Introductory Quantum Mechanics

Introductory Quantum Mechanics “Quantum mechanics is God's way of saying 'trust me'” Historical The full story of the early development of quantum mechanics is fascinating and full of opinionated characters, triumphs, tragedy and irony. There was much back-and-forth discussion and several avenues of thought as to how to develop the new physics. It was by no means a straightforward progression to modern quantum physics. In fact, the arguments about the meaning of the equations involved is still ongoing. We begin with J.J. Thompson. He, of course, discovered the electron in 1897 at Cambridge University. This was quite an disturbing thing to find out since it was thought at the time that atoms were indivisible (as indeed, the very word 'atom' means) and yet here was a small particle coming out of an atom. Evidently, atoms are divisible. Since the electron was determined to have a negative charge and since atoms are neutral it was thought that the electrons were embedded in a sphere of positive charge to make up the atom. This was the short-lived 'plum pudding' model of the atom. Ironically, J.J. Thompson's son George Thompson later showed that the electron can behave as a wave rather than a particle. We return to the electron shortly. At the end of the nineteenth century, physics had achieved astounding success with the application of Newton's laws to the movement of bodies in space and the elucidation of the characteristics of electromagnetic radiation via Maxwell's equations. In the sciences and most especially, in physics we like to model physical phenomena mathematically as it allows us to predict what will happen in the future. For two centuries Newton's laws were the fundamental starting point for modelling all sorts of physical phenomena and were very successful and are still fully adequate for macroscopic phenomena. It was then, perhaps natural for physicists to think that there wasn't much of a fundamental nature left to discover but rather small things that would fit into the big picture as described by the physics of the time. There were of course some minor problems but they would ultimately be solved and all would be well. One of these was the problem of blackbody radiation. All bodies in the universe radiate electromagnetic energy of a spectrum of frequencies. Measurement of this spectrum of frequencies

shows two main things. First, for very low and very high frequencies the intensity of the radiation is low and at intermediate frequencies there is a frequency at which the intensity is at a maximum, nm. Second, the maximum nm changes as the temperature changes. When T is increased nm increases. We are all familiar with this phenomenon in connection with electric heaters that glow as they warm up. A plot of typical measurements of this spectrum looks like:

T =4000K

I

T =3000K T =2000K  Figure 13-1: Blackbody radiation spectrum Attempts to model this resulted in the Rayleigh-Jeans law which worked for large values of wavelength but increasingly deviated from the observed spectrum as wavelength was decreased:

I

 Figure 13-2: Rayleigh Jeans law (dashed line) and experimental data (solid lines) As the Rayleigh-Jeans law was based on the physics of the day this was very much a puzzle to be solved. The idealized absorber/emitter of radiation is the blackbody and has spectral intensity plots as shown in the previous figures. It was thought at the time of Max Planck, in 1900, that the radiation of electromagnetic energy was due to the oscillation of charges in the surface of the blackbody and that there was no restriction on the energies of the oscillations. The Rayleigh-Jeans law was built on this assumption. Max Planck however, took the very bold step of saying “what if the energies are not continuous but rather restricted to certain values?”. Using this idea he put together a new equation to predict radiation intensities as a function of wavelength and lo and behold it worked. Using Planck's new equation it was possible to replicate exactly the observed spectra of figure 7-1. Central to his derivation was the idea that the energy of the oscillators is proportional to their frequency:

E=h ν

[13-1]

where h is the constant of proportionality now known as Planck's constant. Planck wrote that he thought of this as merely a 'trick' in order to derive a workable equation and did not put much further thought into it. In 1905 Albert Einstein wrote a paper discussing another curious phenomenon observed earlier by Phillip Lenard in 1902, the photoelectric effect. When a metallic surface under vacuum is exposed to electromagnetic radiation above a certain threshold frequency, the

light is absorbed and electrons are emitted. The salient observations are: 1. For a given metal and frequency of incident radiation, the rate at which photoelectrons are ejected is directly proportional to the intensity of the incident light. 2. For a given metal, there exists a certain minimum frequency of incident radiation below which no photoelectrons can be emitted. This frequency is called the threshold frequency. 3. Above the threshold frequency, the maximum kinetic energy of the emitted electron is independent of the intensity of the incident light but depends on the frequency of the incident light. These were very curious observations with respect to the understanding of electromagnetic waves (light) at the time. Maxwell's equations indicate that the intensity of the electromagnetic wave is directly related to the energy of the electromagnetic wave. Thus it was reasonable to expect that increasing the intensity of the light would eventually result in the removal of electrons from the metal regardless of the frequency. Observation 2 counters this idea. There is a minimum frequency required for ejection of electrons. If the frequency of light is below this threshold frequency it does not matter what the intensity is, no electrons will be emitted. Also flying in the face of conventional understanding was observation 3. Again, one would expect that the higher the intensity of the incident radiation the larger the kinetic energy of the emitted electrons would be. Not so. Instead, increasing the intensity resulted in an increase in the number of emitted electrons not their kinetic energy. In order to explain this Einstein postulated that the incident light was not doing so in a wave fashion but as particles which we now call photons. Furthermore, he took Planck's idea of the oscillators with energies proportional to their frequencies and applied it to these particles. Thus each photon is deemed to have an energy of E=h ν and the intensity of the incident light on the metal is related to the number of photons, not the energy of the light. Each photon then interacts with the electrons in the metal and, if the photon has enough energy, possibly ejects an electron from the metal. If the frequency of the photon is below the threshold frequency this is equivalent to saying that the photon does not carry enough energy to eject the electron. The basic idea is that a photon with energy hn interacts with an electron and transfers its energy to the electron. If the energy is sufficient the electron will be removed from the surface of the metal and have a kinetic energy which

is directly related to the energy of the photon:

h ν=E k +W Ek =h ν−W where W is the work function or the energy required to remove an electron from the metal atom in question. The particle idea combined with Planck's equation explains the photoelectric effect exceptionally well whereas the old electromagnetic wave ideas of Maxwell do not. Equation [13-1], known as Planck's equation, leads us to a new way of thinking about things. We can, when we hear the word 'frequency', think 'energy' and vice versa. Further evidence for the particle nature of light came in 1923 from Arthur Compton. His observation was that when x-rays are beamed at electrons they interact in such a way as to impart momentum to the electrons and scatter the x-rays. Furthermore, the scattered x-rays are at longer wavelengths or lower frequencies. The wave picture of electromagnetic radiation cannot explain this but the particle picture of light does so very nicely. An x-ray photon has momentum some of which it can impart to the electron. In doing so it loses energy and will, according to Planck's equation, decrease in frequency.

In 1908, Ernest Rutherford's lab staff, recent PhD Hans Gieger and undergraduate student Ernest Marsden performed an experiment in which alpha rays were directed at a very thin sheet of gold foil. The expected result based on Thompson's plum pudding model of the atom was that most of the alpha rays would pass directly through the foil with some minimal scattering a few degrees off the path of the beam. To their surprise a very small amount of alpha rays were deflected almost backwards towards the source. Since alpha rays are positively charged particles this means that they encountered a small region of concentrated positive charge in the gold atoms in the foil. In other words the nucleus! This lead to the Rutherford model of the atom with negatively charged electrons orbiting a positively charged nucleus and is the visual picture still used in many situations today (unfortunately) such as company logos1. Another perplexing problem at the time was that of the hydrogen emission spectrum. When subjecting an evacuated tube containing a bit of hydrogen to a high voltage light is emitted. When this light is 1 They probably think that it is a 'modern' looking thing to display a Rutherford atom in their logo .. it isn't.

passed through a spectrometer prism it is split into discreet lines. In the visible region of the light spectrum these lines are known as the Balmer series:

656.5nm 486.3nm 434.2nm Figure 13-3: The Balmer series of the hydrogen emission spectrum.

Balmer was able to empirically fit the positions of these lines into an equation:

1 1 1 ν= =109680 2 − 2 λ 2 n

(

)

There are of course other lines in the hydrogen spectrum in the ultra violet and infrared regions. The Swiss spectroscopist Johannes Rydberg produced a general equation for all lines in the hydrogen spectrum:

1 1 1 ν= λ =109680 2 − 2 n 1 n2

(

)

This is a beautifully simple equation but it doesn't really tell us anything about why we can do this calculation. Enter Neils Bohr. He had been working in the Rutherford laboratory at about the same time as the scattering experiment and was very familiar with the Rutherford model of the nuclear atom. He was puzzling over the hydrogen atom emission spectrum in 1913 and came up with the idea that the orbits of the electrons might be fixed. The essence of his idea is: 1. The electrons can only travel in certain orbits at a certain discrete set of distances from the nucleus with

specific energies. 2. The electrons do not continuously radiate energy as they orbit the nucleus. Normally, an accelerating charged particle radiates electromagnetic energy. They can only gain and lose energy by switching from one allowed orbit to another, absorbing or emitting light with a frequency ν determined by the energy difference of the levels according to the Planck equation. This was a very crude model of the atom and left many questions to be answered. However it was successful in explaining and reproducing the Rydberg equation for the emission spectrum of hydrogen and perhaps more importantly pointed the way for those who followed Bohr. n=3 h

n=1

Figure 13-4: The Bohr model of the atom.

Prince Louis de Broglie of the French nobility and a trained physicist had the very brilliant idea in 1924 that if light could exhibit the characteristics of waves under some circumstances and particles under others then might not the same be possible for

particles such as electrons. In a brilliant synthesis of Planck's equation and Einstein's matter/energy equation from special relativity he derived what the wavelength of the particle must be. Starting with Einstein's (now) famous equation:

E=mc2 E =mc c and Planck's equation:

E=h ν= E h = c λ

hc λ

and:

h =mc λ h λ= mc This equation applies to light but might it not carry over into the world of matter? If so then we simply write:

λ=

h mv

[13-2]

This was, of course, a very big leap and was not shown to be true conclusively for some years (by George Thompson in Britain and Davidson and Germer in the USA). Nonetheless it paved the way for further developments, specifically, Erwin Schroedinger's quantum mechanics. But first .. In early 1926, while thinking about the Bohr atomic model and transitions of electrons between states, Werner Heisenberg came up with a method of doing quantum calculations based on matrix methods. One of the most unusual things to come out of this was the fact that the multiplication of matrices is generally non-commutative (as we saw in chapter 2). Thus, with p being the matrix representation of momentum and q the matrix form of position coordinates: p q−q p≠0

and was to prove the genesis of the Heisenberg uncertainty principle.

Also in 1926, Erwin Schroedinger produced a method of doing quantum calculations based on differential calculus founded on the de Broglie idea of constrained 'matter waves'. His quantum mechanics is the more intuitive of the two main methods (if that can be said of either), the other of course being Heisenberg's matrix mechanics. Erwin Schrodinger viewed his functions (see below) as referring to 'matter waves' however, an alternate interpretation suggested by Max Born was that they represented the probability of finding the electron in a region of space. Born's view has prevailed. The development of quantum mechanics took place during the first thirty years of the 20th century with the findings of those up to Heisenberg's time referred to as 'old quantum mechanics' and those of Heisenberg and afterwards 'new quantum' mechanics. In the old quantum mechanics physicists were trying to hold on to the classical pictures of microscopic matter developed over the previous centuries. The orbiting electron of Rutherford and Bohr is an example. With Heisenberg and those afterwards these pictures were abandoned in favour of a more abstract vision founded in the mathematical description of microscopic events. There was, of course, much discussion and argument about what the new equations mean. Einstein famously quipped “God does not play dice” in connection with the probabilistic interpretation of the wave equation. The birth of modern quantum mechanics can be dated from the 5th Solvay conference in Brussels in 1927 where most of the major figures in European physics gathered to hammer out the details of the new physics (which are still being argued about).

Figure 13-4:Attendees at the fifth Solvay Conference, 1927. Back: A. Piccard, E. Henriot, P. Ehrenfest, Ed. Herzen, Th. De Donder, E. Schroedinger, J.E. Verschaffelt, W. Pauli, W. Heisenberg, R.H. Fowler, L Brillouin Middle:P. Debye, M. Knudsen, W.L. Bragg, H.A. Kramers, P.A.M. Dirac, A.H. Compton, L. de Broglie, M. Born, N. Bohr Front:I. Langmuir, M. Planck, M. Curie, H.A. Lorentz, A. Einstein, P. Langevin, Ch. E. Guye, C.T.R. Wilson, O.W. Richardson

We have not discussed the property of matter called 'spin' which was discovered in conjunction with the development of quantum mechanics. This we will leave until a later chapter as it is to us as nmr spectroscopists, a rather special property. The Basics of Quantum Mechanics This was always a very confusing topic in school ... and probably rightly so. For the beginner, there is so much in the way of unfamiliar mathematics that is difficult to relate to 'reality' that it sometimes seems hopeless to even begin to comprehend the topic. You will not end up understanding quantum mechanics but you will be in good company such as the physicist Richard Feynman who said “nobody understands it”. Nonetheless, to the reader we say “stay the

course” and keep reading. Hopefully we shall be able to dispel some or even all of your misgivings. We usually begin talking about modern quantum mechanics by referring to the stationary Schroedinger equation:

Ĥ Ψ=E Ψ

[13-3]

which is not derived but rather postulated (see below). It is 'stationary' in the same sense as the 'stationary' part of the classical wave equation of chapter 7. H is the Hamiltonian operator, E is the total energy (a scalar quantity) and Y is the 'wavefunction' that is postulated to contain all of the knowable information about the system of interest. It is therefore, as you can see, an eigenvalue equation. We cannot derive it but we can rationalize2 it by thinking about the three-dimensional wave equation starting with a 3D version of a stationary wave equation [7-7]: ∇ 2 Ψ=−α2 Ψ where: 2 2 2 ∇ 2= ∂ 2 + ∂ 2 + ∂ 2 ∂x ∂ y ∂z and 2π α= λ

Now, using deBroglie's postulate concerning the wavelength of a particle: h h λ= = p mv 2π p 2 π mv α= = h h and our wave equation now becomes: ∇ 2 Ψ=

−4 π2 m2 v 2 Ψ h2

Using the Hamiltonian expression from classical mechanics (see chapter 8):

2 Following J. Baggot's exposition. See reference 8.

E=T+U=

p2 1 +U= mv 2+U 2m 2

we rearrange to get: mv 2=2(E−U)

and then insert into our current wave equation: ∇ 2 Ψ=

−8 π 2 m ( E−U ) Ψ h2

−h2 ( 2 ∇ 2+U ) Ψ=E Ψ 8π m or H Ψ=E Ψ where −h2 H= 2 ∇ 2+U 8π m to get the Schroedinger wave equation. 'H' is an operator that operates on Y. One will frequently see a modification of this: ( ℏ=

h 2π

−ℏ2 2 ∇ +U ) Ψ=E Ψ 2m (reduced Planck constant )

This version is helpful for seeing that the first term is a kinetic energy term: 2

2

−ℏ ∇ 2= −ℏ ( ∂2 + ∂2 + ∂ 2 ) 2m 2m ∂ x 2 ∂ y 2 ∂ z 2 where: −

p2x ℏ 2 ∂2 → 2m ∂ x 2 2m

[13-4]

corresponds to the kinetic energy in the x-coordinate. Accordingly, when we construct a quantum mechanics Hamiltonian for a particular problem we generally replace the classical px with i ℏ ∂ . Why the i, you ∂x ask? To generate the minus sign in equation [13-4]. We will use this expression for the momentum operator when we (finally) begin to talk about nmr spectroscopy.

We proceed now with the postulates of quantum mechanics upon which everything is based. Postulates are either 'self-evident' in the same sense that it is self-evident that heat flows from hot objects to cold objects, or they are taken on trust and must survive experimental test if they are to be kept. So we can only state them here, not prove them since they are 'unprovable' except by repeated experimental verification. We can however try to show why they were postulated in the first place. The usual introductory exposition of the postulates uses the wavefunctions of Erwin Schroedinger from above however, for the purposes of this document, it is useful to cast them in the more general form of Dirac. That is, we shall consider our wavefunctions as state vectors (which they are by definition, see chapter 3) and use the algebra of vectors and Dirac's notation to develop our equations. This has the advantage that, even though the concept of the wavefunction and working with it is somewhat foreign to the unitiated, working with vectors is usually not. One can always refer to the familiar concepts of Euclidean vector algebra and apply them to state vectors; the fundamental algebra is identical. Thus, we can rewrite the Schroedinger equation ([13-3]) in its Dirac form:

Ĥ | Ψ >=E | Ψ > Postulate I: Any state of a system at a given time is fully described by a state vector, |Y(>, in a Hilbert space. |Y(> is the 'ket' form of the vector. The corresponding 'bra' form is =(< Ψ | Ψ >)* (see equation [3-30b]) and the scalar is therefore real. Since we are dealing with a vector we must be able to express it as a linear combination of any other vectors in the Hilbert space. In particular we can represent the vector as a combination of orthonormal basis vectors:

| Ψ >=∑ c i | ϕi >=∑ | ϕi > i

i

where =ci

See equation [3-23]. This is the well-known superposition principle of quantum mechanics and since (obviously): | Ψ >=| Ψ >

we can say that:

∑ | ϕi >< ϕi |=1 i

[13-5]

which is known as the closure relation. This is very helpful in developing further equations. Postulate II: To every physical observable there corresponds a linear, Hermitian operator, A, acting on |Y(>. In our discussion of vectors in chapter 3 we saw that operators can be represented by matrices. This is also so here and we will develop this in our consideration of nmr spectroscopy. Postulate III: The only possible values that can result from measurements of a physical observable A are the eigenvalues ai of the equation:

̂ |ϕi >=ai |ϕi > A where A is the operator corresponding to the observable and |fi> is the state eigenvector. The operator A is, from postulate II, a Hermitian operator which means the the eigenvalues, ai, are real, as we saw must be the case in chapter 3. Postulate IV: If |Y>> is the normalized state vector of a system, then the average value (or expectation value) of a physical observable, A is:

̂ |Ψ> < A >=< Ψ | A We can rationalize this using the Born postulate: the square of the modulus of | Ψ > , ∣| Ψ >∣2 , is the probability (probability density actually) of observing a particle in a small volume of a Hilbert space. This originally was proposed by Max Born (Olivia Newton-John's

grandfather!) to explain the diffraction of electrons via Schroedinger's wave-function formulation of quantum mechanics. Of course, the probability of finding the particle anywhere from −∞ to +∞ must be 1 so the normalized wave function is written:

∫∣| Ψ >∣2 d τ=∫ < Ψ | Ψ >d τ=1 The Born postulate is useful for explaining where the average value postulate comes from. We use ∣| Ψ >∣2 in order to guarantee a positive number, which of course we must have if it is to be interpreted as a probability. We expand | Ψ > in its (orthonormal) basis set: | Ψ >=∑ c i ϕi i * i

< Ψ | Ψ >=1=∑ c i

j

= ∑ ∑ c c j i

* i

j

= ∑ ∑ c *i c j δij i

j

2

= ∑ ∣c i∣ =1 i

where δij is the Kronecker delta from chapter 3. We see that the sum of the squares of the moduli of the basis vector coefficients is one. Similarly:

̂ |Ψ> < A >=< Ψ | A * ̂ ∑ c j |ϕ j > = ∑ ci , over a number of measurements is:

< A >=∑ Pi ai i

Where Pi is the probability of obtaining measurement ai . Also, recall that:

∑ Pi=1 i

Evidently, the square of the moduli of the basis coefficients correspond to probabilities. Postulate V: the time dependence of the state vector is expressed in the time-dependent Schroedinger equation: ̂ | Ψ (t)>=i ℏ ∂ | Ψ(t )> H ∂t or ∂ | Ψ(t) >= −i H ̂ | Ψ (t)> ℏ ∂t This is of the same form as the time-dependent wave function, equation [8-7]. We will use this later in our discussion of the time evolution of a set of spins. Vector Notation Recalling the vector information from above, we note that both our wave functions and vectors can be referred to as being orthogonal (or can be made so). This leads to the idea that we can think of and represent our wave functions as vectors via Dirac notation. Indeed, much of the literature refers to wave functions as vectors. So, for a superposition of basis states:

| Ψ >=∑ c i | ϕi > i

we recall from chapter 3 that we can represent this ket vectorially as a column matrix:

[]

c1 c | Ψ >= 2 c3 ⋮

and the corresponding bra is represented as a row matrix:

< Ψ |=( c*1 c *2 c *3 ⋯)

and we can calculate our inner product by matrix multiplication:

[]

c1 c * * * * < Ψ | Ψ >=[ c 1 c 2 c 3 ⋯] 2 =∑ c i ci =1 c3 i ⋮ From chapter 3, we can represent our operator with a matrix: ̂ | ϕ j >= Â ij < ϕi | A

[

Â11 Â12 Â21 Â22 ̂ A= ⋮ ⋮ ̂ A n1 Ân2

]

⋯ ⋯ ⋱ ⋯

[13-6]

Projection operators (also discussed in chapter 3) are constructed from the outer products of members of the basis vector set:

P̂ i=| ϕi >=∑ x k pk k=1 t

< x 2 >=∑ x 2k p k k

σ=Δ x =√ x−< x >

where pk is the probability of measurement x k . Let's consider something similar for a pair of non-commuting Hermitian operators, A and B. First we define the standard deviation for each operator in a fashion analogous to that of equation [5-5]:

̂ A ̂ −< A > Δ A= ̂ B−< ̂ Δ B= B> Since A is Hermitian and < A > is a real scalar, we can say that Δ A is also Hermitian. Assume that the action of these operators on the state vector ket, | Ψ > , is:

Δ Â |Ψ >=| Φ> Δ B̂ |Ψ >=| Θ> Similarly, the bra operations will be: ̂ < Ψ |Δ A=< Ψ |(Δ Â T )*=< Φ | ̂ < Ψ |Δ B=< Ψ |( Δ B̂ T )*==< Ψ |Δ Â Δ A ̂ | Ψ >=< Φ |Φ > =< Ψ |(Δ A) 2 2 ̂ | Ψ >=< Ψ | Δ B̂ Δ B| ̂ Ψ >=< Θ|Θ > =< Ψ |(Δ B)

and with reference to chapter 3, we invoke the Cauchy-Schwartz inequality (equation [3-10]): ̂ ΔB ̂ | Ψ >∣2 < Ψ |( Δ Â 2) | Ψ >< Ψ |(Δ B̂ 2) | Ψ > ⩾ ∣< Ψ |Δ A or 2 ∣∣∣∣ ⩾ ∣∣

Now, since the left side of the equation is real and positive (equation [3-30]): ⩾ 0 and < Θ|Θ > ⩾ 0 the right side must also be real and positive: |< Φ|Θ >|2 ⩾ 0

We cannot say the same about alone .. it will generally be a complex number. However, since and are conjugates of each other, we can write:

|< Φ |Θ >|2 = which guarantees that the product of the two inner products, and , is real and we can then extend the inequality a bit: =∣∣∣< Θ|Θ >∣ ⩾ ∣∣2 = < Θ|Φ > or 2 2 ⩾ < Φ |Θ > Now from our definitions of F and Q we write: ̂ ΔB ̂ | Ψ >= =< Ψ | Δ A ̂ −< A >)( B−< ̂ =< Ψ |( A B >)|Ψ > ̂ B−< ̂ ̂ A< ̂ B >+< A >| Ψ > =< Ψ | A A > B− ̂ B ̂ |Ψ >−< Ψ | < A > B| ̂ Ψ >−< Ψ | A ̂ | Ψ >+< Ψ |< A >| Ψ > =< Ψ | A ̂ B ̂ |Ψ >−< A > < Ψ | B ̂ | Ψ >−< Ψ | A ̂ | Ψ >+< A >< Ψ || Ψ > =< Ψ | A = < AB >−< A >−< A >+< A > =< AB >−< A > and similarly: ==−< A >

When looking at all of these symbols we can sometimes be overwhelmed and forget that something such as is simply a number, in our case a complex number, and must follow all of the algebraic rules for complex numbers. For clarity, we will temporarily indicate this by substituting z for and z* for where the asterisk means, of course, the complex conjugate. Thus we can write: z z *=∣z∣2 from equation [1-4a]. From equation [1-4b] we have: z−z *=2i⋅Im ( z ) or * z−z Im (z )= 2i

and we can rewrite our inequality: ⩾ Re( z)2 +Im( z )2

This being the case, since the left hand part of the inequality is greater than or equal to the right hand part, we can write that the left hand side is greater than or equal to the real or imaginary

parts alone. We choose the imaginary part because we need to develop an expression with a commutator in it. Choosing the real part would give us an anticommutator. ⩾ Im (z)2 2 z−z * 2 2 ⩾ 2i < Φ |Θ>−< Θ| Φ> 2 2 ⩾ 2i

(

)

(

2

)

Now, using our calculations from above we write: < AB >−< A >−< BA >+< A > 2i 2 < AB>−< BA > ⩾ 2i < AB−BA > 2 2 2 ⩾ 2i 2 2 2 ⩾ 2i or ⩾ 2i

⩾

(

(

2

)

)

(

)

(

[13-8]

)

This final result (after much work!) is the generalized version of Heisenberg's famous uncertainty principle. For the operators x and p the commutator is :

[ x̂ , ̂p ] | Ψ >=( x̂ ̂p − ̂p ̂x )| Ψ >=( x)(i ℏ ∂ )| Ψ >−(i ℏ ∂ )(x )| Ψ >

∂x ∂x ∂| Ψ > =i ℏ x −( ∂ )( x )| Ψ > ∂x ∂x ∂|Ψ> ∂| Ψ > =i ℏ x −| Ψ >−x (using the product rule) ∂x ∂x =i ℏ | Ψ >

(

(

)

)

and:

[ ̂x , p̂ ] =i ℏ From equation [13-8]:

Δ x Δ p⩾

[ x̂ , ̂p ] 2i

or Δ x Δ p⩾

ℏ 2

which is the common expression of the Heisenberg uncertainty principle with respect to position and momentum. The commutator is useful for telling us whether or not two operators have common eigenvectors (eq. 3-38). This being the case the commutator of the operators will be zero and from [13-8] there can potentially be no uncertainty in measurements corresponding to these operators: ⩾ 0

If the commutator is non-zero, as in the case of operators without simultaneous eigenvectors, then equation [13-8] applies. Problems

References 1. S. Jones, The Quantum Ten, Thomas Allen Publishers, 2008. 2. E.G. Steward and S.M. McMurry, Quantum Mechanics: Its Early Development and the Road to Entanglement, Imperial College Press, 2008. 3. D.A. McQuarrie, Quantum Chemistry, University Science Books, Sausalito, 2008. 4. I.N. Levine,Quantum Chemistry, Allyn and Bacon Inc., 1974. 5. R. Shankar, Principles of Quantum Mechanics, Plenum Press, 1994. 6. P.A.M. Dirac, Quantum Mechanics, Oxford University Press, 1947. 7. R.I.G. Hughes, The Structure and Interpretation of Quantum Mechanics, Harvard University Press, 1989.

8. J. Baggot, The Meaning of Quantum Theory, Oxford University Press (1992).