Elements of Modern Physics
70
3.3 POSTULATES OF QUANTUM MECHANICS
The discussion so far has been for the purpose of introducing the ideas of wave
function and wave equation, and making them plausible. These ideas are now
formalized and generalized in terms of some postulates of quantum mechanics.
Postulate 1: Every state of n particles is described by a wave function
ψ (r i , t), i = 1, ..., n where r i are the coordinates of the n particles, such that the
probability at time t, of finding the particles in respective volumes d
3
r i about the
points r i , is
dP = |ψ (r i , t)|
2
d
3
r 1 ...d
3
r n
(3.16)
It is relevant to comment here that:
1. Since the particles must be found somewhere, the total probability must
be 1,
∫dP = ∫ |ψ (r i , t)|
2
d
3
r 1 ...d
3
r n
= 1
(3.17)
This defines the normalization of the wave function. It is also clear from
the definition of probability that the average value of a function f (r i ) is
〈 f (r i ) 〉 = ∫ |ψ (r i , t)|
2
f(r i ) d
3
r i ... d
3
r n
(3.18)
2. For a single particle system, the results simplify to
dP = |ψ (r 1 , t)|
2
d
3
r 1
1 = ∫ |ψ (r 1 , t)|
2
d
3
r 1
(3.19)
〈 f (r 1 ) 〉 = ∫ | ψ (r 1 , t)|
2
f(r 1 ) d
3
r 1
(3.20)
In particular, the average position is given by
〈 r 1 〉 = ∫ |ψ (r 1 , t)|
2
r 1 d
3
r 1
(3.21)
Postulate 2: The wave function ψ (r i , t) satisfies the Schrödinger equation
( , )
i
r t
i
t
∂ψ
∂
= ( ,
) ( , )
− ∇ ψ
i
i
i
H
i
t
r
r
(3.22)
where H is the Hamiltonian or the energy operator obtained from the classical
expression for the total energy by replacing p i byi
i ∇
. For a single article in
the presence of an interaction potential, the total energy is E =
2
1
( )
2
p V r
m
+
which leads to the equation
( , )
∂ψ
∂
t
i
t
r
=
2
2
( , )
( ) ( , )
2
−
∇ ψ
+
ψ
t V r
t
m
r
r
(3.23)