result is again in accord with the classical definition of the velocity. However, for energy eigenvalues (or stationary states), the timedependence is null such that: (4-24d) This means that the Hamiltonian Ĥ is commuted with energy eigenvalues ( ) and [Ĥ, ]=0. In this case, the average value of the operator is said to be constant of motion and obeys the conservation rules. Note 4-5. Hermitian Operators Some of the above quantum operators are said to be Hermitian. A Hermitian operator H has the property that ∫(Hψ)*ψ.dx = ∫ψ*(Hψ).dx. The average value of the operator H in a state ψ is < H > = ∫ψ*Hψ.dx. The complex conjugate of the average value is * = ∫ψ(Hψ)*dx= ∫(Hψ)*ψdx = if H is Hermitian, so the average value is real. We can show that the operators x and px = -j.d/dx are Hermitian. This is obvious for x because it is real. Also, for px we have: ∫(pxψ)*ψdx = ∫(-j.dψ/dx)*ψdx = j.∫(dψ*/dx)ψdx. Integrate by parts with u = ψ, dv = (dψ*/dx)dx, so that du = (dψ/dx)dx and v = ψ*. The integrated part vanishes because ψ, ψ* both vanish at x = ±∞, and the result is -j∫ψ*(dψ/dx)dx = ∫ψ*pxψdx. Therefore, px (as will as any function of x and px) is also Hermitian. Note 4-6. Canonical Commutation The commutator of two operators A and B is defined as follows: [A, B] = AB - BA. -210Prof. Dr. Muhammad EL-SABA


Chapter 4

When [A, B] = 0, the operators A and B are said to be commuted. Noncommutation of two observable quantities means that observables cannot be measured simultaneously. In this case, we have [A, B] = j. For instance, the displacement x and momentum px operators obey the canonical commutation relation [x, px]= j. because they are noncommuted and hence cannot be measured simultaneously. Note 4-7. Conjugate Variables Conjugate variables are the variables that don’t commute. For instance, the displacement x and momentum px operators are non-commuted and obey the canonical commutation relation [x, px]= j.

4-13. Probability Current Density The probability current density J(r,t) is defined as follows:

J ( r ,t )  



j *    (  * ) 2m



(4-25a)

The probability current density J(r,t) expresses the particle current or flux, which is associated to a wave function (r,t). It is related to the probability density Pd(r,t) by the following continuity equation:

Pd ( r , t )  .J (r , t )  0 t

(4-25b)

which means that there is no spontaneous creation of particles. The above differential equation has the familiar form of conservation of charges:

( r ,t )  .J ( r , t )  0 t

(4-25c)

where the probability density P(r, t) replaces the charge density (r,t) and the probability current density replaces the current density J(r,t).

4-14. Schrödinger’s Equation The most famous differential equation that describes the position probability of a particle is known as the Schrödinger equation. The Schrödinger equation is simply a translation of the physical equality: -211Prof. Dr. Muhammad EL-SABA


p2 + V(r) E = K.E + P.E = 2m

Chapter 4

(4-26)

where V(r) represents the potential energy in which the particle is moving7. The Schrödinger equation can be obtained by substituting the postulates of quantum operators into (4-26) and applying both sides on :

 - 2 2    + V r   j =  t  2m 

(4-27)

This is the time-dependent Schrödinger equation. This relation can be rewritten in a simple form using the Hamiltonian operator Ĥ as follows:

j ∂/∂t = Ĥ 

(4-28a)

E = Ĥ

(4-28b)

or

where the Hamiltonian differential operator Ĥ is given by:

Ĥ

 p2   - 2 2   + V r  =  + V r  =   2m   2m 

(4-29)

If the wave function (r, t) is monochromatic, it may be written as follows:

 

 r,t  =  r  exp  -j

E t   

(4-30)

Therefore, the Schrödinger equation may be written in the following time- independent form: 2





  2 + E -V r   = 0 2m

(4-31)

where the potential energy V(r) is assumed to be only space-dependent. 7

Here we neglected the rest energy m c2 in the expression of the total energy E of the particle.



Chapter 4

The time-independent Schrödinger equation can also be derived starting from the well-known Helmholtz wave equation: 2 2   +k  = 0

(4-32.a)

where we make use of the relation: k =

p 2m( E -V) =  

(4-32.b)

It should be noted that the basic Schrödinger equation, which we presented above, is based on certain assumptions, which are sometimes drastic. Among these assumptions one can cite: o The phenomena of generation-recombination (or creationannihilation) of particles are neglected. o The particles are assumed to move with velocities much smaller than the light velocity. Therefore, particles are described in nonrelativistic manner. 4-15. Dirac’s Equation In order to take the special relativity into account and to describe the emission and absorption of light quanta (photons), the basic Schrödinger equation has to be a little bit modified8. The so-called Dirac’s equation is equivalent to the Schrödinger equation, but with taking the special relativity into account. The Dirac equation states:

j ∂/∂t =Dirac 

(4-33.a)

where the Dirac Hamiltonian Dirac is given by:

Dirac  mo2c4 + c2p2) 1/2 = mo2c4 - c2 ħ2∇2) 1/2

(4-33.b)

Here mo is the particle rest mass and c is the speed of light. In presence of external magnetic and electric fields, the Dirac Hamiltonian reads:

Dirac m c2+ . (p - eA)c+ e

8

(4-33.c)

The Schrödinger equation contains a second spatial derivative and first time derivative, thus, it is not correct from the point of special relativity (Lorentz covariant). It changes its form, when viewed by a uniformly moving coordinate system.



Chapter 4

where A is the vector potential and  is the electrical potential. Also, the parameters  and are given by: 0 0 1   0  1

0 0 1 0

0 1 0 0

1 0 0 0  j  0 0    0 0 0 0 0 j 0  , 2   , 3   0  j 0 0  1 0 0     0 j 0 0 0  0  1

1 0 1  0 0  1 ,   1 0 0   0 0 0

0 0 1 0 0 1 0 0

0 0  0  1

(4-33.d) The signs of the operator β distinguish particles and antiparticles; it has two 1’s and two -1’s to take care of the 2 possible spin directions. With this Hamiltonian operator, a wavefunction for a particle has vanishing antiparticle components, and vice versa. The Hamilton operator yields the velocity operator v in the same relation that is valid in classical physics:

v = dx/dt = p / (mo2c4 + c2p2 )1/2.

(4-33.e)

As a consequence of the solution of this equation, spin and antiparticles could be mathematically explained. The Dirac Hamiltonian describes how charged particles move by electromagnetic fields. A similar Hamiltonian (called Maxwell-Hamiltonian) can describe how fields move by charged particles. Together, they form what is called quantum electrodynamics (QED). However, nobody so far has managed to combine the Schrödinger equation with the general theory of relativity; the two even appear to be antagonistic. Therefore, it must be kept in mind that the basic Schrödinger equation is not a fundamental theory but rather, a successful approximation that results in good solutions, when applied to atoms and molecules. Note 4-8. Special Relativity In the special relativity theory, the relativistic energy is given by the famous Einstein relationship: E = m.c2 = moc2 + KE

(i)

where E includes both the rest mass energy and kinetic energy for a particle (½ mv2) . Note that mo with the zero subscript is the rest mass, and m without a subscript is the relativistic mass. The relativistic energy of a particle can also be expressed in terms of its momentum in the expression: -214Prof. Dr. Muhammad EL-SABA


E  mc 2 

Chapter 4

p 2 c 2  mo2c 4

(ii)

Note also that the rest energy moc2 of an electron is 0.51 MeV. So accelerating energies of a few hundreds keV are necessary before relativity may be taken into account for electrons in free space. Note 4-9. Proof of the de-Broglie Duality Relation From the above relations we have:

p

mo v

(iv)

1  v / c 

2

By setting the rest mass equal to zero (for Bosons) and applying the Planck relationship (E = h) we get the De-Broglie momentum expression! p = E/c = h/c = h/(v)

4-16. Solution of the Schrödinger Equation For time-independent Hamiltonians the solution to Schrödinger equation (SE) can be cast as an eigenvalue problem. This allows us to solve SE in many cases exactly. Therefore, the solution of the time-independent SE9 for a particular system gives the sets of eigenfunctions (n) of the Hamiltonian Ĥ and the corresponding eigenvalues En such that: Enn = Ĥn

(4-34)

The starting point in the solution is to determine the Hamiltonian operator of the system for which we seek the wave functions. The potential energy V(r) of the system will specify the physical problem in hand. For an electron in an isolated hydrogen atom the potential energy is as simple as V(r) = -e2/(4or), wherever for a crystal it is a complicated periodic function and there is a considerable mathematical complexity in solving for . For the matter of demonstration, we shall study the solution of the Schrödinger equation for some simple cases, like a particle moving in a square potential well. This case resembles the motion of an electron inside a metal. The electron is bound inside the metal unless it has not supplied by energy equal to or greater than the metal work function. Thus, Vo represents the work function of the metal containing the electron 9

The solution of the time-dependent SE is treated in the next section §4-17.



Chapter 4

and the potential well dimensions represent the metal size. At first we shall solve the problem of a one-dimensional potential well and then we shall extend our solution to the real 3-dimensional case. Then, we’ll attack the hydrogen atom problem in three dimensions. 4-16.1. Case A: 1-Dimensional Square Potential Well Assume an electron of average energy E is moving inside a potential well. The potential well height Vo is assumed to be greater than the average electron energy E. Assume the potential well is symmetric, as shown in the beside figure, such that: V(x) = 0 = Vo

for for

|x|  a/2 |x| > a/2

(4-35a)

Assume a particle of energy E < V o V(r) moving inside this potential well. Calculate the wave eigen-functions associated to the particle as well as Vo the energy eigen-values (allowed energy values) which E may take). - a/2 0 a/2 The Schrödinger equation inside and outside the potential well are then given by: 2 2    + E= 0 2m  x 2

x 

f or

2 2    + (E -V )  = 0 o 2m  x 2

f or

a 2

x 

(4-35b) (4-35c)

a 2

As the potential well is symmetric around x=0, the solution of these two equations may be symmetric or asymmetric as follows: i) Symmetric solution

 ( x ) = A exp (  x )

for

x  -

 ( x ) = B cos (  x )

for

x 

 ( x ) = A exp (- x )

for

a 2 a x  2


a 2

(4-36.a) (4-36.b) (4-36.c)


Chapter 4

where 

2m E 

2m (Vo - E)



2



(4-37)

2

Applying the boundary conditions of  (the continuity of  and d /dx) at x =  a/2, results in:

a 2

 (- ) = A exp (-

a 2

) = B cos (

a 2

(4-38.a)

)

d a a a a a (- ) = A exp ()= B sin ( ) dx 2 2 2 2 2

(4-38.b)

Dividing (4-38.b) by (4-38.a) we get:

a

 =  tan (

2

)

(4-39.a)

But we have from (4-37)

2 + 2 =

2mVo 

(4-39.b)

2

The allowed values of  and  (and hence the energy E) can be obtained by solving these two relations graphically as shown in figure. It is interesting to find the values of  and  in the limit when Vo is very large (infinite well). In this case the radius of the circle in figure will be very large and  will be given by:

 a = (2 n +1) 

a

(4-40a) a 0



2

3 4

And hence the energy eigenvalues En (the allowed values of E) are obtained by substituting (4-40a) into (4-37) to get:



En

2 2   = 2m

Chapter 4

2  = 2 n  1 2  2m a 2

2

(4-40b)

ii) Asymmetric solution

 ( x ) = - A exp (  x )  ( x ) = B sin (  x )

for

 ( x ) = A exp (- x )

x  -

for

for

a 2

(4-41a)

x 

a 2

(4-41b)

x 

a 2

(4-41c)

Applying the boundary conditions of  (the continuity of  and d /dx) at x =  a/2, results in:

 = -  cot (

a 2

)

(4-42)

Again the allowed values of  and  (and hence of the energy E) can be obtained by solving this relation simultaneously with equation (4-37) as shown in Fig. 4-7(b). In the limit when Vo is very large a (infinite well), the radius of the circle in Fig. 4-6 will be very large a 2mVo and  will be given by: h 

 a = 2 n

(4-43.a)

Hence the energy eigen-values En (the allowed values of E) are obtained by substituting (4-43.a) into (4-37) to get:

En

2 2   = 2m

a 0



2   = 2 n   2 m a2 2

2

3 4

2

(4-43b)

In general one can write the energy eigen-values (for both symmetric and asymmetric solutions) as follows: -218Prof. Dr. Muhammad EL-SABA


Chapter 4

 En = n2 = Eo n2 2 2m a 2

2

(4-44)

Figure 4-13 depicts the shape of first eigen-functions n (symmetric and asymmetric) as well as their corresponding eigen-energies En. We note that the energy E is allowed to have only certain discrete (or quantized) values En. The index (or the subscript) n is called the quantum number. This quantization is a consequence of applying the boundary conditions on . We note also that, although E E. According to classical physics, the particle could not penetrate into the barrier. However, the wavefunction associated with such a free particle must be continuous at the barrier and show an exponential decay inside the barrier. The wavefunction must also be continuous on the far side of the barrier, so there is a finite probability that the particle will tunnel through the barrier as shown in figure 4-15. As a particle approaches the barrier, it is described by a free particle wave-function. When it reaches the barrier, it must satisfy the Schrödinger equation in the form:   2  2 .  ( U o  E )  0 2m x 2

(4-49a)

where Vo is the barrier height. This equation has the following solution:

(x) = A exp(-x) where A is a constant and  = (1/ħ)√2m(Vo -E) -221Prof. Dr. Muhammad EL-SABA

(4-49b)


Chapter 4

Fig. 4-15. A particle penetrating a finite square potential barrier.

4-16.4. Case D: The Hydrogen Atom The Schrödinger model of the atom is known as the quantum mechanical model. As we’ll see in this section, the Schrödinger model places the electrons in orbitals, not fixed orbits. Orbitals are regions of space. The electrons are like a cloud of negative charge within that orbital. The electron shells proposed by Bohr are still used, but the electrons in each shell are not all equal in energy. The Schrödinger model consists in solving the Schrödinger equation in the spherical coordinates. The Bohr radius ao =5.29Å is a typical atomic distance, and atoms have a diameter on the order of 2ao =10.6 Å. An atom has a small nucleus whose radius is on the order of 10-3 Å, and whose charge is +Ze, where Z is the atomic number. This charge is neutralized by the electrons, so there is no electric field outside the atom. Inside the atom, however, there is a field that increases as the nucleus is approached. This field gives rise to a scattering potential energy V(r)=-Ze2(1/4or). The hydrogen atom is the simplest atom in nature (Z =1). The electron in the hydrogen atom sees a spherically symmetric potential, so we use spherical polar coordinates. The potential energy is simply that of a point charge:  e2 V( r )  4o r

(4-50a)

Therefore, the starting point is the polar form of the Schrödinger equation: -222Prof. Dr. Muhammad EL-SABA

Electronic Engineering Materials & Nanotechnology  2 1 . 2 2  r sin

  2   sin  r  r 

      sin    

1   2   2    sin  

Chapter 4    V ( r )  E 

(4-50b)

where  is the reduced electron mass, which is given by the parallel combination of the electron and proton masses, me, mp , respectively:

= me.mp / (me + mp)

(4-50c)

Fig. 4-16. Model of the Hydrogen atom and its spherical coordinates.

One of the simple approaches for solving such a partial differential equation is to separate it into individual equations for each variable involved. In fact, the hydrogen Schrödinger equation is separable. Solving it involves separating the variables into the form:

r = R(r).P().F()

(4-50d)

Compiling all the radius-dependent terms and setting them equal to a constant Cr, gives the following radial equation:

1 d  2 dR  2 2 2 r  ( E . r  ke )  Cr 2 R dr  dr  

(4-51a)

where k=1/4o. Then the angular parts of the equation can be separated into the following colatitudes and azimuthal equations:

sin d  dP  2 .  sin   Cr sin    C  P d  d 

(4-51b)

1 d 2F .  C F d 2

(4-51c)

where Cr and C are the separation constants. -223Prof. Dr. Muhammad EL-SABA


Chapter 4

The hydrogen atom solution requires the solution of the separated equations (4-51), which obey the constraints on the wavefunction (e.g., the continuity of  and its derivative). For instance, the solution to the radial equation, R(r), can exist only when a certain constant, called the principle quantum number (n), is restricted to integer values (n=1,2,..). The solution of the radial equation takes the form:

Rn,l = rl Ln,l exp(-r /n ao)

(4-52)

Here Ln,l is called the associated Laguerre function10. In the notation of the periodic table, the main shells of electrons are based on the principal quantum number and labeled K (n=1), L (n=2), M (n=3), etc. Similarly, a constant arises in the co-latitude equation, P(), gives the orbital quantum number (l) such that l =0,1,2,3,… n-1. A detailed solution of the co-latitude equation involves conversion of the above equation to a spherical form in which the variable is cos and then solving it by a series expansion (spherical harmonics) method. The orbital quantum number (l) is used as a part of the designation of atomic electron states in the energy levels spectroscopic notation. For instance, if n =2, then l takes two values (0,1), corresponding to two subshells. The first sub-shell (n =2, l = 0) is designated 2s and the second sub-shell (n =2, l = 1) is designated 2p. The orbitals thus are called 1s, 2s, 2p, 3s, 3p, 3d, ..., where the numbers 1, 2, 3, ..., correspond to the values of n, and the letters s, p, d, ..., correspond to the values of l. It should be also noted that the orbital quantum number (l) determines the magnitude of the orbital angular momentum (L = rxp) by the relation:

L2 = l (l +1)ħ2

(4-53a)

Finally, a constraint on the azimuthal equation, F(), gives what is called the magnetic quantum number (m) such that m= - l, - l +1, …+ l F ( )  A.. exp( j.m . )

(4-53b) The direct implication of this quantum number is that the z-component of angular momentum is quantized according to:

Lz = m  ħ 10

(4-53c)

More mathematical details about Laguerre’s functions can be found in Appendix K.



Chapter 4

The solutions of the angular equations (colatitudes and azimuth) can be combined together in one solution, in terms of spherical harmonics. The conventional spherical harmonics, Yl,m(,) are defined as follows:

 2l  1 ( l  m )!  Ylm (  , )  ( 1 )m  .  4  ( l  m )!  

1/ 2

Plm (cos  ). exp( jm ) (4-53d)

where Plm ae the associated Legender polynomials. This solution is valid for m ≥ 0, while the eigenfunctions corresponding to negative values of m are obtained from the identity Yl,-m = (-1)m Y*lm .

Fig. 4-17. Hydrogen atom wavefunctions of first orbitals

The above-indicated constants (n, l, m) are called the electron quantum numbers. In addition to these quantum numbers the experimental evidence suggested an additional 4th quantum number called the spin s (or ms). The spin property of the electron is attributed to the closely spaced splitting of the hydrogen spectral lines (the hydrogen atom fine structure). Also, the Stern-Gerlach experiment showed that a beam of silver atoms directed through an inhomogeneous magnetic field would be forced into two beams, as shown in figure 4-18. Both of these experimental situations were consistent with the possession of an intrinsic angular momentum and a magnetic moment by individual electrons. Classically this could occur if the electron was a spinning ball of charge, and this property was called electron spin. Quantization of angular momentum had already arisen for orbital angular momentum, and if this electron spin behaves the same way, an angular momentum quantum number s = ± ½ is required to give just two states. The spin angular momentum S has the following magnitude:

S 2  S x2  S y2  S z2  s.( s  1). 2


(4-53e)


Chapter 4

Therefore, this intrinsic electron property gives an additional zcomponent for the angular momentum:

Sz = s ħ, with s =  ½

(4-53f)

When the orbital angular momentum and spin angular momentum are coupled, the total angular momentum (J = L+S) will have the following quantized magnitude:

J 2  J x2  J y2  J z2  j.( j  1). 2

(4-53g)

where the total angular momentum quantum number (j) is given by:

j=l± s=l± ½

(4-53h)

This gives the z-component of the total angular momentum of electron:

Jz = mj ħ, mj = -j, -j+1, -j+2, … j-1, j

Fig. 4-18. Illustration of the Stern-Gerlach experiment.


(4-53i)


Chapter 4

Note 4-10. L-S Coupling The notion of electron spin was originally proposed in 1925, by the two graduate students, Goudsmit and Uhlenbeck. They proposed the spin angular momentum, , obeying the same quantization rules as those governing orbital angular momentum of atomic electrons. In electronic systems, the spins of electron si interact among themselves so they combine to form a total spin angular momentum S. The same happens with orbital angular momenta li, forming a single orbital angular momentum L. This is called Russell-Saunders coupling or L-S coupling. The S and L add together and form a total angular momentum J=L+S. This situation is valid as long as external magnetic fields are weak, so the coupling between orbital and spin angular momenta is stronger than with the external magnetic field. Strong magnetic fields cause these two momenta to decouple (Paschen-Back effect) which gives rise to a different splitting pattern in the energy levels. The orbital angular momentum for an electron can be visualized in terms of a vector model, as shown in figure 4-19. Therefore, for each principal quantum number n, all smaller positive integers are possible values for the angular momentum quantum number l.

(a) orbital angular momentum

(b) total angular momentum

Fig. 4-19. Vector model of the electron angular moment.



Chapter 4

The magnetic quantum number m (or shortly m) can take on all integers between l and -l, while s can be ½ or -½. This leads to a maximum of 2 unique sets of quantum numbers for all s orbitals (l=0), 6 for all p orbitals (l =1), 10 for all d orbitals (l=2) and 14 for all f orbitals (l =3). Paul Dirac explained why the electron had spin ½, that is, why it does not look the same if you turned it one complete revolution, but did if you turn it through two complete revolutions. It also predicted that the electron should have a partner, an antielectron, or positron. It is found experimentally that electrons, protons, neutrons and neutrinos have s=±½ , while photons have s =±1 and pions (-mesons) have s=0. For the matter of illustration, the hydrogen first radial probability is shown in figures 4-20, 21, 22. Also figure 4-23 depicts the hydrogen energy spectrum.

Fig. 4-20. Hydrogen atom wavefunctions and probability density of first orbitals



Chapter 4

Fig. 4-21. The hydrogen 1s radial probability (* = 4 r2 R2(r) ).

Fig. 4-22. Potential energy, V(r), in hydrogen atom and first 3 probability densities with l = 0. Probability densities are shifted by the corresponding electron energy.

The basic hydrogen energy level structure we obtained is in agreement with the simple Bohr model. Therefore, the bound state energies of the electron in the hydrogen atom can be calculated using the following formula: -229Prof. Dr. Muhammad EL-SABA


En = - m e4/(8 n2 o2 h2) = -13.6/n2 [eV]

Chapter 4

(4-53.i)

However, when the spectral lines of the hydrogen spectrum are examined at very high resolution, they are found to be closely spaced doublets, as shown in figure 4-25. This level splitting is called the fine structure and was one of the first experimental evidences for electron spin. The small splitting of the spectral line is attributed to an interaction between the electron spin S and the orbital angular momentum L. It is called the spinorbit interaction. A proper understanding of the electron spin did not come until 1928, when Dirac's theory, described in equation (4-33), combined the quantum mechanics, and the special theory of relativity.

Fig. 4-23. Hydrogen energy spectrum



Chapter 4

Fig. 4-24. Hydrogen fine structure

Fig. 4-25. Absolute energy of different orbitals.

Note that the hydrogen atom is the only atom for which exact solutions of the Schrödinger equation exist. For any atom that contains two or more electrons, no solution has yet been discovered (for instance there is no closed-form analytical solution for the helium atom!) and we need to introduce approximations. -231Prof. Dr. Muhammad EL-SABA


Chapter 4

Table 4-3. First ten orbitals and corresponding quantum numbers of a hydrogen atom

Shell Orbitals n K 1s 1 L 2s 2 2p 2 M 3s 3 3p 3 3d 3 N 4s 4 4p 4 4d 4 4f 4 O 5s 5 5p 5 5d 5 5f 5 5g 5

l 0 0 1 0 1 2 0 1 2 3 0 1 2 2 4

m 0 0 -1, 0, 1 0 -1, 0, 1 -2, -1, 0, 1, 2 0 -1, 0, 1 -2, -1, 0, 1, 2 -3,-2,-1,0,1,2,3 0 -1, 0, 1 -2, -1, 0, 1, 2 -3,-2, -1, 0, 1, 2,3 -4,-3, -2,-1 ,0, 1, 2,3,4

:

:

:

:

:

s ½, - ½ ½, - ½ ½, - ½ ½, - ½ ½, - ½ ½, - ½ ½, - ½ ½, - ½ ½, - ½ ½, - ½ ½, - ½ ½, - ½ ½, - ½ ½, - ½ ½, - ½

Elements

:

:

}8

}

18

} }

32

32

Unknown Elements

4-16.5. Case E: Quantum Harmonic Oscillator The quantum harmonic oscillator is analogous to the classical harmonic oscillator. It is one of the most important model systems in quantum mechanics because so many physical problems can be approximated to it. In fact, it is also one of the few quantum mechanical problems, which have a simple closed-form solution. Assume a one-dimensional harmonic oscillator of a particle of mass m, which is subjected to a potential V(x) = ½ m.ω.x2, where  is the oscillation frequency and x is the displacement from equilibrium position. The Hamiltonian of the particle then reads:

Ĥ = p2/2m + ½ m 2 x2

(4-54a)

Substituting Ĥ into the Schrödinger equation and solving, by power series, we obtain:

 m   n( x )  .  2 n n!    1

1/ 4

 m .x 2   m  .exp .x   .H n  2     

(4-54b)

where n is integer and Hn(x) is the Hermite series11, which is given by: 11

More mathematical details about the Hermite series can be found in Appendix K. -232Prof. Dr. Muhammad EL-SABA


Hn(x) (-1)n .exp[x2]. dn(exp[-x2])/dxn

Chapter 4

(4-54c)

The solution gives the wavefunctions (n) for the oscillator. The graph of n(x) for different values of n is shown in figure 4-26. The corresponding energy eigenvalues (En) can be obtained by substituting the solution of n(x) into the Schrödinger equation (Enn=Hn) to obtain:

En = (n + ½ ).ћ 

(4-54d)

Note that the lowest achievable energy is not zero, but ½ω, which is called the ground state energy or zero-point energy of the oscillator. The following figure shows a comparison between the quantum and classical oscillator solutions for two quantum numbers (n = 0, 10). Note that the correspondence, between the classical and quantum predictions for the most probable location, seems far-fetched for ground state. However, for a relatively high quantum number (n=10) the most probable location shifts and the quantum solution begins to look like the classical probability which is shown by the dashed line. Somewhere, along the continuum from quantum to classical, the two descriptions must merge. The idea of merging the quantum and classical descriptions is the basis of the correspondence principle, which is introduced in §4-8 of this chapter.

Fig. 4-26. Wave eigenfunctions ψn(x) for the first four bound eigen-states ( n=0 to3) of a quantum harmonic oscillator -233Prof. Dr. Muhammad EL-SABA


Chapter 4

Fig. 4-27. Comparison between the quantum and classical oscillator for ground state and for n = 10.

4-16.6. Case F: Quantum Anharmonic Oscillator As mentioned so far, a system residing near the minimum of a potential field may be treated as a harmonic oscillator. In this approximation, we expand the potential energy (by Taylor series) around the minimum and discard terms of third or higher order, resulting in an approximate quadratic potential. Once we have studied the system in this approximation, we may wish to investigate the corrections due to the truncated higher-order terms, particularly the third-order term. The anharmonic oscillator Hamiltonian is the harmonic oscillator Hamiltonian with an additional x3 potential:

Ĥ = ½ p2/m + ½ m 2 x2 +  x3

(4-54e)

If the harmonic approximation is valid, the coefficient λ is small compared to the quadratic term. We may therefore use perturbation theory to determine the corrections to the states and energy levels imposed by the anharmonic term. It should be noted that the an-harmonicty of atom vibrations is a key issue to interpret the thermal expansion in solids. Also, anharmonicity of onedimensional chains is important to understand non-linear lattice excitations, such as solitons12. 4-16.7. Case G: Coupled Harmonic Oscillators As we’ll see in Chapter 7, the solitons are a sort of quasi-particles which describe the collectivity of moving wave packets. 12



Chapter 4

Here, we consider N equal masses which are connected to their neighbors by springs, in the limit of large N. The masses form a linear chain in one dimension, or a lattice in two or three dimensions. Here we denote the positions of the masses by x1, x2, ..., as measured from their equilibrium positions. Therefore, xi =0 if the ith particle is at its equilibrium position. The Hamiltonian of the total system is given by the sum:

Ĥ = i pi2/2m + ½ m 2 i,j (xi-xj)2

(4-55)

The potential energy is summed over nearest-neighbor pairs, so there is one term for each spring. It should be noted that there exists a coordinate transformation to turn this problem into a set of independent harmonic oscillators, each of which corresponds to a particular collective distortion of the lattice. These distortions display some particle-like properties, and are called phonons, as we have discussed so far in Chapter 2 of this book.



Chapter 4

4-17. Approximate Methods (for Solving the Schrödinger Equation) As a matter of fact, only few problems in quantum mechanics can be solved exactly. The following figure shows the different methods to solving the time-dependent and time-independent Schrödinger equation. As shown, there exist many approximations to solving the timedependent Schrödinger equation, among which the adiabatic approximation is the most widely used. The method works in cases where the Hamiltonian changes slowly by comparison with the natural internal frequency of the wave function

Fig. 4-28. Different methods to solving the Schrödinger equation

The time-independent approximate methods provide an analytical solution of Schrödinger's equation for certain structures with specific potential energy distributions. Analytical solutions are only available in certain cases, as for constant or linear potentials and potential steps. In the following subsections we present the analytical solution of the Schrödinger equation for linear potential barriers (Gundlach method) and the Wentzel-Kramers-Brillouin (WKB) approximation for the case of quadratic and triangular barriers. For the accurate simulation of transport across arbitrary barriers, there exist other advanced models, such as the transfer-matrix method and the quantum transmitting boundary method (QTBM). They are mainly used to solve the one-dimensional Schrödinger, e.g., to obtain the quantized energies in hetero-structures and metal–oxide–semiconductor (MOS) structures. 4-17.1. The Gundlatch Approximate Method The Gundlach method provides an analytical solution of Schrödinger's equation for a linear potential energy barrier. The one-dimensional timeindependent Schrödinger equation in this case reads: -236Prof. Dr. Muhammad EL-SABA


  2  2 .  ( E  W ( x)).  0 2m x 2

Chapter 4

(4-56a)

where W(x) is a linear potential between the points xo and x1 , which is given by:

 W  Wo   W ( x)  Wo  x  xo  1  x1  xo 

(4-56b)

Therefore, Wo =W(xo), and W1 = W(x1). For xo < x V1 and E > VN (4-65c) For the bound state case, the eigenenergy satisfies the following equation:

T11(E) = 0

for

E denotes the state of the system at any one time t, the following Schrödinger equation holds: (4-89b)



Chapter 4

where H is a self-adjoint operator, called the system Hamiltonian. As an observable, H corresponds to the total energy of the system. The Heisenberg picture does not distinguish time from space, so it is better for relativistic theories than the Schrödinger equation. Moreover, the similarity to classical physics is more obvious: the Hamiltonian equations of motion for classical mechanics are recovered by replacing the commutator above by the Poisson bracket. In Stone-von Neumann theorem, the Heisenberg picture and the Schrödinger picture are unitarily equivalent. However, Schrödinger's formalism is easier to understand. The Interaction picture (or Dirac picture) is an intermediate representation between the Schrödinger picture and the Heisenberg picture. Whereas in the other two pictures either the state vector or the operators carry time dependence, in the interaction picture both carry a part of the time dependence of observables. A state vector in the interaction picture is defined as follows: (4-89c) Where Ho,S is the stationary part of the Hamiltonian of the Schrodinger equation and |(t)> is the same state vector as in the Schrödinger picture. An operator in the interaction picture is defined as follows: (4-89d) Note that As(t) is not typically time dependent, and can be rewritten as As. It only depends on t if the operator has explicit time dependence, for example an external time-varying electric field. As for the density matrix, it can transform to the interaction picture in the same way as any other operator. The following table summarizes the main differences between the three pictures of quantum mechanics. Table 4-3. Quantum pictures Time Evolution of: State vector

Quantum Picture Heisenberg (Matrix Mechanics)

Interaction (Dirac)

constant

Observable Density matrix

Schrödinger (Wave Mechanics)

constant constant -255-

Prof. Dr. Muhammad EL-SABA


Chapter 4

4-21. Quantum Perturbation Theory As we stated so far, there exist only few systems for which the Schrödinger equation (or equation of motion) can be solved exactly. Therefore, the approximate methods are expected to play an important part in all applications of the quantum theory. So far, we presented the WKB approximation, which is used to solve the Schrödinger equation in certain tunneling problems. The perturbation theory is also an approximate method to solve specific quantum problems. In perturbation theory, the Hamiltonian Ĥ of the system under investigation is split into a basic, and exactly solvable part Ho, and a perturbation H’ and written as: Ĥ = Ho + H’. Perturbation theory then uses the knowledge of the solution to Ho (i.e., of its eigenfunctions no and eigenvalues Eno) together with the perturbation H’ to give an approximate solution of the system with full Hamiltonian Ĥ. Here we’ll briefly describe the basis of the quantum perturbation theory for both stationary (bound) states and dynamic (scattering) states. 4-21.1. Stationary (Time-independent) Perturbation Theory The stationary perturbation theory is concerned with finding the deviation in the discrete energy levels (of bound electrons) when a small perturbation, like electric or magnetic fields, is applied to the system of electrons. The Hamiltonian of Schrödinger equation may be written as the sum of two parts: Ho of a simple system for which the Schrödinger equation can be easily solved, and a small term H’, which represents the perturbation:

Ĥ= Ho + H’

(4-90a)

where  is a constant, which is limited between 0 and 1. This parameter allows us to turn on or turn off the perturbation, for instance,  =0 means no perturbation. Assume the unperturbed system has Eno energy levels and no wavefuctions. The perturbed wave function and energy levels, may be expressed in expanded series as follows:

n = no + n1 + n2 + n3 …. + …

(4-90b)

En = Eno +  En1 + 2 En2 + 3 En3 …. + …

(4-90c)

The first terms of this series are illustrated in figure 4-33. Applying the Schrödinger equation to the system, Ĥn =(Ho+H’)n= En, yields: -256Prof. Dr. Muhammad EL-SABA


Chapter 4

0th order terms: (Ho - Eno ).no = 0 1st order terms: (Ho - Eno ).n1 = (En1 –H’)no 2nd order terms:(Ho - Eno ).n2 = (En1 –H’)n1 + En2 no 3rd order terms:(Ho - Eno ).n3 = (En1 –H’)n2 + En2 n1 + En3 no etc…. (4-91) The 0th order terms represent the unpertubed system and the 1 st order terms represent the first-order perturbation and so-on. The solution of the wavefunctions nj of the jth order perturbation needs the knowledge of n,j-1. Also, the knowledge of En,j needs the knowledge of En,j-1. For instance, the first-order perturbation energy value En1 (j = 1) is given by:

En1 = (no , H’n1 ) = < (no | H’| no) >

(4-92a)

This is actually the correction term of the first-order perturbation. Therefore, the first order perturbation energy is given by:

En1 = En0 +  < (no | H’| no) >

(4-92b)

And the corresponding wavefunctions of the first order perturbation: (4-92c)

Fig. 4-33. First order and second order correction terms

4-21.2. Applications of the Stationary Perturbation Theory The stationary perturbation theory is usually applied to study the effect of external static (electric and magnetic) fields on the atom energy levels. i-Simple Example The following figure depicts a simple example of a perturbed system, which has the following Hamiltonian. -257Prof. Dr. Muhammad EL-SABA


Ĥ= Ho + H’ = U(x) + Up(x)

Chapter 4

(4-93a)

where the unperturbed system (whose Hamiltonian (Ho=U(x)) has a wellknown solution, as indicated in §4-16.1 (1-D Square Potential Well).

Fig. 4-35. First order perturbation example

In the case of symmetric solution

 (x) = A exp (x) x < -a (x) = B cos (x) |x| < a (x) = A exp (-x) x > a with  

2m E 

2

, = 

2m(U-E)



2

and

Eno = ħ22/8ma2(2n+1)2 The change in the lowest energy level is:

As the perturbation potential Up(x) outside the interval b = -3 e ζ ao

(4-94b)

where ao is the atom radius. For instance, the hydrogen atom in its 1 st excited state behalf like a permanent dipole moment of magnitude 3e.ao. The amount of splitting and or shifting is called the Stark splitting or Stark shift. In general one distinguishes first- and second-order Stark effects. The first-order effect is linear in the applied electric field, while the second-order effect is quadratic in the field.



Chapter 4

Fig. 4-35. Illustration of the Stark Effect. Regular Rydberg energy level spectra of hydrogen in an electric field near n=15 for magnetic quantum number m=0

4-21.3. Dynamic (Time-dependent) Perturbation Theory The dynamic perturbation theory is concerned with finding the deviation in the energy levels of scattering electrons, due to small perturbations. At first sight, the scattering problem would appear to be a difficult timedependent problem involving the coordinates of two particles that have separate wave functions, and then they move and interact, and finally scatter into two wave packets moving in new directions. This problem is indeed difficult, but it may be transformed into a time-independent, stationary-state problem that is much easier to solve. In this view, we consider a state of one particle corresponding to a plane wave at large distances that interacts with a fixed potential at the origin. Outgoing waves are identified, and they represent the scattered particle. Consider the case of a moving electron, which is scattered by atom vibrations (phonons) in a crystal lattice. Here the electron energy changes with time, when it is affected by the potential perturbation Hdef (r). According to the perturbation theory, the energy eigenvalues in nonequilibrium state (perturbed state) can be obtained by solving the timedependent Schrödinger equation with the perturbed Hamiltonian Ĥ:

Ĥ(r,t) = j (r,t)/t

(4-95a)

Ĥ(r) = Ho (r) + Hdef (r,t)

(4-95b)

where Ho is the unperturbed Hamiltonian of the system (in equilibrium) and Hdef(r,t) = Hdef(r).exp(±jt) is the Hamiltonian perturbation due to system defects (e.g., lattice vibrations or phonons). Assuming that the -261Prof. Dr. Muhammad EL-SABA


Chapter 4

perturbation component is not dominant, the solution can be put in the following series expansion:

(r,t) =  Ci(t) i(r) exp [-j (E/) t]

(4-95c)

where i(r) is the eigen-function of the unperturbed sate (initial state before collision) such that:

Ho  = Ei i

(4-95d)

Here Ei is the energy of the initial unperturbed state, before collision. Substituting (4-95c) into (4-95a) and taking the scalar product with the conjugate of the final state wave-function, f, yields:

j d Ci(t) /dt =  Ci(t) exp [-j t].

(4-96a)

where  =[(Ef -Ei)/ħ]. Solving the above equation with the coefficients Ci(t) = Ci0(t) + Ci1(t) + Ci2(t) +…. yields:

Ci0(t) =  (ti – tf)

(4-96b)

and

1 C i 1 ( t )  2  i H def  *f  2

2

sin 2 t 

2

(4-96c)

Similarly:

1 C f 1( t )  2  *f H def  i  2

2

sin2 t 

2

(4-96d)

On the other hand, we define the probability of finding the particle scattered from an initial state (Ei) to final state (Ef) as:

P( E f , Ei )  lim C f 1 t 

2

(4-96e)

4-21.4. Transition Probability The transition rate of a particle from an initial state (Ei) to final state (Ef) is given by:



S( E f , Ei ) 

P( E f  Ei ) t

1  lim 2 H def t  

Chapter 4

2

sin2 ( t ) ( t )2

(4-97a)

where = [(Ef - Ei)/ħ] = (f -i). If the collisions are not efficient (so that the collision time is negligible with respect to the perturbation duration), then the above relation can be approximated as a Dirac delta function. Keeping only the 1st order time-dependent term in the perturbed wave-function, the transition rate from the initial state (Ei) to final state (Ef) is then given by:

S ( E f , Ei ) 

2  *f H def  i 

2

 ( E f  Ei )

(4-97b)

where we substituted lim [sin(t)/t]2=2()/t. This approximation is t  

sometimes called the first Born approximation. The Born approximation means that the collisions are considered as instantaneous. It should be noted that both Ef and Ei depend on the electron initial and final wave vectors k and k’ as well as the nature of collision process itself.

Fig. 4-36. Plot of probability versus driving frequency, and how it can be approximated to a delta function

For instance, let’s consider the scattering of electrons with lattice vibrations (phonons), as shown in figure 4-37. Collision center


k’

Electron k

k

q


Chapter 4

Fig. 4-37. Schematic diagram indicating the wave vectors before and after collision, inside the first Brillouin zone (q = k - k’).

If  is the energy of the phonon quanta, which is emitted or absorbed during scattering process and the initial and final wave-vectors where situated in the same energy valley (intravalley scattering), then:

Ei = E(k) + Ni  ,

Ef = E(k) + Nf 

(4-97c)

where Ni and Nf are the initial and final phonon distribution (in equilibrium, it is the Bose-Einstein distribution function). As the collision involves either absorption or emission of a phonon, then Nf = Ni ±1. Therefore, the probability of transition per unit time of charge carriers (from state k to state k’), due to a certain scattering mechanism may be expressed by the following a delta function:

S (k , k ' ) 

2 2 M (k , k ' )  ( E (k )  E (k ' )   ) 

(4-98a)

where M(k,k’) is called the transition matrix element (or the dipole moment) of the Hamiltonian perturbation operator Hdef.

M (k, k’) = ∫*(k’) Hdef (k) d3r

(4-98b)

where the wave function(k) involves both electrons and phonons coordinates. The above equation (4-98) is sometimes called the Fermi golden rule. Actually, the momentum conservation rules imply that k and k’ be related to the wave vector of emitted or absorbed quanta (e.g., phonons or photons) q by the following relation:

k = k’ ± q + G

(4-99)

where G is the reciprocal lattice (k-space) translation vector. Note 4-14. Fermi’s Golden Rule & Transition Probability -264Prof. Dr. Muhammad EL-SABA


Chapter 4

One of the prominent failures of the Bohr model for atomic spectra was that it couldn't predict why one spectral line would be brighter than another. The answer came from the quantum perturbation theory, an explanation in terms of wavefunctions, and for situations where the transition probability is constant in time. It is usually expressed in a relationship called Fermi's golden rule. In general, a transition rate depends upon the strength of the coupling between the initial and final state of a system and upon the number of ways the transition can happen (i.e., the density of the final states). In many physical situations the transition probability is given in the following form, which is called the Fermi golden rule:

The transition probability if is also called the decay probability and is related to the mean lifetime  of the state by if = 1/. The general form of Fermi's golden rule applies to atomic transitions, nuclear decay, particle scattering and a large variety of physical transitions. A transition will proceed more rapidly if the coupling between the initial and final states is stronger. The coupling term, |Mif|2, is called the "matrix element" for the transition. This term comes from an alternative formulation of quantum mechanics in terms of matrices rather than the differential equations of the Schrodinger equation. The matrix element can be placed in the form of an integral where the transition interaction may be expressed as a potential V (or a perturbation Hamiltonian) which operates on the initial state wavefunction. The transition probability is proportional to the square of the integral of this interaction over the system space.

4-21.5. Optical Transitions The matter-light interaction can help for a quantum description of the operation of optoelectronic devices. The study of matter-light interaction usually begins with the time-dependent perturbation theory and the -265Prof. Dr. Muhammad EL-SABA


Chapter 4

approach of Fermi’s golden rule for optical transitions. The Fermi’s golden rule gives the rate of transition from a single state to a set of states, which can be described by a density of state function. The transition from a state to state may occur as a result of collision of electrons with lattice vibrations (phonon absorption or emission) or upon interaction with EM radiation or light (photon emission or absorption). As we have seen above, the probability of a transition from an initial state |i> to a final state |n> can be written, according to Fermi, as follows:

(4-100)

where ni is dipole moment13, which is involved in the light-matter interaction process.

Fig. 4-38. Schematic illustration of an electromagnetically induced transition from an initial state i to one of final states n

13

The classical and quantum mechanical descriptions of the matter–field interaction therefore, both incorporate the dipole at a fundamental level. In the classical theory, absorption occurs when an incident wave induces a dipole moment along the direction of the wave polarization and then the surrounding medium dissipates the energy. Emission occurs when an excited dipole synchronously radiates energy with its motion. Maxwell’s wave equation incorporates the dipoles in terms of the polarization vector or susceptibility. While quite successful, this classical description does not account for the quantum nature of matter and light, and does not explain basic phenomena such as the spontaneous emission of light from matter. The quantum theory of the matter–field interaction uses the operator form of the dipole. -266Prof. Dr. Muhammad EL-SABA


Chapter 4

Fig. 4-39. Schematic illustration of an electromagnetically induced transition from an initial state i to one of final states n

i. Light Absorption Probability We first consider the case for absorption where ni. Then we have the component of the wave function parallel to the |n> axis

(4-101)

We can find the transition probability for absorption from the expression:

or

(4-102)

ii. Light Emission Probability The case for emission obtains when ni

(4-103)

This gives:



Chapter 4

(4-104)

and

(4-104)



Chapter 4

4-22. Second Quantization The first forms of quantum mechanics given by Schrödinger’s wave equation and Heisenberg's matrices though originally developed for the description of systems with single particles, can also be applied to a system with a fixed number N of identical particles. Nevertheless this is very cumbersome, mainly because of the Pauli principle forcing the wave functions to be anti-symmetric (fermions) or symmetric (bosons) when exchanging two particles. Anti-symmetry wave functions (done by the Slater determinant) is rather complicated and error-prone. Therefore, a different representation for quantum mechanics, the second quantization, has been developed to remove the problems just mentioned. Indeed, second quantization removes the problem of anti-symmetry (or symmetrizing) wave functions. The formalism of second quantization automatically takes care of that. Additionally, it allows for the treatment of systems with varying number of particles, such as phonon or photon systems, or for the treatment of the superconducting state as in the superconductivity (BCS) theory. Quantum mechanics, in its original formulation, deals with operators acting on wave functions. This is also the case in the formulation provided by second quantization. But the operators used in second quantization are rather different from those in the original formulation which we call the first quantization. In first quantization we have learned how to describe a given physical system by means of the Hamilton operator. The description of the physical system in second quantization is based as well on a Hamilton operator, but its form is completely different from that of the former. One of our goals here is to show how to construct the Hamiltonian used to describe a given physical system in second quantization when the corresponding Hamiltonian for first quantization is given. Let us first summarize the treatment of a system of a fixed number N of identical particles in first quantization. If one of the particles is described by the Hamiltonian: (4-105) The collection of N identical and not interacting particles is represented by the N-particle Hamiltonian: (4-106)



Chapter 4

where the operators pi and ri are acting on the ith particle. The wave functions of the N-particle system are: (4-107) which can be written as a linear combination of the functions: . (4-108) That is, products of single particle wave functions k(r) which are usually chosen as eigenfunctions of the Hamiltonian h0, and therefore given by: (4-109) where k is a quantum number denoting a stationary state of the single particle Hamiltonian h0. In a translationally invariant system or a crystal, this may be the momentum or quasimomentum, respectively, together with a spin index if the particles under discussion carry such property. Note that these N-particle wave functions have to be antisymmetrized (we focus on fermion systems; for bosonic systems, the wave function has to be symmetrized), and therefore, all the single particle states ki have to be different. Otherwise the wave function vanishes, expressing the fact that a single-particle state can be occupied only once in fermionic systems. A translationally invariant two-particle interaction as for instance the Coulomb interaction is represented in the N-particle system by the Hamiltonian:

(4-110)

Where the sum is taken over all combinations of two particles. The factor ½ compensates for a double-counting in the sum over i and j. 4-22.1 Creation and Annihilation (destruction) Operators In second quantization, we usually introduce some new operators, called the creation and destruction operators. Given a state |>, the creation operator c+|> adds a particle in the single-particle state  to |>. If |> is an N-particle state, then c+|> is an (N+1)-particle state. If the state |> -270Prof. Dr. Muhammad EL-SABA


Chapter 4

already contains a particle in state, the expression c+|> vanishes. The destruction operator c works in a similar way. It removes a particle in the single-particle state from the (many-body) state |>, and destroys the entire many-body state, if such particle was not there. As the notation suggests, the creation and destruction operators are mutually Hermitian conjugate. For bosons, creation and destruction operators b+ and b are also defined. They work similarly to their fermion-colleagues, but allow for multiple occupancy of the single-particle states. That is, the creation operator b+ never annihilates a state |>, whether it already contained particles in the single-particle state or not. These properties of the Fermions and Bosons creation and destruction operators are guaranteed by anti-commutation and commutation rules, respectively. 4-22.2. Fermion Operators For Fermions, the anti-commutator operators read: (4-111)

guarantee antisymmetry between Fermion particles (like electrons), because cc = -cc, and prevent from double occupancy. The second relation, taken for  determines the eigenvalues of the creation and destruction operators. Applied to a state |>, it yields cc+|> + c+ c=|>. If the single-particle state  is occupied in |>, the first term in the sum will vanish whereas the second one reproduces ji with a factor of one. If the state is unoccupied, the first term yields a factor of one, while the second vanishes. 4-22.3. Boson Operators For boson, operators have to express the fact that many particle states of a system composed of identical boson particles are symmetric upon exchange of two particles. Then the bosons commutation relations read: (4-112)

These relations describe the fact that single-particle boson states can be occupied multiply. Note that the creation and destruction operators are -271Prof. Dr. Muhammad EL-SABA


Chapter 4

not hermitic and therefore, are not observables. The importance of these operators lies in the fact that all other operators can be expressed as linear combinations of products of creation and destruction operators. An example is the particle number operator N, and the Hamiltonian H0, which are given by: (4-113) where nk is the number operator counting the number of particles in the single-particle state k, and k is the single-particle dispersion relation. Note 4-15. Many-Body Problems One interesting challenge in quantum mechanics has been the manybody, or N-body problem. The problem is that while we have a good understanding of what happens to individual independent electrons that are not interacting with each other, electrons do interact with one another via simple electrical repulsion. Apparently all electrons, unlike people, are the same, although like people (who are composed of electrons) not two electrons may occupy the same space. Incorporating the interaction of electrons into our understanding has been a big problem. Professor Walter Kohn in his Nobel Prize Lecture estimated that the number of parameters needed to describe a system of N interacting electrons varies at least as e3N. That means that understanding the effect of one additional electron requires a factor of e3 = 22 increase in computer capacity. If computers continue to double power every 1½ years (like Moore’s law), it will take about 8 years to improve our understanding by one new electron. In order to extend our present level of understanding in helium with 2 electrons it will take about 32 years. To finish the periodic table of 100 elements it will take 800 years, a single strand of DNA will take about 1,000 times the age of the universe. This is a problem. According to our present knowledge, there are almost 1080 electrons in the discovered universe. To understand complex systems we need to wait for a new theory! or we try simplifying assumptions.



Chapter 4

4-23. Correlation Between Solid-State Physics, Nuclear Physics & Cosmology It’s well known that the nuclear physics has been developed in parallel with solid-state physics on the basis of quantum theory. The nuclear physics has been interested in the discovery of atomic structure of matter as well as the interaction between different atomic particles and constituents. The electron was discovered by J. J.Thomson at the end of 19th century. In 1911 E. Rutherford has analyzed the experiments of H. Geiger and E. Marsden, and suggested that the atom is composed of a positive nucleus surrounded by a cloud of electrons. Then, in 1913, N. Bohr established his famous model describing the quantized nature of electron motion around the nucleus. Using this model, Bohr was able to describe the spectrum of the hydrogen atom. The discovery of the nucleus constituents and the nuclear forces that glues them together has been a fruitful subject of investigation along the last half century. So, the nucleus was considered to be composed of positive protons and neutral neutrons. Neutrons were discovered by Chadwick in 1932. Both protons and neutrons share so many physical characteristic (except for charge). So, they may be considered as two charge cases of a single particle called nucleon. The number of protons is called the atomic number Z and the number of nucleons A (number of both protons and neutrons) is called the atomic weight of the atom. The similar charge nucleons are confined together in a tiny space (the nucleus) by the aid of other quasi particles called mesons, which act as the glue between them. So, mesons are the nuclear field particles, as photons are the electromagnetic field particles. Actually, it has been shown experimentally that the nucleon has a well defined volume and maybe considered as a sphere of nuclear material whose radius is equal to: ro A1/3, where ro = 1.2 x 10-13 cm (1.2 Fermi). The strong nuclear forces are short-range forces, whose field decays so rapidly (exponentially) outside the nucleus. The nuclear potential energy between two nucleons may be expressed as a function of the distance r, between them, as follows:

U(r) = C (m / r) exp ( - r/m )

(4-114)

where C is a constant in the order of 10 MeV. Also, m is the Compton wavelength of the nuclear field particles, (m= /mc =1.4 Fermi). -273Prof. Dr. Muhammad EL-SABA


Chapter 4

Fig. 4-40. Illustration of the internal nuclear force.

A peculiar and very important property of the nucleus is that its mass is not the sum of the masses of the protons and neutrons that make up the nucleus, it is less! This is because the strong nuclear force binds neutrons and protons in the nucleus together and energy is required to separate them. This energy is called the binding energy, and by E=mc2 gives the nucleus less mass than the sum of the masses of its constituents. The strong binding forces are only effective on short distances r 10-24 sec). Also some particles share so many characteristics. For instances, the so-called muons, are fatter but shortlived brothers of electrons. The so-called anti-particles have the same characteristics of their counterpart particles, except for charge. Thus antiparticles have opposite baryon number ‘b’. For instance, positrons, which are the anti-electrons have baryon number b = +1, while electrons have baryon number b = -1. When a particle collides with its anti-particle, they annihilate, and burn out completely. In this collision, all their mass is transformed into high energy photons (electromagnetic  rays).

Fig. 4-41. Types of Particles. The baryon number ‘b’ indicates the particle charge. -275Prof. Dr. Muhammad EL-SABA


Chapter 4

The quarks, which are the constituents of nucleons (like protons and neutrons) have fractional baryon numbers (b = 1/3), i.e., a fractional charge. Actually, there exist 6 types of quarks and anti-quarks, with different baryon numbers but all have spin s = ½. Quarks are named as up/down, charm/strange and top/bottom quarks. The following table depicts the above indicated particles. The existence of quarks was assumed by M. Gell Mann, and has been confirmed experimentally in the 1990’s. Today, quarks and leptons, and their antiparticles, are candidates for being the fundamental building blocks from which all else is made. For instance, protons and neutrons are made up of 3 quarks. Particle physicists call them the "fundamental" or "elementary" particles -- both names denoting that, as far as current experiments can tell, they have no substructure. The figure 4-42 depicts the fundamental particles, which are structure-less, up to our current knowledge today. The particles divide into two classes, quarks and leptons. There are 6 particles of each class and 6 correspond-ing antiparticles. All other particles, like neutrons and protons are made-up of these 12 elemental particles. In addition, there are 4 force carrier particles (gluons, photons, W & Z bosons), that are responsible for strong, electromagnetic, and weak interactions. These force carriers are also fundamental particles.

Fig. 4-42. Table of fundamental matter particles (current state of knowledge)



Chapter 4

Fig. 4-43. Evolution of the new discoveries of elemental particles

4-23.2. Physical Forces between Particles There are four basic forces at work in nature, namely:    

Gravity Force Strong nuclear Force Weak nuclear Force Electromagnetic Force

Therefore, the physical reactions between particles may be classified into one of the following types:    

Gravitational reactions (massive bodies, with gravitons). Strong nuclear reactions, (hadrons, like mesons, with gluons), Weak nuclear reactions, (unstable particles, with W & Z ) Electromagnetic reactions, (electrons, with photons)

All these reactions are obeying conservation and symmetry rules. For instance, the total charge and energy conservation, of interacting particles, are always obeyed in all interactions. Particles that interact by the strong interaction are called hadrons. This general classification includes mesons and baryons but specifically excludes leptons, which do not interact by the strong force.



Chapter 4

The weak interaction acts on both hadrons and leptons. Hadrons are viewed as being composed of quarks, either as quark-antiquark pairs (mesons) or as three quarks (baryons), as shown in figure 4-41. There is much more to the picture than this, however, because the constituent quarks are surrounded by a cloud of gluons, which are the exchange particles for the color force. The charge associated with strong interactions is called color charge (R/G/B and their combinations). Table 4-4. Fundamental forces in Nature

Force Strong Nuclear Force which ties nucleons Electromagnetic

Strength Range Particle 1 Short , others -15 10 m mass > 0.1 GeV 1/137 10-9

Weak Nuclear

Gravity

10-38

Infinite Photon, mass = 0, spin = 1 Very W+, W-, Z, Short Mass > 80 -17 10 m GeV, spin=1 Infinite Graviton?, mass = 0, spin = 2

Quarks and gluons have color charge and consequently participate in strong interactions. The other type of fundamental particles is leptons. There exist 6 types of leptons, 3 of which are charged (electrons, muons and tau particles) and 3 are not charged (neutrinos-e, neutrinos-v and neutrinos-). Leptons, photons and W and Z bosons do not have color charge and therefore do not participate in strong interactions. The W and Z particles are the massive exchange particles which are involved in nuclear weak interactions, between electrons and neutrinos. They’re predicted by Abdul-Salam, the Pakistanis scientist. Abdul-Salam has proved that the weak nuclear forces have an electromagnetic nature, and was awarded the Nobel Prize for this discovery. The prediction included the masses of these particles as a part of the unified theory of the electromagnetic and weak forces, that was confirmed and measured at CERN laboratory in 1982. The fact that the observed strengths seem different is attributed to the masses of the W and Z particles. Under certain conditions a force of -278Prof. Dr. Muhammad EL-SABA


Chapter 4

large strength can have the appearance of a force of small strength if the particle that carries the force is very massive. Theoretical calculations show that at a fundamental level the weak and electromagnetic forces have the same strength if the W and Z particles have masses of 80 and 90 GeV respectively. The masses measured at CERN laboratories were 82 GeV and 93 GeV, a brilliant confirmation of the electroweak unification. Table 4-5. Fundamental building blocks (particles) of the universe.

Fig. 4-44. Hadrons and their types.



Chapter 4

4-23.3. Standard Model The Standard Model is a hugely-complex and mathematically-arcane theoretical construct painstakingly pieced together over last 40 years by particle physicists to explain the ever-larger bestiary of subatomic particles. Until recently (July 2012), a prime missing link in the standard model was the Higgs Bosons. Higgs are field particles, which are recently discovered during electrons positrons collision experiments. Higgs are believed to have a huge equivalent mass (about 115 GeV). On December 2013, the Nobel Prize in Physics was awarded to Peter Higgs and François Englert, for their work and prediction. The existence of Higgs explains why fundamental particles have mass and why their masses are widely disparate. In fact, the idea of Higgs particles derives from the solid-state physics, where we consider that electrons have an effective mass, which is different from their mass in free space, due to the surrounding field of lattice vibrations (phonons)., particle Physicists presume that empty space is similarly permeated with a Higgs field. Therefore, this background field of Higgs is locally distorted when a particle moves through it and clusters around the particle and creates its mass. Thus, the Higgs Bosons in free space is presumed to behave like phonons, in solid-state materials.

Fig. 4-45. The standard model of particles.

The so-called “Super-Symmetry Theory” is an extension to the Standard Model, which brings the gravitational force within the model -280Prof. Dr. Muhammad EL-SABA


Chapter 4

rubric. It presumes that each of the matter particles has its own antimatter super-symmetric counterpart. When matter and anti-matter collide together they vanish and energy is released in the form of gamma  rays (high energy photons) are released. It’s believed, according to current cosmic theory, that when our universe, began (after a Big Bang14) an equal quantities of matter and antimatter were produced.

Fig. 4-46. The history of the universe, according to the Big Bang Theory.

14

The current model of the universe is based on the 'Big Bang' concept, whereby the original matter in the universe was contained in an infinitely small concentration that somehow exploded, sending material in an outward radial direction, as shown in figure 4-18. This 'hot big-bang' concept has been validated by observations over the past fifty or more years. The Hubble Constant, which defines the rate of expansion of the universe, has been refined many times as data became more abundant and more precise. The presently accepted value for the Hubble Constant is 71 km/sec/mpc yielding an age for the universe of 13.7 billion years (Harwood, 2003). Observations of stellar redshifts support a model of the universe that is expanding in all directions, leaving behind the telltale sign of a 2.73K background radiation (according to Nicolson, 2003). The main missed point in the Big-Bang theory is the justification of the emission of billions of billions of billions of electrons and other particles with such incredible precision of charge and mass. This indeed should be done by a smart force. Interestingly, the idea of this smart Big-Bang theory has been pointed to explicitly in the Holy Quran (the last revealed authenticated book, more than 1400y ago, since) in chapter 21 (Al-ANBYA) verset 30. Muslims believe that Qurán is an immortal miracle of the prophet Muhammad, and that it replete with so many scientific facts that have been discovered and have become scientifically established and and those to be discovered in the course of time. Muslims ask; How did prophet Muhummad, who was unlettered, know these amazing facts that has been discovered recently?



Fig. 4-47. Illustration of the big-bang theory


Chapter 4


Chapter 4

Universe is mostly light (photons). Apparently, it was light that formed the dominant constituent of the universe. The ordinary matter came after. Electrons and positrons created from light (pair-production) and destroyed by annihilation at about equal rates. The pair-production threshold is 1 MeV, so the thermal energy kT=8.6 MeV was well above that. Protons and neutrons being changed back and forth, so about equal numbers. The energy difference between neutron and proton is 1.29 MeV, so protons can be freely changed to neutrons at this temperature. Only about one baryon for 109 photons, as inferred from the 3K background and density estimates. Since the conservation of baryon number is a strong conservation principle, it is inferred that the ratio of photons to baryons is constant throughout the process of expansion. Until recently, cosmology physicists believed that matter and antimatter15 are mirrors of each other. We wouldn’t actually be able to tell whether we live in a matter or antimatter world. However, some experimental observations, in 1960’s, have shown that the decay of some sort of mesons called kaons (or k-mesons), has a rate which is different from their counterparts. This asymmetrical decay has enabled the Russian scientist A. Sakharov to show why one type of matter prevail over the other in the universe. Sakharov was awarded Nobel Prize for this discovery. Technically speaking, the asymmetry of matter and antimatter decay is referred to as “Charge conjugation/Parity” or CP. The CP is measured, for a given symmetric pair of particle, as a factor which changes between –1, 1. Nevertheless, there is no strong evidence of the amount of antimatter left in the universe now. Actually, there is billion times as much energy in terms of radiation (compared to the existence of matter) and this tiny bit of matter that is left over—that’s us, the whole universe. Concepts related to the idea of other parallel universes exist in physics, in philosophy and theology. Many experiments have been made on the decay of massive particles (like muons) and their counterparts to discover the reason beyond this big difference. Unfortunately, these particles are 15

Dirac introduced the concept of antiparticles. Now we know that for every particle there is an antiparticle. However some particles could be self-conjugate, in the sense that particle and antiparticle could be the same. Of course such particles have to be electrically neutral.The possibility of a self-conjugate fermion was first pointed out by Majorana, and hence they are called Majorana fermions while the other fermions (with distinct particles and antiparticles) are called Dirac fermions. Among the fermions of the Standard Model, only neutrinos are electrically neutral and hence qualify to be Majorana particles. But it is still an open question whether neutrinos are Majorana particles or Dirac particles. -283Prof. Dr. Muhammad EL-SABA


Chapter 4

not so massive enough (kaons = 970 times heavier than electrons, and muons are 1870 times heavier) to account for the big discrepancy between matter and antimatter. However, the discovery of Higgs of such huge masses (about 115 GeV) may be a key issue for a better understanding of this problem. 4-24. Quantum Field Theory (QFT) The quantum field theory (QFT) is an advanced theory that tries to explain the matter-wave interactions, in terms of the mechanics of the field particles. For instance, the electromagnetic field particles are photons. Therefore, the interaction between two electrons involves the exchange of photons. Also, the nuclear field particles are some sort of mesons. Therefore, the interaction between two nucleons (nucleus particles) involves the exchange of mesons. In fact, the classical electromagnetic field theory, which has been developed in the 19 th century, is entirely based on the solution of Maxwell’s equations. For example, one can solve the problem of field propagation between two antennas, by solving the Helmholtz equation (which can be derived from the set of Maxwell’s equations), without even noting the nature of the thing which is vibrating during wave propagation. In the past, physicists were conventionally using the word “ether” to refer to the vibrating medium during electromagnetic wave (EMW) propagation. Now, we know that EMW are propagating via the mechanical vibration of photons. The quantum electrodynamics (QED) is an example of QFT that describes the properties of electromagnetic radiation and its interaction with matter. As we have argued so far in §4-15, this theory has proved spectacularly successful in describing the interaction of light with matter. QED deals with processes involving the creation of elementary particles from electromagnetic energy, and with the reverse processes in which a particle and its antiparticle annihilate each other and produce energy. The fundamental equations of QED apply to the emission and absorption of light by atoms and the basic interactions of light with electrons and other elementary particles. Charged particles interact by emitting and absorbing photons, the particles of light that transmit electromagnetic forces. For this reason, QED is also known as the quantum theory of light. Both QFT and QED are based on the elements of quantum mechanics laid down by P. Dirac and W. Heisenberg, during the 1920's, when photons were first postulated, as well as the Einstein theory of special relativity. In 1928, Dirac formulated an equation describing the motion of electrons that incorporated both the requirements of quantum theory and the theory of special relativity. -284Prof. Dr. Muhammad EL-SABA


Chapter 4

The Dirac equation (4-33), which is sometimes called the relativistic Schrödinger equation, was capable of describing the spin motion of electrons, for the first time. However, during the 1930s, it became clear that QED gave the wrong answers for some relatively elementary problems. For example, although QED correctly described the magnetic properties of the electron and its antiparticle, the positron, it proved difficult to calculate specific physical quantities such as the mass and charge of the particles. It was not until the late 1940’s, when experiments conducted during World War II that these difficulties were resolved. The development of the theory was the basis of the 1965 Nobel Prize in physics, awarded to R. Feynman, J. Schwinger and S. Tomonaga. They showed that two charged particles can interact in a series of processes of increasing complexity, and that each of these processes can be represented graphically through Feynman diagrams.

Fig. 4-48. Some Feynman diagrams used in the calculation of the amplitude of a moving electron from point A to B. Solid lines with arrows correspond to electrons (positron arrows point in the opposite direction). Wavy lines correspond to photons. Black dots correspond to interaction vertices. Virtual particles are drawn in red

In QFT, instead of calculating a wave function, we calculate the amplitude. Just like the wave function, the amplitude has the magnitude, with its probabilistic interpretation. Amplitude of a transition from an initial state to some final state is calculated in QFT by summing up the amplitudes of all possible paths between the two states. The paths not only correspond to different trajectories of particles, but they involve the creation and annihilation of multitudes of intermediate virtual particles. The physics of virtual particles is different than that of ordinary particles. -285Prof. Dr. Muhammad EL-SABA


Chapter 4

A virtual photon or electron may have any mass, including imaginary (real photons are massless). They still have to obey the fundamental laws of physics, like the conservation of energy and momentum, conservation of charge, etc. The various possible intermediate paths (or histories) are described using Feynman diagrams. For instance, to calculate the amplitude of an electron going from point A to point B, you have to sum up the amplitudes corresponding to: 

A single non-virtual electron moving from A to B  The electron emitting a virtual photon and re-absorbing it  The electron emitting two virtual photons and re-absorbing them  The electron emitting a virtual photon which turns into an electronpositron pair, which then annihilates, creating another virtual photon that is absorbed by the electron, and so on. Taking the example of the force between two electrons, the classical theory of electromagnetism would describe it as arising from the electric field produced by each electron at the position of the other. The force can be calculated from Coulomb's law. However, the quantum field theory approach visualizes the force between the electrons as an exchange force arising from the exchange of virtual photons. It is represented by a series of Feynman diagrams, the most basic of which is shown in the figure above. With time proceeding upward in the diagram, this diagram describes the electron interaction in which two electrons enter, exchange a photon, and then emerge. Using a mathematical approach known as the Feynman calculus, the strength of the force can be calculated in a series of steps, which assign contributions to each of the types of Feynman diagrams associated with the force. Figure 4-49 depicts the Feynman diagram for electron-electron interaction.

Fig. 4-49. Feynman diagrams for electron-electron interaction (charge repulsion).



Chapter 4

The QED applies to all electromagnetic phenomena associated with charged particles such as electrons, and the associated phenomena such as pair production, electron-positron annihilation, Compton scattering, etc. It was used to precisely model some quantum phenomena which had no classical analogs, such as the Lamb shift and the anomalous magnetic moment of the electron. QED was the first successful quantum field theory, incorporating such ideas as particle creation and annihilation into a self-consistent framework. The quantum field theory has been applied to understand the strong interactions between quarks and protons, neutrons, and other baryons and mesons. It is sometimes called quantum chromo-dynamics (QCD). More details about Feynman diagrams can be found in Chapter 9, when we talk about the quantum transport theory.

4-25. Unified Field Theory The unified field theory is the theory that attempts to combine any two or more of the known interaction types (strong, electromagnetic, weak and gravitational) in a single theory so that the two distinct types of interaction are seen as two different aspects of a single mathematical structure. The ‘grand unified' theory (GUT) unifies three of the four force types (strong, weak and electromagnetic interactions) in this way. The benefit is that the unification gives a simpler overall theory and predicts relationships between parameters that are otherwise independent.

4-26. Gauge Field Theory All forces are known to be governed by gauge symmetries, descriptions of which are called gauge theories. The gauge field is the phase of the wave function. The gauge field is not unique, so it allows the electromagnetic field to be calculated, but we cannot calculate the electromagnetic field from the gauge field. In fact, there are many gauge fields that will produce the same electromagnetic field and there is no way to find which it is. The wave function needs to include the effect of local changes in phase. To do this we add the potential A(r) term, which accounts for the electromagnetic force field, to the Hamiltonian in the Schrödinger equation. 4-27. The Final Theory and Theory of Everything (ToE) The so-called final theory shows that the simple reality of our universe has been distorted by a long succession of misunderstandings. It began with Newton’s mistake that gravity is attracting force, which neither he, nor anyone, has ever been able to explain. Similarly, Franklin’s mis-287Prof. Dr. Muhammad EL-SABA


Chapter 4

understanding of observations gave us the equally unexplained electric charge concept. Also, Einstein committed a further mistake by considering gravity as a mysterious warping of 4-dimensional space-time, which also remains completely unexplained. However, other theories followed, such as the concept of the nuclear forces holding the atom together, quantum theory, special relativity theory, the big bang concept, and the theory of everything (ToE). The later theory claims that the extension of Einstein’s Space-time concept with quantum physics explains all phenomena of the universe: gravity, quarks, antimatter.. etc. However, all of these ideas are useful working models of observations, but they are all based on unexplained hypotheses. For instance, physicists have never posed a fundamental question about the origin of the universe concerning the enigma of identical electrons. Electrons and positrons have extremely precise masses: 0.510998918 MeV. How can it be that all electrons and positrons of the universe have strictly the same mass and volume? measured with an extraordinary precision of less than 0,0000086%. This enigma needs a rational explanation. In our search for satisfying answers and better understanding it is crucial that we ask our scientists about their beliefs.

Fig. 4-50. Forces and Theories



Chapter 4

4-28. Summary Quantum mechanics is necessary for the description of nature on the atomic scale. Quantum theory tells us about the nature of the microscopic constituents of matter, from molecules and atoms to subatomic particles and quarks. These tiny particles behave in a totally different way from objects in our ordinary experience. Our knowledge about matter on atomic scales has led to technological advances in nuclear physics and materials science. Somewhere along the continuum from quantum to classical, the two descriptions must merge. On the other hand, quantum theory has shown that light is not just a wave, but behaves like particles (photons). This insight resulted in the field of quantum optics, which has led to the invention of laser and optoelectronic devices. Applications to cryptography and computing are still in an experimental stage.

There is an important difference between classical (Newtonian) mechanics and quantum mechanics. In quantum mechanics, we do not talk about what will happen in a certain circumstance to a certain particle. Rather, we speak about the probability that a particle will arrive in a given circumstance. Therefore, in quantum mechanics, the only thing that we can calculate is the probability of different events. The quantum theory is based on certain pioneering concepts and principles. According to Planck, the vibration energy of atoms is not continuous. Rather, the frequencies of vibration are fixed and in order to change from one frequency to another, the solid should absorb or emit discrete packets of energy (quanta). The energy of a quantum is proportional the frequency of oscillation: E=nhf where h is the Planck constant, n is an integer. -289Prof. Dr. Muhammad EL-SABA


Chapter 4

In 1924, De Broglie claimed that all matter has a wave-like nature and has a wavelength given by: =h/p, where p is the particle momentum and  is the wavelength of its associated wave. Accordingly, the so-called crystal momentum vector is related to the wave vector by: p =  k

The wave-particle duality holds that light and matter exhibit properties of waves and of particles. The wave nature of light is demonstrated by diffraction, and interference of light and the particle nature of light is demonstrated by the photoelectric effect. On the other hand, the wave -290Prof. Dr. Muhammad EL-SABA


Chapter 4

properties of electrons were suggested by the De Broglie hypothesis and the subsequent experiments of Davisson and Germer. Heisenberg introduced the uncertainty principle, which states that: It is impossible to specify precisely and simultaneously the value of both members of particular pairs of physical variables that describe the behavior of an atomic system. The members of these pairs are canonically conjugated to each other in the Hamiltonian sense. According to Heisenberg statement we have:

 p x .  x   / 2 ,  E.  t   / 2 According to Heisenberg, if you make the measurement on an object, and you can determine the x-component of its momentum with an uncertainty px, you cannot, at the same time know its x-position more accurately than x =  /2px , where h is a definite fixed constant given by nature. Therefore, "The more precisely the position is determined, the less precisely the momentum is known". In quantum mechanics, particles have a wave nature and can be described by wave functions. Our knowledge of a wave is defined by its eigenfunctions. The probability of finding a particle somewhere is given by:

P  r2  r < r1  =

r2



 * d 3r

r1

A milestone in the early development of quantum theory came with the work of E. Schrödinger who proposed his famous equation (1926), describing quantum systems in terms of a complex wavefunctions ψ=ψ(x). In his exposition, he proposed a physical interpretation of the unknown function described by his equation, where he regarded the wave intensity ψ∗ψ as the density of the electric charge. This approach gives a unique image of the electron but was wrong. According to his interpretation, a charge would spread out very rapidly and without any limit, . That is, of course, in contrast with the experimental evidence: indeed a particle is always found in a small region of space. Subsequently, a heuristic interpretation of the wave-function in terms of probability amplitudes was given, known nowadays as the Born rule (1926). In this explanation16, the wave intensity is viewed as the The Born rule. A (complex, normalized) wave-function ψ = ψ(x) represents the most complete description of a given system which squared modulus ψ2(x)dx is the probability of finding a particle around the position x in the interval dx. 16



Chapter 4

probability of finding a particle rather than its density. In other words, the best possible description of a quantum system is of probabilistic nature. The Schrödinger equation and the Born rule, along with the following further prescriptions constitute what is, nowadays, known as the standard or Copenhagen interpretation of quantum mechanics: every system is completely represented by a wave-function ψ(x; t); when a measurement is performed, the wave-function collapses; the wave-particle duality of matter must be invoked to explain experimental results (complementarity principle). Starting from the quantum end and noting that energies have discrete values or quantum numbers, one can anticipate that for high energies or high-enough quantum numbers, the quantum theory merge with the classical. This merging of the quantum and classical is called the correspondence principle. There exist two cases or two types of particles: Bosons and Fermions. Fermions and Bosons are different at the quantum nature. According to Pauli's exclusion principle, identical fermions cannot occupy the same quantum state at the same time. Bosons, however, can share quantum states. In order to observe this fundamental difference, gases of Bosons or Fermions have to be chilled to ultra-low temperatures, where quantum states have a high occupation probability. At low temperatures, Bosons will fall into a single quantum state to form a Bose-Einstein condensate, whereas Fermions tend to fill energy states from the lowest up, with one particle per quantum state. At high temperatures, Bosons and Fermions spread out over many states with, on average, one particle per state. It should be noted that Fermions do not undergo a sudden phase transition in the ultra-low temperature regime. Instead, the quantum behavior emerges gradually as the Fermion gas is cooled below the Fermi temperature TF = EF / kB, where EF is the Fermi energy.



Chapter 4

The Pauli Exclusion Principle is a quantum mechanical principle which states that no two identical fermions may occupy the same quantum state. This principle was formulated by Wolfgang Pauli in 1925. The Pauli Exclusion Principle plays a role in a large number of physical phenomena. One of the most important, and the one for which it was originally formulated, is the electron shell structure of atoms. An atom contains bound electrons equal in number to the protons in the nucleus. Since electrons are fermions, the Pauli Exclusion Principle forbids them from occupying the same quantum state. The time-dependent Schrodinger equation (in non-relativistic form) can be found out as follows:

This equation may be also written as follows:

The Hamiltonian operator is linear and Hermitian. The Schrödinger equation can be solved to find out the eigenfunctions (wave functions) and eigenvalues (allowed energy levels) of particles or electrons in a quantum system, under the influence of a potential energy V.



Chapter 4

The analytical solution is only possible for a few simple systems such as the hydrogen atom. However, for more complicated systems, one may use the numerical solution, together with a series of approximations. Schrödinger’s model of the atom is known as the quantum mechanical model. Quantum mechanics places the electrons in orbitals, not fixed orbits (like Bohr’s model). Orbitals are regions of space. The Schrödinger model consists in solving the Schrödinger equation in the spherical coordinates. The following figure depicts the initial orbitals, which result from the solution of this equation.

The electron shells proposed by Bohr are still used, but the electrons in each shell are not all equal in energy. The shell has subshells. Hence, Quantum physics describes the states of electrons in an atom according to a four-fold scheme of quantum numbers (n, m, l and s). The quantum numbers describe the allowable states electrons may occupy in an atom. The quantum numbers have the following significances  Principal number (n) - describes the energy level.  Azimuthal number (l) - how fast the electron moves in its orbit (angular momentum); this is related to the shape of the orbital.  Magnetic (m) - its orientation in space.



Chapter 4

To fulfill the Paul exclusion principle (no two electrons can be in the same state), a fourth quantum number (s) was added. This number describes the direction of the electron spin (up or down). If n, m and l, are the same, then s is different so that the electrons have opposite spins.

Quantum theory successfully predicts the shell structure of atoms and can explain the observed features of the periodic table. The shell structure and the ionization energies of atoms can be probed by photoelectron spectroscopy. In this technique we use incident EM radiation (usually Xrays or UV rays) to knock electrons out of individual atoms, thereby leading to measures of the energy levels. The so-called Fermions have widely disparate masses and charges, which distinguish them from Bosons (field particles). There exist some fundamental particles and other composite ones, like hadrons. For instance, there exist Boson hadrons (Mesons) and Fermion hadrons (Baryons). One can classify the matter particles into three major groups:   

Leptons, (electrons, neutrinos, .. etc) Bosons, or field force carriers, (mesons, glueons, photons.. etc). Quarks, or the constituents of nucleons (protons, neutrons).

The following figure depicts the main groups of well-known particles.




Chapter 4


Chapter 4

4-29. Problems 4-1) Give a list of things that motivate the need for quantum mechanics. For each of these, write two to three sentences explaining how classical mechanics fails. 4-2) Prove that the shortest time measurable in nature is √(4G ħ/c5), where G is the gravitational constant and c is the speed of light. 4-3) For the particle in a box, a single measurement of the energy at time t=0 gives E=4Eo. Can a single measurement of the energy of the particle in the same box at time t=2 possibly give a value of E = Eo? 4-4) State if each of the following are True or False. Give reason(s). [1]1. The momentum of a photon is proportional to its wavevector [2]2. The energy of a photon is proportional to its phase velocity [3] We do not experience the wave nature of matter in everyday life because the wavelengths are too small [4] The state of a particle is determined by a unique continuous complex valued function. [5] The description of the state of a quantum mechanical particle always requires a continuous complex valued function. [6] Every Hermitian operator commutes with the Hamiltonian operator [7] The position operator for the particle in a box commutes with the Hamiltonian operator [8] For the particle in a box, a single measurement of the energy at time t=0 gives E=Eo (Eo is the ground-state energy). This means that the state of the particle is the ground state. [9] The expected value of momentum for any state for the particle in a box is zero. [10] Only classical thermodynamic systems can be irreversible; quantum systems are not. [11] Momentum of a particle in a box is conserved since the momentum operator commutes with the Hamiltonian. [12] Motion backwards in time is possible over microscopic times and distances [13] If humans were not macroscopic, they could neither observe nor study motion. [14] Because of the finite accuracy of microscopic motion, faster than-light motion is possible in the microscopic domain -297Prof. Dr. Muhammad EL-SABA

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] [ ] [ ]


Chapter 4

4-4) A potential well has a height of 0.05 eV. What should be the width of the well so that the binding energy of the electron (m* = 0.063mo) would be equal to 0.025 eV. 4-5) Calculate the de Broglie wavelength for the following particles: a) an electron with kinetic energy 1 eV. b) a photon with energy 10 eV. c) a car of 1 ton weight and moving with velocity 100 km/hr. Discuss why the wave-like nature of particles is only significant for microscopic particles. 4-6) Using the Bohr model of the hydrogen atom, a) prove that the wavelengths corresponding to various lines in the hydrogen spectrum are given by:  n 2 .n 2  n2n1 = 911  22 1 2  Å  n2  n1  where n1, n2 are integers (n2 > n1) corresponding to different levels.



Chapter 4

The series of spectrum lines in which n1=1 are called the Layman series while those in which n1=2 are called the Balmer group, etc. b) The transition λ21 for hydrogen – the shortest wavelength possible – is called the Lyman-alpha line. Verify that its wavelength is 121.6 nm, which lies in the ultraviolet. Note that this line is easily observed with telescopes, since most of the visible stars consist of excited hydrogen. c) Calculate the Bohr radius ro and energy in the ground state Eo. 4-7) If the position of a moving electron is determined with error of 1 Angstrom, calculate the minimum error to be considered in the measurement of its wave vector k. 4-8) The de-Broglie Wavelength of a particle at average kinetic energy kBT is known as the thermal de Broglie wavelength T. a) Calculate T of an electron at 300K. b) What is the number of valence electrons in a thermal de Broglie cube of silicon whose edge is thermal de Broglie wavelength? c) What is the fluctuation in electron number in such a cube? d) What is the significance of localization of an electron by a wave packet in a thermal de Broglie cube? Hint: The density of states in the valence band of Si, is Nv = 10-19 cm-3. Also, the thermal velocity of carriers in Si is vth = 107 cm/s. 4-9) Prove the Heisenberg uncertainty principle  p x .  x   / 2 . Let x = x - and px = px - . For simplicity assume zero averages. 4-10) Find the wavefunctions (eigenfunctions) and allowed energy values (eigenvalues) of a particle of mass m in one dimension is in the potential 4-11) Consider the square potential barrier shown in figure. A free electron of energy E, which is less than the barrier height Vo, is assumed to be i approaching the barrier from left and tunneling through it. 

V(x) Vo

t

E

r x o

a

The incident electron wave i and the reflected wave r are assumed to have the same wave vector k, as the transmitted wave t so that the solution of Schrödinger equation outside the barrier is given by: -299Prof. Dr. Muhammad EL-SABA


 i  A exp  j k x  r  A 1

i

r

Chapter 4

exp  j k1 x 

 t  A exp  j k x t

1

Also the solution of  inside the barrier may be written as a sum of two exponential terms

  Bi e  k 2 x  B r e k 2 x

o

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