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 quantummechanical theory.

2 Develop an understanding of key quantum mechanical concepts such as thewavefunction, Heisenberg’s uncertainty principle, Schrödinger’s equation etc.

3 Use the Schrödinger equation to solve simple 1D problems that can be related tophysical phenomena.

Course structure

13 lectures + 1 review2 exams

Homework 25%Mid term exam 30%Final exam 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 Contents1 10/6 Course outline, introduction, double slit experiment2 10/13 Black body radiation, photoelectric effect3 10/20 Compton scattering, Franck and Hertz experiment4 10/27 Bohr model of the hydrogen atom, de Broglie theory5 11/17 Schrödinger equation, wavefunctions6 11/24 Operators, eigenvalue equations, expectation values7 12/1 The infinite potential well8 12/8 Pauli exclusion principle, Heisenberg uncertainty principle9 12/15 Scattering and tunnelling

10 12/22 Band gaps, angular momentum11 1/5 One dimensional barrier problems12 1/19 The harmonic oscillator13 1/26 Free particles and wavepackets14 1/30 Review

2/2 Final exam

Course outline (J)

No. 月/日 内容

1 10/6 量子力学の歴史、２スリットの実験

2 10/13 黒体輻射、光電効果

3 10/20 コンプトン散乱

4 10/27 ボーア原子モデル、ド．ブロイ理論

5 11/17 シュレディンガー方程式、波動関数

6 11/24 演算子、固有値方程式、期待値

7 12/1 一次元井戸形ポテンシャル中の粒子

8 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

9 12/15 トンネル効果

10 12/22 バンドギャップ、角運動量

11 1/5 一次元段階ポテンシャル

12 1/19 調和振動子

13 1/26 自由粒子、波束運動

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 Bcourses.

“Quantum mechanics” by Alistair I M Rae (IOP 2002). Compact, cheap.

“Introduction to quantum mechanics” by David J Griffiths (Prentice Hall 1994). Wellwritten, 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 andwaves.

Note that electrons and other sub-atomic particles behave like waves in the doubleslit experiment.

What is quantum mechanics?

In the 17th century Newton developed an explanation of how things move, this isknown as classical mechanics.

Newton’s laws of motion relate force, momentum and acceleration and are excellentat describing the motion of everyday objects and even apply to planets orbiting starsetc.

But when we come to look at very small objects, like atoms and molecules, we findthat Newton’s laws no longer apply.

Quantum mechanics was developed to explain the behaviour of these very smallobjects, 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

F = mav = u + at

s = ut + ½at2

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 velocitiesalong 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 , px , p y , pz =1

2m px

2 p y2 p z

2 f x , y , z , where H is

the Hamiltonian and p is the momentum.

FM

Quantum mechanics

For very small particles we cannot know both x and px at the same time due toHeisenberg’s uncertainty principle.

x px≥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 negligibleon macroscopic objects, but for electrons, protons and neutrons etc, theuncertainties x and px become significant.

Instead of x and px, for a free particle we can know px and E. So the state of quantummechanical objects can be described by

= f p x , E or = f E , L2 , Lx

where L is the angular momentum. We can know one component of the angularmomentum, 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 whichcould not be explained using classical theory.

e.g.

Light can behave as both a wave (interference) and a particle (photoelectriceffect).

Electrons orbit atomic nuclei, but they never collapse into the nucleus aspredicted by classical mechanics. Light emitted from atoms forms discrete linespectra.

Theory based on the wave nature of light cannot account for the observedfrequency 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 Scientist Discovery1898 Curie Radioactive polonium and radium1901 Planck Blackbody radiation1905 Einstein Photoelectric effect1911 Rutherford Model of the atom1913 Bohr Quantum theory of spectra1922 Compton Scattering photons off electrons1924 Pauli Exclusion principle1925 De Broglie Matter waves1926 Schrödinger Wave equation1927 Heisenberg Uncertainty principle1927 Davisson & Germer Experiment on wave properties of electrons1927 Born Interpretation of the wavefunction1928 Dirac Relativistic wave equation

Double slit experiment - Balls

Consider three double slit experiments. In the first experiment balls are fired at a wallin which there are two slits. The slits and the balls are collected in buckets placedbehind the slits.

Experiment with balls

If we count the number of balls in eachbucket we find a distribution similar tothat shown in the figure.

The number of balls, or “intensity”, in eachbucket is simply the total of the ballsarriving from slits 1 and 2.

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 probabilitydistribution for balls which pass through slit 1. We can then close slit 1 and open slit2 and get the distribution P2(x) for balls which pass through slit 2.

With both slits open we can measure P12(x) . In the experiment using balls we findthat 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 asbuoys in the water that bob up and down.

Experiment with waves

The intensity of waves at the detectorsforms an interference pattern. At somepoints the amplitudes of the wavescombine to produce a peak, at otherpoints the waves cancel out.

In this case, the intensity distribution isnot simply the sum of the intensitiesobtained from individual measurementsof I1 and I2.

Double slit experiment – Waves II

We measure the wave intensity, I(x), at each point along the screen.Plane waves can be written as ψ(x , t)=Acos(kx−ω t )=Aei (kx−ω t )

The phase varies along the screen and the waves passing through slits 1 and 2 havethe same frequency so we simplify this to ψ=Aeiα , where α represents the phase.

The intensity of a wave is equal to the absolute square of ψ , i.e. I (x)=|ψ|2=ψψ* .(Intensity means energy density, or rate of energy transported per unit area).

For waves passing through slit 1 (slit 2 is closed) we have I 1(x)=|ψ1|2 . Similarly,

when slit 2 is open and slit 1 is closed we have I 2(x)=|ψ2|2 .

When both slits are open we have ψ=ψ1+ψ2 at the screen, i.e.I 12(x)=|ψ|

2=|ψ1+ψ2|2=(ψ1+ψ2)(ψ1+ψ2)

*

I 12(x)=|ψ1|2+|ψ2|

2+|ψ1ψ2|(ei (α1−α2)+e−i(α1−α2))=I 1(x)+ I 2(x)+2√ I 1(x) I 2(x)cos(α1−α2)

(because cos = eie−i

2). So I 12(x)≠I 1(x)+ I2(x) .

Double slit experiment – Waves III

The intensity I (x) is given by the Fraunhofer diffraction conditions.

In general I (x)= I 0

sin2(δ/2)(δ/2)2

, and =2 asin , where a is the slit width and is

the diffraction angle. sin is proportional to y.

For multiple (N) slits I (x)= I 0

sin2(δ/2)(δ/2)2

sin2(N δ/2)sin2(δ /2)

The separation between the peaks is approximately x=mλ Dd

where D is the slit to

screen separation, d is the slit separation, m is an integer and is the wavelength ofthe light used.

Wave diffraction - Figures

Diffraction from multiple slits is containedwithin the Fraunhofer diffraction envelopefor a single slit.

Calculation of peak separation usingthe 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 aphosphor-coated screen, so they can be considered as particles.

Experiment with electrons

We find an interference pattern, similar tothe pattern produced by waves.

The pattern is not a simple addition of theintensities from slits 1 and 2 as it was inthe experiment with balls.

A similar result is obtained if a laser is used, i.e. photons also produce a wave-likeinterference pattern.

Double slit experiment – Electrons II

We can count the number of electrons arriving at each point along the screen andproduce a probability function, P(x).

We can describe the state of the electrons by a wavefunction, x ,t . As with thewave experiment, the probability of finding an electron at a particular location alongthe screen is given by P x=∣ x ,t ∣2 .

And, as in the wave experiment, P12 x=∣ 1 x ,t 2 x , t ∣2≠P1 x P 2 x .

Individually, electrons behave as particles. They can be emitted one at a time andthey are detected at a single point on the screen. If only one slit is open then P(x) issimilar to the experiment with balls. But with two slits open we obtain a diffractionpattern, 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 wecan 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 singleslit and never through both slits at once. We also find that the functions P1(x) andP2(x) are almost the same as when the experiment is performed with a single slit. Butthe function P12(x) is no longer an interference pattern, it is simply P1(x)+P2(x), thesame as in the balls experiment.

The act of observing which slit the electrons passed through has changed the resultof the experiment.

Observing the electrons

Suppose the distance between two slits along the x axis is d. If we want to knowwhich slit an electron passed through then we must measure its position with anerror 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

maxima i.e.x2=mλ D

2d= hD

2 p d(using λ=h / p (de Broglie wavelength)). The angular

difference measured from the slit is with D tan = x2

. In terms of momentum

p y tan= px . This givesx2=

D px

p y

= h D2 p d

. If p y≫ px then px=h2d

.

If we are to keep the interference pattern then Δ px≪h/2d , otherwise the peaks andtroughs of the interference pattern will be indistinguishable.

Putting the two inequalities together we find x px≪h/4 , whereas Heisenberg’suncertainty principle says x px≥h /4 .Strictly speaking, these are notincompatible; in practice observation of the electrons destroys the diffraction pattern.

The wavefunction, Ψ

Suppose we add more slits to the electron experiment

We can write P(x) as P x=∣∑N

n x , t ∣2

. As N→∞ we find that in order to calculate

the probability of an electron arriving at a point x we must consider all possible pathsthat 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 anythingabout 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 determineposition, the less we know about momentum and vice-versa.

The double slit experiment demonstrates that electrons sometimes behave likewaves and sometimes like particles.

The wavefunction represents the state of a particle, but it cannot be determined by asingle measurement.