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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 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||
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 *
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