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Textbook
Claude Cohen-Tannoudji
(born 1933)
Bernard Diu
(born 1935)
Franck Laloë
(born 1940)
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
Isaac Newton
(1642 – 1727)
Thomas Young
(1773 – 1829)
1.A.1
Kindergarten stuff
• Is light a wave or a flux of particles?
1.A.1
Kindergarten stuff
• Is light a wave or a flux of particles?
1.A.1
Kindergarten stuff
• Is light a wave or a flux of particles?
1.A.1
Kindergarten stuff
• Is light a wave or a flux of particles?
• 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:
E  hv  


p  k
• h – Planck’s constant, 6.626 × 10-34 J∙s
Max Karl Ernst Ludwig Planck
1858 – 1947
Albert Einstein
1879 – 1955
1.A.2
Wave-particle duality
1.A.2
Wave-particle duality
1.A.2
Wave-particle duality
1.A.2
Wave-particle duality
1.A.2
Wave-particle duality
1.A.2
Wave-particle duality
1.A.2
Wave-particle duality
1.A.2
Wave-particle duality
• 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
Wave-particle duality
• 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
P( x)  I ( x)  E ( x)
E( x)  E1 ( x)  E2 ( x)
I ( x)  I1 ( x)  I 2 ( x)
2
1.A.2
Wave-particle duality
• 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:

 2
P(r , t )  E (r , t )
James Clerk Maxwell
1831-1879
1.A.3
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
Étienne-Louis Malus
1775 – 1812
1.A.3
Principle of spectral decomposition
• 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
 

E (r , t )  E0e p exp[ i(kz  t )]
 

E ' (r , t )  E0 ' ey exp[ i(kz  t )]
1.A.3
Principle of spectral decomposition
• 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


e p  ex


e p  ey
• 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



e p  ex sin   e y cos 
1.A.3
Principle of spectral decomposition
• 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
sin 2   cos 2   1
• Measurement disturbs the photons in a fundamental
fashion



e p  ex sin   e y cos 
1.B.1
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
h
h
dB  
p mv
• The frequency of matter waves is
E
f 
h
Louis de Broglie
1892 – 1987
1.B.1
Wave properties of particles
• 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
h
h
dB  
p mv
• The frequency of matter waves is
E
f 
h
Louis de Broglie
1892 – 1987
1.B.1
Wave properties of particles
• 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
Clinton Joseph Davisson (1881 – 1958) and Lester Halbert Germer (1896 – 1971)
1.B.2
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;

 2 3
C – normalization constant):
dP(r , t )  C  (r , t ) d r
1.B.2
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:


 (r , t0 )   a (r )
• The probability of measuring an eigenvalue a at t0
can be found by performing spectral decomposition:


 (r , t0 )   ci i (r )
i
Pa 
ca
2
c
i
i
2
1.B.2
The wave function
• Important relationships:

 dP(r , t )  1

 2 3
dP(r , t )  C  (r , t ) d r
 2 3
1
  (r , t ) d r  C
1.B.2
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




2
2
2
2


 (r , t )
    (r , t )   (r , t )   (r , t ) 


i



 V (r , t ) (r , t )
2
2
2


t
2m  x
y
z

• V – potential energy (“potential”)
• Superposition principle applies
Erwin Rudolf Josef Alexander
Schrödinger
1892 – 1987
1.C.1
Schrödinger equation
• For a free particle:




2
2
2
2

 (r , t )
    (r , t )   (r , t )   (r , t ) 
2

  
i



 (r , t )
2
2
2
t
2m  x
y
z
2m




 2
2
 (r , t )  A
• Solution:  (r , t )  A exp[ i(k  r  t )]
2
2
p
h
h
E
k
E

 
f 

2m
p mv
h
2m
• Using the superposition principle:

 

1
3
 (r , t ) 
g
(
k
)
exp[
i
(
k

r


t
)]
d
k
3/ 2 
2 
• For 1D
 ( x, t ) 

1
g (k ) exp[ i (kx  t )]dk

2 
1/ 2

1.C.1
Wave packet
1
2
• Wave packet:  ( x, t ) 
• At t = 0:
 ( x,0) 
1
2

i ( kx t )
g
(
k
)
e
dk



ikx
g
(
k
)
e
dk


1
• Using Fourier transformation: g (k ) 
2
• If:
g (k )   (k  k0 )
 ( x, t )  Aei ( kx t )

ikx

(
x
,
0
)
e
dx


1.C.2
Wave packet
• Wave packet:
Re ( x)
k 
k 



i
k

x
i
k




g (k0 ) ik0 x 1  0 2  1  0 2  x 
 ( x,0) 
 e
e  e

2
2
2 

g (k0 ) ik0 x 
 k 

e 1  cos
x 
2
 2 

 k 
xd   0
• Interference is destructive when 1  cos
 2

k
 k 
kx  4
xd   x  2 xd
cos
xd   1
2
 2

1.C.2
Wave packet
• More waves in a packet:
1.C.2
Wave packet
• More waves in a packet:
kx  1
1.C.3
Wave packet
• More waves in a packet:
 ( x, t ) 
1
2

p  k
i ( kx t )
g
(
k
)
e
dk



ipx
1

 ( x,0) 

(
p
)
e
dp

2 

• From Fourier calculus:
  ( x,0)

2

dx    ( p) dp  C
2

• Probability of finding a particle between x and x+dx :
1
2
dP( x)   ( x,0) dx
C
• Probability of measuring a momentum between p
and p+dp:
1
2
 dP ( p) 
C
 ( p) dp
1.C.3
Heisenberg uncertainty principle
kx  1
p  k
px  
• 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
Werner Karl Heisenberg
1901 – 1976
1.C.4
Evolution of a free packet
i ( kx t )

(
x
,
t
)

Ae
• For a single wave:
2

k


• Phase velocity: v ( k ) 
k
2m
k
v (k ) 
2m
• Three-wave packet:


k  
  
k  
  

i
k

x



t
i
k

x










0
0
0
0



g (k0 ) i k0 x 0t  1  2   2   1  2   2 t  
e

 ( x, t ) 
 e
 e
2
2
2 

g (k0 ) i k0 x 0t  
 
 k

e
1  cos 2 x  2 t 
2



• Maximum occurs when:
 
 k
cos
xM 
t 1
2 
 2
k

xM 
t 0
2
2

xM 
t
k
1.C.4
Evolution of a free packet
• For multiple waves in a packet :
 ( x, t ) 
• Group velocity:
k 2

2m
1
2

i ( kx t )
g
(
k
)
e
dk


 d 
vG (k0 )   
 dk  k  k0
k
 2v (k0 )
vG (k0 ) 
m
1.D.1
Particle in a time-independent scalar
potential
• Schrödinger equation:




 (r , t )
2
i

 (r , t )  V (r ) (r , t )
t
2m
• Separating variables:


 (r , t )   (r )  (t )
 d (t )

 
2
i (r )

 (t ) (r )  V (r ) (r )  (t )
dt
2m


i d (t )
2

  (r )  V (r )  
 (t ) dt
2m (r )
1.D.1
Particle in a time-independent scalar
potential
2


i d (t )


  (r )  V (r )  
 (t ) dt
2m (r )
d (t )
i
  (t )
dt
 (t )  e it

 

2

 (r )  V (r ) (r )   (r )
2m
 it


 (r , t )   (r )  (t )   (r )e
• This is called a stationary solution for a stationary
eigenstate
E  
 2
 2
 (r , t )   (r )
1.D.1
Particle in a time-independent scalar
potential
2

 



 (r )  V (r ) (r )  E (r )
2m
• Introducing linear differential operator

2
H 
  V (r )
2m


H (r )  E (r )
• This is an eigenvalue equation for H


Hn (r )  Enn (r )

 iEnt / 
 n ( r , t )   n ( r )e
• Linear superposition of solutions is a solution

 iEnt / 
 (r , t )   cnn (r )e
n


 (r ,0)   cn n (r )
n
1.D.2
1D square potentials
 2 d 2

 V  ( x)  E ( x)

2
 2m dx

 d 2 2m

 2  2 E  V  ( x)  0

 dx
