Quantum Mechanics: Historical Perspective Spring 2012 Mark F. Horstemeyer Amitava Moitra ICME 4990/6990.

Quantum Mechanics: Historical Perspective

Spring 2012

Mark F. HorstemeyerAmitava Moitra

ICME 4990/6990

Reference TextsPrinciples of Quantum Mechanics by R ShankerModern Quantum Mechanics by J.J. Sakurai

Other Books:D. J. Griffiths， Introduction to Quantum Mechanics, Sara M. McMurry. Quantum mechanics. Paul Roman, Advanced Quantum Theory-An outline of the fundamental ideas, J. J. Sakurai，Modern quantum mechanics. R. Shankar. Principles of quantum mechanics. Feynman, The Feynman Lectures on Physics, Vol. 3; Quantum MechanicsCohen-Tannoudji， Quantum Mechanics, Vol. 1&2；Leonard I. Schiff. Quantum mechanics. L. D. Landau and E. M. Lifshitz, Quantum Mechanics R. Robinett’s Quantum mechanics: classical results, modern systems and visualized examplesJ. Townsend, A modern approach to quantum mechanics, D. Park, Introduction to quantum theory, P.A.M.Dirac, The Principles of Quantum Mechanics Leslie E. Ballentine， Quantum Mechanics: A Modern Developmentter Haar, Problems in Quantum Mechanics

The Beginning• What is a Physical Law?

• Based upon Bacon’s (1260) Scientific Method: observation, hypothesis, experiments

• A statement of nature that must be experimentally validated• Experiments done in different frames must yield same results• Describes the physical world

• Why should truth be a function of time?• Laws formulated with observations• Observations depends on accuracy of the instruments• Advancement of technology leads to better instrumentation• Laws that remain true gain in stature, those which don’t may lose

their impact (example of Newton’s Laws of Gravity and Relativity)• Domain of physical law and the general abstraction

Matter & Radiation

• Classical Mechanics• Formulated by Galileo, Newton, Euler, Lagrange, Hamilton• Remained unaltered for three centuries

• Some History• Beginning of 1900’s two entities---

– Matter & Radiation• Matter described by Newtons laws (mass and particles) 1687• Radiation wave equations by Maxwell’s equation (Field Theory)

1864 (light consists of transverse undulations which cause the electric and magnetic phenomenon)

• It was thought that we now understood all…• First breakthrough came with radiations emitted by a black body

The Black Body RadiationWhat was the observation

mT = Constant

Total Power radiated T4

Raleigh Jeans Law

Raleigh Jeans Law

Why Black Body?

Black Body… continued

• At a more fundamental level why should there be three laws that apparently have no relation with each other and yet describe one physical phenomenon?

• Why is it a physical phenomenon?• Planck (1899)solved the mystery by

enunciating that it emitted radiation in quantas of photons with energy h

Enter Einstein (1905) Nobel Prize Idea!

• If radiation is emitted in quantas, they should also be absorbed in quantas

• He could explain photo electric effect using this…

• Light is absorbed in quanta of h• If it is emitted in quantas of h

Must consist of quantas

)1(2

' Coscm

h

e

• What was the importance of Compton effect?

• Collision between two particles– Energy-momentum must both be conserved simultaneously

• Light consist of particles called photons

• What about phenomenon of Interference & diffraction?

• Logical tight rope of Feyman

• Light behaves sometimes as particles (Newton) sometimes as waves (Maxwell)

Enter Compton (1923)

Enter de Broglie (1923)

• Radiation behaved sometimes as particles sometimes as waves

• What about matter?

• De Broglie’s hypothesis• Several questions cropped up!

What is it? --Particle or Waves

What about earlier results?• What is a good theory?

Need not tell you whether an electron is a wave or a particleif you do an experiment it should tell

you whether it will behave as a wave or particle.

Second Question brings us to the domain of the theory

Domain of a theory

• Domain Dn of the phenomenon described by the new theory

• Subdomain D0 where the old theory is reliable.

• Within the sub domain D0 either theory may be used to make quantitative predictions

• It may be easier and faster to apply the old theory

• New theory brings in not only numerical changes but also radical conceptual changes

• These will have bearing on all of Dn

velocity

Inve

rse

size

RelativityClassicalMechanics

QuantumMechanics

QuantumFieldTheory

Experimental Observations

• Thought Experiments• Stern-Gerlach Experiments (1921);

measurements of atomic magnetic moments• Analogy with mathematics of light• Feynman’s double slit thought experiment

Thought Experiments• We are formulating a new theory!

• Why are we formulating a new theory?

• Already motivate you why we need a new theory?

• How?

• Radiation sometimes behaves as• Particles• Waves

• Same is true for Matter

Thought Experiments• We must walk on a logical tight rope

• What is Feyman’s logical tightrope?

• We have given up asking whether the electron is a particle or a wave

• What we demand from our theory is that given an experiment we must be able to tell whether it will behave as a particle or a wave.

• We need to develop a language for this new theory.

• We need to develop the Mathematics which the language of TRUTH which we all seek

• What Kind of Language we seek is the motivation for next few lectures.

Stern-Gerlach Experiment (1921)

Oven containing Ag atoms

Collimator Slits InhomogeneousMagnetic Field

detector

Classically oneWould expect this

Nature behavesthis way

Stern Gerlach Experiment unplugged• Silver atom has 47 electrons where 46 electrons

form a symmetrical electron cloud with no net angular momentum

• Neglect nuclear spin

• Atom has angular momentum –solely due to the intrinsic spin of the 47th electron

• Magnetic moment of the atom is proportional to electron spin

• If z < 0 (then Sz > 0) atom experiences an upward force & vice versa

• Beam will split according to the value of z

Scm

e

e

BEnergy

.

z

BF zz

Stern-Gerlach Experiment (contd)

• One can say it is an apparatus which measures the z component of Sz

• If atoms randomly oriented • No preferred direction for the orientation of • Classically spinning object z will take all possible values

between & -•

Experimentally we observe two distinct blobs•

Original silver beam into 2 distinct component• Experiment was designed to test quantisation of

space

• Two possible values of the Z component of S observed SZ

UP & SZdown

• Refer to them as SZ+ & SZ

- Multiples of some fundamental constants, turns out to be

• Spin is quantised• Nothing is sacred about the z direction, if our

apparatus was in x direction we would have observed Sx

+ & Sx- instead

What have we learnt from the experiment

SG Z

This box is the Stern Gerlach Apparatus with magneticField in the z direction

Thought Experiments startZ+

Z-Source SG Z

Z+

Z-Source SG Z

BlockedZ+

Source SG Z

Blocked

SG Z

SG Z

Thought Experiment continues

• No matter how many SG in z direction we put, there is only one beam coming out

• Silver atoms were oriented in all possible directions

• The Stern-Gerlach Apparatus which is a measuring device puts those atoms which were in all possible states in either one of the two states specific to the Apparatus

• Once the SG App. put it into one of the states repeated measurements did not disturb the system

Conclusions from our experiment

• Measurements disturb a quantum system in an essential way

• The boxes are nothing but measurements

• Measurements put the QM System in one of the special states

• Any further measurement of the same variable does not change the state of the system

• Measurement of another variable may disturb the system and put it in one of its special states.

Complete Departure from Classical Physics

• Measurement of Sx destroys the information about Sz

• We can never measure Sx & Sz together– Incompatible measurements

• How do you measure angular momentum of a spinning top, L = I

• Measure x , y , z

• No difficulty in specifying Lx Ly Lz

• Consider a monochromatic light wave propagating in Z direction & it is polarised in x direction

• Similarly linearly polarised light in y direction is represented by

• A filter which polarises light in the x direction is called an X filter and one which polarises light in y direction is called a y filter

• An X filter becomes a Y filter when rotated by 90

Analogy

An Experiment with Light

• The selection of x` filter destroyed the information about the previous state of polarisation of light

• Quite analogous to situation earlier• Carry the analogy further

– Sz x & y polarised light– Sx x` & y` polarised light

Source X Filter Y Filter

Source X’ Filter Y FilterX’ Filter

No LIGHT

NoLIGHT

LIGHT

PREFACE

Newton’s mechanics

Maxwell’s electrodynamics

Einstein’s relativity

Quantum mechanics

Not by one individual

Characteristic of Quantum mechanics

No general consensus

Can “do”, but can’t tell what we are doing.

Niels Bohr:: “If you are not confused by quantum physics then you haven’t really understood it”.

Richard Feynman: ”I think I can safely say that nobody understands quantum mechanics”.

rudiments of linear algebra

Knowledge requirements for students

complex numbers

calculus and partial derivatives

Fourier analysis

Elementary classical mechanics

Maxwell’s electrodynamics

mathematical physics

Dirac delta function

Math

Physics

2

2

dt

xdmaF

x),( txF

om)(tx

dt

dxv

dt

dva

)(tx)(tx

)(xV

x

VF

mvp 2

2

1mvT

(1) Classical mechanics:

maF

Initial conditions:

00 )0(,)0( xxvv

)(xV

The Schrödinger Equation (1926)

Classical mechanics Quantum mechanics

(2) Quantum mechanics

)(tx ),( tx Wave function

maF

Vxmt

i2

22

2

xm

)(xV

sJh

3410054572.12

)0,(x ),( tx)(tx)0(x)0(v

Schrödinger Equation

The Statistical Interpretation

?),( txWhat is the Wave function?

2|),(| tx

Born’s (1926) statistical interpretation:

gives the probability of finding the particle at

point x, at time t－ or more precisely,

b

adxtx 2|),(|

Probability of finding the particle between a and b, at time t.

2|),(| tx

xA B Ca b

a b

The shaded area represents the probability of finding the particle between a and b, at time t. The particle would be relatively likely to be found near A, and unlikely to be found near B.

The statistical interpretation introduces a kind of indeterminacy into quantum mechanics.

Quantum mechanics offers statistical information about the possible results.

Question? Where was the particle just before I made the measurement?

Do measure the position of the particle

Is it a fact of nature or a defect of theory??

Which is the right answer?

In 1964, John Bell

Showing that it will make an observable difference

The experiments show that the orthodox answer is right.

Immediate repetition of measurement

Zeno effect = remains on its initial state

Continuous evolution == no measurement

discontinuous collapse == measurement

Probability

Discrete Variables

Age Number

14 1

15 1

16 3

22 2

24 2

25 5

)( jNj

1)14( N

N

Number(age)

1)15( N

3)16( N

2)22( N

2)24( N

5)25( N

The total number of people: 14)(0

j

jNN

)( jN

j10 11 1412 13 15 16 17 18 19 20 21 2322 27262524

(1) The probability that the person’s age would be j ?

N

jNjP

)()( 1)(

0

j

jP

(2) What is the most probable age ?

14

5)( jPMax 25j

14

1

14

114

3

14

2

14

2

14

5

(3) What is the median age ?

(4) What is the average (or mean) age ?

2114

2552422221631514

)( jN

j10 11 1412 13 15 16 17 18 19 20 21 2322 27262524

14

1

14

114

3

14

2

14

2

14

5

14

5

14

2

14

2

14

3

14

1

14

1

Jj

J

j

jPjP )()(0

23J

23J

the average value of age j is

0

0 )(

)(

j

j jjPN

jjN

j

(5) What is the average of the squares of the ages ?

0

20

2

2 )(

)(

j

j jPjN

jNj

j

0

0 )()(

)()(

)(j

j jPjfN

jNjf

jf

Discussions: 2j2j

22 jj

8

6

4

1

)( jN

j1 2 53 4 6 7 8 9 10

)( jN

j1 2 53 4 6 7 8 9 10

8

1

8

1

4

1 4

2

The statistical quantities:

Average value j

the most probable value

Median value

How to discriminate them ?

J

the number of elements 7,6,5

jjj

)( jPjjj )()( jPjjPj

0)( jPjj

22)( jjj

222 )( jjj )(2jPjj

22 jj

22 jj 022 jj

0 22 jj

Deviation from the average

Continuous Variables

age 16 years, 4 hours, 27 minutes,…… Continuous Value

interval : [ 16, 17 ]

The probability in a sufficiently short interval is proportional to the length of the interval.

Infinitesimal intervals dx

Probability that an individual (chosen at random) lies between x

and x+dxdxx)(

The proportionality factor, , is called probability density.)(x

dxxxPdxxPxdP )()()()(

b

aab dxxP )(

The probability that x lies between a and b is given by the integral

dxx)(1

dxxxx )(

dxxxfxf )()()(

2222 )( xxx

Normalization

2|),(| tx is the probability density for finding the particle at point

x, at time t.

1|),(| 2

dxtx

Vxmt

i2

22

2

),( tx ),( txA

A complex constant

We can choose complex constant A to meet above equation

Normalization

)(x

Proof:

The Schrödinger Equation automatically preserves the normalization of the wave function

dxtxt

dxtxdt

d 22 |),(||),(|

ttttx

t

***2 )(|),(|

Vi

xm

i

t

2

2

2*

2

*2*

2

Vi

xm

i

t

xxm

i

xxxm

i

t

**

2

*2

2

2*2

22||

xxm

i

xtx

t

**2

2,

dxtx

dt

d 2|),(|

xxm

i **

2

0),(lim

txx

0|),(| 2

dxtx

dt

d

Normalizable

Momentum

For a particle in state Ψ, the expectation value of x is

dxtxxx

2),(

The expectation value is the average of repeated measurements on an ensemble of identically prepared systems.

dxt

xdxtxxdt

d

dt

xd 22),(

dxxxx

xm

i **

2

0

dt

dx

dxxxx

xm

i **

2

Using integration-by-parts method

xx

xdxxxm

i **

**

2

dx

xdx

x

*2*

dxxm

i *or

dx

xm

i *

*d x idx

dt m x

d x

vdt

*( )d x

p m i dxdt x

*x x dx

*p i dxx

momentum

operator

position

All classical dynamical variables can be expressed in terms of position and momentum.

For example:

Kinetic energy:2

21

2 2

pT mv

m

Angular momentum: L r mv r p

Classical dynamical variables Quantum operators

( , )Q x p ( , )Q x ix

*( , ) ,Q x p Q x i dxx

2 2*

22T dx

m x

The Uncertainty Principle (Heisenberg 1926)

wavelengthposition

The wavelength of Ψ is related to the momentum of the particle by the de-Broglie formula:

2hp

2x p

The Uncertainty Principle

wavelengthposition

nonlocal

local nonlocal

local