Download 5.0. Wave Mechanics

5.0. Wave Mechanics
One cornerstone of quantum theory is the particle-wave duality [the other is the
principle of uncertainty]. For example, in optical phenomena such as diffraction and
interference, light behaves like waves. In collision processes such as photo-electric
and Compton effects, light behaves like particles. Another example is the electron,
which behaves like a particle when moving freely, but acts like standing waves when
bounded in an atom. These 2 aspects are connected by the relations
(5.1)
E  h  
p
h

 k
(5.2)
where E and p are the energy and momentum of the particle, respectively;  and  are
the frequency and wavelength of the wave, respectively;  and k are the angular
frequency and wavenumber of the wave, respectively;
h

h
and
2
6.6256  1034 J s
is called the Planck’s constant.
Historically, eq(5.1) was 1st proposed by Planck as an empirical fix for the breakdown
of the theory of classical statistical mechanics when applied to the problem of black
body radiation. It was later applied by Einstein to the photo-electric effect, thus
revealing the particle-like aspect of “waves”. Later on, de Broglie proposed the
general validity of eqs(5.1-2). The implied wave-like aspects of “particles” were 1st
demonstrated by Thomson and Davisson using the diffraction of electrons by a crystal
lattice.
The particle-wave duality is best described by the wave mechanical formulism of
quantum theory invented by Schrodinger. Thus, the state of a “particle” is
represented by a (complex) wave function   x,t  so that the probability of finding
the particle in an infinitesimal volume d 3 x about x at time t is proportional to
P  x, t  d 3 x    x, t 
2
d 3x
(5.4)
where P is called the probability density if it satisfies the normalization condition
 P  x, t  d
3
x     x, t 
2
d 3x  1
(5.4a)
For wave functions that cannot be normalized, (5.4) indicates the relative probabilities.
One example of this is a free particle described by the plane wave
  x, t   exp i  k  x   t  
(5.3)
A quantity that is represented as a function A  A  x, p in classical mechanics is
represented in wave mechanics as a differential operator Aˆ  A  xˆ, pˆ  .
In the
so-called r-representation, we have
xˆ  x
and
pˆ 
i

(5.6)
The dynamics of the particle is described by the time-dependent Schrodinger
equation
i

  x, t   Hˆ   x, t 
t
(5.7a)
where Ĥ is the Hamiltonian operator.
For a particle moving in a potential
V  V  x  , we have
1 2
p  V x
2m
so that (5.7a) becomes
H

Hˆ  
 2 2


i
  x, t    
  V  x     x, t 
t
 2m

2
2m
2  V  x 
(5.7)
Note that although some generalizations of (5.7), e.g., to the case of a system of many
particles, are straightforward, some generalizations are not readily tractable, e.g.,
writing (5.7) in terms of generalized coordinates. Finally, eq(5.7a) can be interpreted
as an additional rule supplementing those in (5.6):
Ĥ  i

t
(5.7)
This will be useful when relativistic generalization of the theory is considered.