Lecture 1 - Introduction, double slit experiments

Quantum mechanics A Objectives of the course

1 Understand the differences between quantum mechanics and classical mechanics. Examine some early experiments which led to the development of quantum mechanical theory. 2 Develop an understanding of key quantum mechanical concepts such as the wavefunction, Heisenberg’s uncertainty principle, Schrödinger’s equation etc. 3 Use the Schrödinger equation to solve simple 1D problems that can be related to physical phenomena.

Course structure 13 lectures + 1 review 2 exams Homework Mid term exam Final exam

25% 30% 45%

Lecture notes and homework assignments available from www.kiroku.riec.tohoku.ac.jp/simon/quantum Office hour:

Wednesday mornings, RIEC, room M317.

Course outline No.

Date

Contents

1

10/6

2

10/13 Black body radiation, photoelectric effect

3

10/20 Compton scattering, Franck and Hertz experiment

4

10/27 Bohr model of the hydrogen atom, de Broglie theory

5

11/17 Schrödinger equation, wavefunctions

6

11/24 Operators, eigenvalue equations, expectation values

7

12/1

The infinite potential well

8

12/8

Pauli exclusion principle, Heisenberg uncertainty principle

9

12/15 Scattering and tunnelling

10

12/22 Band gaps, angular momentum

Course outline, introduction, double slit experiment

11

1/5

One dimensional barrier problems

12

1/19

The harmonic oscillator

13

1/26

Free particles and wavepackets, review

14

1/30

Review

2/2

Final exam

Course outline (J) No. 1 2 3 4 5 6 7 8 9 10

内容

月/日 10/6 量子力学の歴史、２スリットの実験 10/13 黒体輻射、光電効果 10/20 コンプトン散乱 10/27 ボーア原子モデル、ド．ブロイ理論 11/17 シュレディンガー方程式、波動関数 11/24 演算子、固有値方程式、期待値

12/1 一次元井戸形ポテンシャル中の粒子 12/8 パウリの排他原理、固体の電子状態、ハイゼンべルグの不確定性原理 12/15 トンネル効果 12/22 バンドギャップ、角運動量 一次元段階ポテンシャル

11

1/5

12

1/19 調和振動子 1/26 自由粒子、波束運動

13 14

1/30 復習 2/2 最後試験

Text books There are many quantum mechanics textbooks. Here are some options: “Quantum mechanics” by B.H. Bransden and C.J. Joachain (Pearson educational, Edinburgh 2000). Reasonably cheap, also used in some Quantum Mechanics B courses. “Quantum mechanics” by Alistair I M Rae (IOP 2002). Compact, cheap. “Introduction to quantum mechanics” by David J Griffiths (Prentice Hall 1994). Well written, mathematical. “Introductory quantum mechanics” by R L Liboff (Addison Wesley 2003). Big, comprehensive, expensive. “量子力学基礎”、末光眞希、枝松圭一、電気-電子基礎シリーズ 15、朝倉書店。

Lecture 1 History of quantum mechanics Double slit experiments

Objectives Learn about the origins of quantum mechanics. Understand the different results of the double slit experiment for particles and waves. Note that electrons and other sub-atomic particles behave like waves in the double slit experiment.

What is quantum mechanics? In the 17th century Newton developed an explanation of how things move, this is known as classical mechanics. Newton’s laws of motion relate force, momentum and acceleration and are excellent at describing the motion of everyday objects and even apply to planets orbiting stars etc. But when we come to look at very small objects, like atoms and molecules, we find that Newton’s laws no longer apply. Quantum mechanics was developed to explain the behaviour of these very small objects, but on the larger scale its predictions are identical to those of Newton.

Classical mechanics The motion of an object is governed by Newton’s laws

M

F = ma v = u + at s = ut + ½at2

F

The state of the object in 3-dimensions can be described by six variables : = f  x , y , z , v x , v y , v z  where x, y and z are the co-ordinates of the object and vx, vy and vz are the velocities along the x, y and z axes. For a free particle we could also write E  x , y , z , v x , v y , v z = H  x , y , z , p x , p y , p z = the Hamiltonian and p is the momentum.

1  p2x  p2y p 2z  f  x , y , z  , where H is 2m

Quantum mechanics For very small particles we cannot know both x and px at the same time due to Heisenberg’s uncertainty principle.  x  p x ≥ℏ / 2 , where ℏ=h/2  and h is Planck’s constant. ħ = 1.055×10-34 Js, so the influence of Heisenberg’s uncertainty principle is negligible on macroscopic objects, but for electrons, protons and neutrons etc, the uncertainties  x and  p x become significant. Instead of x and px, for a free particle we can know px and E. So the state of quantum mechanical objects can be described by = f  p x , E  or = f  E , L 2 , L x  where L is the angular momentum. We can know one component of the angular momentum, e.g. Lx, and the total angular momentum L, but in that case Ly and Lz must be unknown, otherwise Heisenberg’s principle will be violated.

History At the beginning of the 20th century there were many experimental results which could not be explained using classical theory. e.g. Light can behave as both a wave (interference) and a particle (photoelectric effect). Electrons orbit atomic nuclei, but they never collapse into the nucleus as predicted by classical mechanics. Light emitted from atoms forms discrete line spectra. Theory based on the wave nature of light cannot account for the observed frequency distribution of radiant energy from a black body. These problems were solved by quantum mechanics.

History II Main events in the development of quantum mechanics Year 1898 1901 1905 1911 1913 1922 1924 1925 1926 1927 1927 1927 1928

Scientist Curie Planck Einstein Rutherford Bohr Compton Pauli De Broglie Schrödinger Heisenberg Davisson & Germer Born Dirac

Discovery Radioactive polonium and radium Blackbody radiation Photoelectric effect Model of the atom Quantum theory of spectra Scattering photons off electrons Exclusion principle Matter waves Wave equation Uncertainty principle Experiment on wave properties of electrons Interpretation of the wavefunction Relativistic wave equation

Double slit experiment - Balls Consider three double slit experiments. In the first experiment balls are fired at a wall in which there are two slits. The slits and the balls are collected in buckets placed behind the slits.

If we count the number of balls in each bucket we find a distribution similar to that shown in the figure. The number of balls, or “intensity”, in each bucket is simply the total of the balls arriving from slits 1 and 2. Experiment with balls

Double slit experiment - Balls We define the probability of a ball arriving at location x as P( x) . If we cover up slit 2 and measure P( x) we will obtain P1 ( x) , i.e. the probability distribution for balls which pass through slit 1. We can then close slit 1 and open slit 2 and get the distribution P 2 ( x) for balls which pass through slit 2. With both slits open we can measure P 12 (x) . In the experiment using balls we find that P12 (x)= P1 ( x)+ P2 (x) .

Double slit experiment - Waves What happens if we use a wave source instead of balls? Think of the detectors as buoys in the water that bob up and down. The intensity of waves at the detectors forms an interference pattern. At some points the amplitudes of the waves combine to produce a peak, at other points the waves cancel out. In this case, the intensity distribution is not simply the sum of the intensities obtained from individual measurements of I1 and I2. Experiment with waves

Double slit experiment – Waves II We measure the wave intensity, I(x), at each point along the screen. i (kx−ωt ) Plane waves can be written as ψ(x , t)= Acos(kx−ω t )= Ae The phase varies along the screen and the waves passing through slits 1 and 2 have the same frequency so we simplify this to ψ= Aei α , where α represents the phase. 2 * The intensity of a wave is equal to the absolute square of ψ , i.e. I ( x)=|ψ| =ψ ψ . (Intensity means energy density, or rate of energy transported per unit area). 2

For waves passing through slit 1 (slit 2 is closed) we have I 1 (x)=|ψ1| . Similarly, 2 when slit 2 is open and slit 1 is closed we have I 2 (x)=|ψ2| . When both slits are open we 2 2 * I 12 (x)=|ψ| =|ψ1 +ψ2| =( ψ1 +ψ2 )( ψ1 + ψ2 )

have ψ=ψ1 + ψ2 at

the

screen,

) =I 1 (x)+ I 2 (x)+2 √ I 1( x) I 2 (x)cos(α1−α 2) I 12 (x)=|ψ1| +|ψ2| +|ψ1 ψ2|( e +e e i e−i (because cos  = ). So I 12 (x)≠I 1 ( x)+ I 2 ( x) . 2 2

2

i (α1 −α 2 )

−i(α 1 −α2 )

i.e.

Double slit experiment – Waves III The intensity I (x) is given by the Fraunhofer diffraction conditions. 2

sin (δ/2) 2  asin  = In general I ( x)= I 0 , and , where a is the slit width and  is 2  (δ/2) the diffraction angle. sin   is proportional to y. 2

2

sin (δ/2) sin ( N δ/2) For multiple (N) slits I ( x)= I 0 (δ/2)2 sin 2 (δ /2) mλ D where D is the slit to d screen separation, d is the slit separation, m is an integer and  is the wavelength of the light used. The separation between the peaks is approximately x=

Wave diffraction - Figures

Diffraction from multiple slits is contained within the Fraunhofer diffraction envelope for a single slit.

Calculation of peak separation using the small angle approximation.

(Here the axis along the screen is shown as y instead of x)

Double slit experiment – Electrons Let’s repeat the experiment with electrons. We can detect individual electrons using a phosphor-coated screen, so they can be considered as particles.

We find an interference pattern, similar to the pattern produced by waves. The pattern is not a simple addition of the intensities from slits 1 and 2 as it was in the experiment with balls.

Experiment with electrons A similar result is obtained if a laser is used, i.e. photons also produce a wave-like interference pattern.

Double slit experiment – Electrons II We can count the number of electrons arriving at each point along the screen and produce a probability function, P(x). We can describe the state of the electrons by a wavefunction,   x ,t  . As with the wave experiment, the probability of finding an electron at a particular location along 2 the screen is given by P  x=∣  x ,t ∣ . 2

And, as in the wave experiment, P 12  x=∣ 1  x ,t  2  x , t ∣ ≠ P 1  x  P 2  x . Individually, electrons behave as particles. They can be emitted one at a time and they are detected at a single point on the screen. If only one slit is open then P(x) is similar to the experiment with balls. But with two slits open we obtain a diffraction pattern, like that found in the waves experiment. The same argument applies to photons.

Double slit experiment – Electrons III

Results of a double slit experiment with electrons

Double slit experiment – Electrons IV Perhaps, when two slits are open, the electrons somehow interact with both slits. Maybe a single electron can pass through both slits at the same time. We can check this by placing a light behind the slits. When an electron is emitted we can determine which slit it passed through, or if it passed through both slits at once. When we do this experiment we find that the electrons always pass through a single slit and never through both slits at once. We also find that the functions P1(x) and P2(x) are almost the same as when the experiment is performed with a single slit. But the function P12(x) is no longer an interference pattern, it is simply P1(x)+P2(x), the same as in the balls experiment. The act of observing which slit the electrons passed through has changed the result of the experiment.

Observing the electrons Suppose the distance between two slits along the x axis is d. If we want to know which slit an electron passed through then we must measure its position with an error of less than d/2, i.e. Δ x≪d /2 . The distance between maxima and minima at the screen is half the distance between x m λ D hD = maxima i.e. = (using λ=h / p (de Broglie wavelength)). The angular 2 2d 2 pd x difference measured from the slit is  with D tan = . In terms of momentum 2 h x D px h D p y tan = p x . This gives = = . If p y ≫ p x then p x = . 2d 2 py 2 pd If we are to keep the interference pattern then Δ p x ≪h/2 d , otherwise the peaks and troughs of the interference pattern will be indistinguishable. Putting the two inequalities together we find  x  p x ≪ h/ 4 , whereas Heisenberg’s uncertainty principle says  x  p x ≥h/ 4 .Strictly speaking, these are not incompatible; in practice observation of the electrons destroys the diffraction pattern.

The wavefunction, Ψ Suppose we add more slits to the electron experiment

∣

2

∣

We can write P(x) as P  x= ∑  n  x , t . As N→∞ we find that in order to calculate N

the probability of an electron arriving at a point x we must consider all possible paths that it can take. In classical mechanics there is only one path between two points. The wavefunction can only be determined by multiple measurements on the system. If we carry out a double slit experiment with just one electron we won’t learn anything about the interference pattern. Many measurements are necessary to determine  to any degree of accuracy

Double slit experiments

Heisenberg’s uncertainty principle

 x  p x ≥ℏ / 2  x = Uncertainty in position  p x = Uncertainty in momentum ℏ=h/ 2  and h is Planck’s constant.

ħ = 1.055×10

-34

Js

Conclusions We cannot know the position and momentum of very small particles simultaneously. The Heisenberg uncertainty principle tells us the more accurately we can determine position, the less we know about momentum and vice-versa. The double slit experiment demonstrates that electrons sometimes behave like waves and sometimes like particles. The wavefunction represents the state of a particle, but it cannot be determined by a single measurement.