Page 225 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

216 Systems of identical particles

system. We begin by labeling the N particles, with each particle i having
coordinates q i . For identical particles, the Hamiltonian operator must be
symmetric with respect to particle permutations
^ ^ ^
H(1, 2, ... , N) ˆ H(2, 1, ... , N) ˆ H(N,2, ... ,1) ˆ
There are N! possible permutations of the N particles. If Ø(1, 2, ... , N)is a
È
solution of the time-independent Schrodinger equation
^
H(1, 2, ... , N)Ø(1, 2, ... , N) ˆ EØ(1, 2, ... , N) (8:28)
then Ø(2, 1, ... , N), Ø(N,2, ... , 1), etc., and any linear combination of
these wave functions are also solutions with eigenvalue E.
^
We next introduce the set of linear hermitian exchange operators P áâ
^
(á 6ˆ ⠈ 1, 2, ... , N). The exchange operator P áâ interchanges the pair of
particles in positions á (with coordinates q á ) and â (with coordinates q â )
^
P áâ Ø(i, ... , j, ... , k, ... , l) ˆ Ø(i, ... , k, ... , j, ... , l) (8:29)
á â á â
^
As in the three-particle case, the order of the subscripts on P áâ is immaterial.
Since there are N choices for the ®rst particle and (N ÿ 1) choices for the
second particle (á 6ˆ â) and since each pair is to be counted only once
^
^
^
(P á⠈ P âá ), there are N(N ÿ 1)=2 members of the set P áâ .
^
Applying the same arguments regarding the exchange operator P for the
two-particle system, we ®nd that P 2 á⠈ 1, giving real eigenvalues 1. We also
^
^
®nd that P áâ and H commute
^
^
[P áâ , H] ˆ 0, á 6ˆ ⠈ 1, 2, ... , N (8:30)
so that they possess simultaneous eigenfunctions. However, the members of the
^
set P áâ do not commute with each other. There are only two functions, Ø S and
^
Ø A , which are simultaneous eigenfunctions of H and all of the pairwise
^
exchange operators P áâ . These two functions have the property
^ á 6ˆ ⠈ 1, 2, ... , N (8:31a)
P áâ Ø S ˆ Ø S ,
^ á 6ˆ ⠈ 1, 2, ... , N (8:31b)
P áâ Ø A ˆÿØ A ,
and may be constructed from Ø(1, 2, ... , N) by the relation
X
^
Ø S,A ˆ (N!) ÿ1=2 ä P PØ(1, 2, ... , N) (8:32)
P
^
In equation (8.32) the operator P is any one of the N! operators, including the
identity operator, that permute a given order of particles to another order. The
summation is taken over all N! permutation operators. The quantity ä P is
always ‡1 for the symmetric wave function Ø S , but for the antisymmetric
^
wave function Ø A , ä P is ‡1(ÿ1) if the permutation operator P involves the