Page 229 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics
P. 229
220 Systems of identical particles
Application of equations (8.39) and (8.40) to (8.41) gives
" #
X
F(Q) (N!) 1=2 f (Q9) Ø (Q9)Ø í (Q) dQ9 (8:42)
í
í
Since f (Q9) is a completely arbitrary function of Q9, we may compare
equations (8.34) and (8.42) and obtain
X ÿ1 X
^
Ø (Q9)Ø í (Q) (N!) ä P ä(PQ ÿ Q9) (8:43)
í
í P
where ä(Q ÿ Q9) is the Dirac delta function
N
Y
ä(Q ÿ Q9) ä(r i ÿ r9 i )ä ó i ó 9 i (8:44)
i1
Equation (8.43) is the completeness relation for a complete set of symmetric
(antisymmetric) multi-particle wave functions.
8.4 Non-interacting particles
In this section we consider a many-particle system in which the particles act
independently of each other. For such a system of N identical particles, the
^
Hamiltonian operator H(1, 2, ... , N) may be written as the sum of one-
^
particle Hamiltonian operators H(i) for i 1, 2, ... , N
^
^
^
^
H(1, 2, ... , N) H(1) H(2) H(N) (8:45)
^
In this case, the operator H(1, 2, ... , N) is obviously symmetric with respect
^
to particle interchanges. For the N particles to be identical, the operators H(i)
must all have the same form, the same set of orthonormal eigenfunctions ø n (i),
and the same set of eigenvalues E n , where
^
H(i)ø n (i) E n ø n (i); i 1, 2, ... , N (8:46)
As a consequence of equation (8.45), the eigenfunctions Ø í (1, 2, ... , N)of
^
H(1, 2, ... , N) are products of the one-particle eigenfunctions
Ø í (1, 2, ... , N) ø a (1)ø b (2) ... ø p (N) (8:47)
^
and the eigenvalues E í of H(1, 2, ... , N) are sums of one-particle energies
E í E a E b E p (8:48)
In equations (8.47) and (8.48), the index í represents the set of one-particle
states a, b, ..., p and indicates the state of the N-particle system.
The N-particle eigenfunctions Ø í (1, 2, ... , N) in equation (8.47) are not
properly symmetrized. For bosons, the wave function Ø í (1, 2, ... , N) must be
symmetric with respect to particle interchange and for fermions it must be
antisymmetric. Properly symmetrized wave functions may be readily con-
