Page 221 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

212 Systems of identical particles

has coordinates q 1 and which q 2 . Thus, only the linear combinations Ø S and
Ø A are suitable wave functions for the two-identical-particle system. We note
in passing that the two probability densities are not equal, even though Ø S and
Ø A correspond to the same energy value E. We conclude that in order to
incorporate into quantum theory the indistinguishability of the two identical
particles, we must restrict the allowable wave functions to those that are
symmetric and antisymmetric, i.e., to those that are simultaneous eigenfunc-
^
^
tions of H(1, 2) and P.

Three-particle systems
The treatment of a three-particle system introduces a new feature not present in
a two-particle system. Whereas there are only two possible permutations and
therefore only one exchange or permutation operator for two particles, the
three-particle system requires several permutation operators.
We ®rst label the particle with coordinates q 1 as particle 1, the one with
coordinates q 2 as particle 2, and the one with coordinates q 3 as particle 3. The
^
Hamiltonian operator H(1, 2, 3) is dependent on the positions, momentum
operators, and perhaps spin coordinates of each of the three particles. For
identical particles, this operator must be symmetric with respect to particle
interchange
^
^
^
^
^
^
H(1, 2, 3) ˆ H(1, 3, 2) ˆ H(2, 3, 1) ˆ H(2, 1, 3) ˆ H(3, 1, 2) ˆ H(3, 2, 1)
È
If Ø(1, 2, 3) is a solution of the time-independent Schrodinger equation
^
H(1, 2, 3)Ø(1, 2, 3) ˆ EØ(1, 2, 3) (8:16)
then Ø(1, 3, 2), Ø(2, 3, 1), etc., and any linear combinations of these wave
functions are also solutions with the same eigenvalue E. The notation
Ø(i, j, k) indicates that particle i has coordinates q 1 , particle j has coordinates
q 2 , and particle k has coordinates q 3 . As in the two-particle case, we seek
^
eigenfunctions of H(1, 2, 3) that do not specify which particle has coordinates
q i , i ˆ 1, 2, 3.
^
We de®ne the six permutation operators P áâã for á 6ˆ â 6ˆ 㠈 1, 2, 3 by the
relations
^ 9
P 123 Ø(i, j, k) ˆ Ø(i, j, k) >
>
^ >
>
P 132 Ø(i, j, k) ˆ Ø(i, k, j) >
>
^ =
P 231 Ø(i, j, k) ˆ Ø(j, k, i)
^ i 6ˆ j 6ˆ k ˆ 1, 2, 3 (8:17)
P 213 Ø(i, j, k) ˆ Ø(j, i, k) >
>
^ >
>
P 312 Ø(i, j, k) ˆ Ø(k, i, j) >
>
^ ;
P 321 Ø(i, j, k) ˆ Ø(k, j, i)
^
The operator P áâã replaces the particle with coordinates q 1 (the ®rst position)