textbook claude cohen-tannoudji (born 1933) franck laloë (born 1940) bernard diu (born 1935)

Textbook

Claude Cohen-Tannoudji (born 1933)

Franck Laloë (born 1940)

Bernard Diu(born 1935)

Chapter 1

Introduction

1

Quantum physics

• Problems at the end of XIX century that classical physics couldn’t explain:

• Blackbody radiation – electromagnetic radiation emitted by a heated object

• Photoelectric effect – emission of electrons by an illuminated metal

• Spectral lines – emission of sharp spectral lines by gas atoms in an electric discharge tube

1

Quantum physics

• Phenomena occurring on atomic and subatomic scales cannot be explained outside the framework of quantum physics

• There are many phenomena revealing quantum behavior on a macroscopic scale, e.g. enables one to understand the very existence of a solid body and parameters associated with it (density, elasticity, etc.)

• However, as of today, there is no satisfactory theory unifying quantum physics and relativistic mechanics

• In this course we will discuss non-relativistic quantum mechanics

1.A.1

Kindergarten stuff

• Is light a wave or a flux of particles?

• Newton vs. Young

Thomas Young (1773 – 1829)

Isaac Newton (1642 – 1727)

Kindergarten stuff


1.A.1

Kindergarten stuff


• However:

• 1) Blackbody radiation

• 2) Photoelectric effect

• 3) Spectral lines

• 4) Etc.

1.A.1

Wave-particle duality

• EM waves appear to consist of particles – photons

• Particle and wave parameters are linked by fundamental relationships:

• h – Planck’s constant, 6.626 × 10-34 J∙s

1.A.1

Albert Einstein1879 – 1955

Max Karl Ernst Ludwig Planck1858 – 1947

hvE kp

Wave-particle duality1.A.2


• The results of this experiment lead to a paradox:

• Since the interference pattern disappears when one of the slits is covered, why then this phenomena changes so drastically?

• Crucial: the process of measurement

• When one performs a measurement on a microscopic system, one disturbs it in a fundamental fashion

• It is impossible to observe the interference pattern and to know at the same time through which slit each photon has passed

1.A.2


• Light behaves simultaneously as a wave and a flux of particles

• The wave enables calculation of particle-related probabilities; e. g., when the photon is emitted, the probability of its striking the screen is proportional to light intensity, which in turn is proportional to the square of the field amplitude

1.A.2

2)()()( xExIxP

)()()( 21 xExExE

)()()( 21 xIxIxI


• Predictions of the behavior of a photon can be only probabilistic: information about the photon at time t is given by the electric field, which is a solution of the Maxwell’s equations – the field is interpreted as a probability amplitude of a photon appearing at time t at a certain location:

1.A.2

2),(),( trEtrP

James Clerk Maxwell1831-1879

Principle of spectral decomposition

• Malus’ Law: the intensity of the polarized beam transmitted through the second polarizing sheet (the analyzer) varies as I = Io cos2 θ, where Io is the intensity of the polarized wave incident on the analyzer

1.A.3

Étienne-Louis Malus1775 – 1812


• What will happen, when intensity is low enough for the photons to reach the analyzer one by one?

• NB: the detector does not register “a fraction of a photon”)

• We cannot predict which photon can pass the analyzer

1.A.3

)](exp[),( 0 tkzieEtrE p

)](exp['),(' 0 tkzieEtrE y


• The analyzer and detector can give only certain specific results – eigen (proper) results: either a photon passes the analyzer or not

• To each of the eigen results there is an eigenstate

• When the state before measurement is arbitrary, only the probabilities of obtaining the different eigen results can be predicted

• To find these probabilities, the state has to be decomposed into a linear combination of eigenstates

1.A.3

xp ee

yp ee

cossin yxp eee


• The probability of an eigen result is proportional to the square of the absolute value of the coefficient of the corresponding eigenstate

• The sum of all the probabilities should be equal to 1

• Measurement disturbs the photons in a fundamental fashion

1.A.3

cossin yxp eee

1cossin 22

Louis de Broglie 1892 – 1987

Wave properties of particles

• In 1924, Louis de Broglie postulated that because photons have wave and particle characteristics, perhaps all forms of matter have both properties

• Furthermore, the frequency and wavelength of matter waves can be determined

• The de Broglie wavelength of a particle is

• The frequency of matter waves is

1.B.1

mv

h

p

hdB

h

Ef

Louis de Broglie 1892 – 1987


• The de Broglie equations show the dual nature of matter

• Each contains matter concepts (energy and momentum) and wave concepts (wavelength and frequency)

• The de Broglie wavelength of a particle is

• The frequency of matter waves is

1.B.1

mv

h

p

hdB

h

Ef

Clinton Joseph Davisson (1881 – 1958) and Lester Halbert Germer (1896 – 1971)


• Davisson and Germer scattered low-energy electrons from a nickel target and followed this with extensive diffraction measurements from various materials

• The wavelength of the electrons calculated from the diffraction data agreed with the expected de Broglie wavelength

1.B.1

The wave function

• In quantum mechanics the object is described by a state (not trajectory)

• The state is characterized by a wave function, ψ, which depends on the particle’s position and the time

• The wave function is interpreted as a probability amplitude of quantum object’s presence (recall electric field as a probability amplitude of photon’s presence)

• The probability density (probability of finding the object at time t inside an elementary volume dxdydz; C – normalization constant):

1.B.2

rdtrCtrdP 32),(),(

The wave function

• The principle of spectral decomposition applies:

• The outcome of a measurement at t0 must belong to a set of eigen results {a}

• An eigenstate (eigenfunction) ψa(r) is associated with each eigenvalue a, if the measurement yields a:

• The probability of measuring an eigenvalue a at t0 can be found by performing spectral decomposition:

1.B.2

)(),( 0 rtr a

i

ii rctr )(),( 0

ii

aa

c

cP 2

2

The wave function

• Important relationships:

1.B.2

1),( trdP rdtrCtrdP 32

),(),(

Crdtr

1),( 32

Schrödinger equation

• In 1926 Schrödinger proposed an equation for the wave function describing the manner in which matter waves change in space and time

• Schrödinger equation is a key element in quantum mechanics

• V – potential energy (“potential”)

• Superposition principle applies

1.B.2

Erwin Rudolf Josef Alexander Schrödinger1892 – 1987

),(),(),(),(),(

2

),(2

2

2

2

2

22

trtrVz

tr

y

tr

x

tr

mt

tri

Schrödinger equation

• For a free particle:

• Solution:

• Using the superposition principle:

• For 1D

1.C.1

),(2

),(),(),(

2

),( 2

2

2

2

2

2

22

trmz

tr

y

tr

x

tr

mt

tri

)](exp[),( trkiAtr

m

k

2

2

h

Ef

mv

h

p

h

m

pE

2

2

22),( Atr

kdtrkikgtr 32/3 )](exp[)(

2

1),(

dktkxikgtx )](exp[)(2

1),( 2/1

Wave packet

• Wave packet:

• At t = 0:

• Using Fourier transformation:

• If:

1.C.1

dkekgtx tkxi )()(2

1),(

dkekgx ikx)(2

1)0,(

dxexkg ikx)0,(2

1)(

)()( 0kkkg )(),( tkxiAetx

Wave packet

• Wave packet:

• Interference is destructive when

1.C.2

xk

kixk

kixik eee

kgx 220

000

2

1

2

1

2

)()0,(

xk

ekg xik

2cos1

2

)(00

)(Re x

02

cos1

dxk

12

cos

dxk

dx

k

2dxx 2 4 xk

Wave packet

• More waves in a packet:

1.C.2

Wave packet


1.C.2

1 xk

Wave packet


• From Fourier calculus:

• Probability of finding a particle between x and x+dx :

• Probability of measuring a momentum between p and p+dp:

1.C.3

dkekgtx tkxi )()(2

1),(

kp

dpepxipx

)(

2

1)0,(

Cdppdxx

22)()0,(

dxxC

xdP2)0,(

1)(

dppC

pPd2)(

1)(

Heisenberg uncertainty principle

• In 1927 Heisenberg introduced the uncertainty principle: If a measurement of position of a particle is made with precision Δx and a simultaneous measurement of linear momentum is made with precision Δpx, then the product of the two uncertainties can never be smaller than h/2

1.C.3

1 xk kp xp

Werner Karl Heisenberg1901 – 1976

Evolution of a free packet

• For a single wave:

• Phase velocity:

• Three-wave packet:

• Maximum occurs when:

1.C.4

)(),( tkxiAetx

kkv

)(

m

k

2

2

m

kkv

2)(

txk

kitxk

kitxki eee

kgtx 22220

000000

2

1

2

1

2

)(),(

txk

ekg txki

22cos1

2

)(000

122

cos

txk

M

0

22

tx

kM

tk

xM

Evolution of a free packet

• For multiple waves in a packet :

• Group velocity:

1.C.4

0

)( 0kk

G dk

dkv

m

k

2

2

m

kkvG

)( 0

dkekgtx tkxi )()(2

1),(

)(2 0kv

Particle in a time-independent scalar potential

• Schrödinger equation:

• Separating variables:

1.D.1

),()(),(2

),( 2

trrVtrmt

tri

)()(),( trtr

)()()()()(2

)()(

2

trrVrtmdt

tdri

)()()(2

)(

)(

2

rVrrmdt

td

t

i


• This is called a stationary solution for a stationary eigenstate

1.D.1

)()()(2

)(

)(

2

rVrrmdt

td

t

i

)()(

tdt

tdi

)()()()(2

2

rrrVrm

tiet )(

)()(),( trtr tier )(

22)(),( rtr

E


• Introducing linear differential operator

• This is an eigenvalue equation for H

• Linear superposition of solutions is a solution

1.D.1

)()()()(2

2

rErrVrm

)(2

2

rVm

H

)()( rErH

)()( rErH nnn

/)(),( tiEnn

nertr

n

tiEnn

nerctr /)(),( n

nn rcr )()0,(

1D square potentials1.D.2

)()(2 2

22

xExVdx

d

m

0)(222

2

xVE

m

dx

d

textbook claude cohen-tannoudji (born 1933) franck laloë (born 1940) bernard diu (born 1935)

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