Page 220 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

8.1 Permutations of identical particles 211
^ (8:11a)
PØ S ˆ Ø S
^
PØ A ˆÿØ A (8:11b)
The factor 2 ÿ1=2 in equations (8.10) normalizes Ø S and Ø A if Ø(1, 2) is
normalized. The combination Ø S is symmetric with respect to particle
interchange because it remains unchanged when the two particles are ex-
changed. The function Ø A , on the other hand, is antisymmetric with respect to
particle interchange because it changes sign, but is otherwise unchanged, when
the particles are exchanged.
The functions Ø A and Ø S are orthogonal. To demonstrate this property, we
note that the integral over all space of a function of two or more variables must
be independent of the labeling of those variables

f (x 1 , ... , x N )dx 1 ... dx N ˆ f (y 1 , ... , y N )dy 1 ... dy N (8:12)
In particular, we have

f (1, 2) dq 1 dq 2 ˆ f (2, 1) dq 1 dq 2
or
hØ(1, 2)jØ(2, 1)iˆhØ(2, 1)jØ(1, 2)i (8:13)

where f (1, 2) ˆ Ø (1, 2)Ø(2, 1). Application of equation (8.13) to hØ S jØ A i
gives
^ ^
hØ S jØ A iˆhPØ S jPØ A i (8:14)
Applying equations (8.11) to the right-hand side of (8.14), we obtain

hØ S jØ A iˆÿhØ S jØ A i
Thus, the scalar product hØ S jØ A i must vanish, showing that Ø A and Ø S are
orthogonal.
If the wave function for the system is initially symmetric (antisymmetric),
then it remains symmetric (antisymmetric) as time progresses. This property
È
follows from the time-dependent Schrodinger equation
@Ø(1, 2)
^
i" ˆ H(1, 2)Ø(1, 2) (8:15)
@t
^
Since H(1, 2) is symmetric, the time derivative @Ø=@t has the same symmetry
as Ø. During a small time interval Ät, therefore, the symmetry of Ø does not
change. By repetition of this argument, the symmetry remains the same over a
succession of small time intervals, and by extension over all time.
Since Ø S does not change and only the sign of Ø A changes if particles 1 and

2 are interchanged, the respective probability densities Ø Ø S and Ø Ø A are
S
A
independent of how the particles are labeled. Neither speci®es which particle