Page 222 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

8.1 Permutations of identical particles 213

by the particle with coordinates q á , the particle with coordinates q 2 (the second
position) by that with q â , and the particle with coordinates q 3 (the third
position) by that with q ã . For example, we have
^ (8:18a)
P 213 Ø(1, 2, 3) ˆ Ø(2, 1, 3)
^ (8:18b)
P 213 Ø(2, 1, 3) ˆ Ø(1, 2, 3)
^ (8:18c)
P 213 Ø(3, 2, 1) ˆ Ø(2, 3, 1)
^ (8:18d)
P 231 Ø(1, 2, 3) ˆ Ø(2, 3, 1)
^ (8:18e)
P 231 Ø(2, 3, 1) ˆ Ø(3, 1, 2)
^
The permutation operator P 123 is an identity operator because it leaves the
function Ø(i, j, k) unchanged. From (8.18a) and (8.18b), we obtain
^
P 2 213 Ø(1, 2, 3) ˆ Ø(1, 2, 3)
^ 2
so that P equals unity. The same relationship can be demonstrated to apply
213
^
^
^
to the operators P 132 and P 321 , as well as to the identity operator P 123 , giving
^ 2 213 ˆ P 2 132 ˆ P 2 321 ˆ P 2 123 ˆ P 123 ˆ 1 (8:19)
^
^
^
^
P
^ ^
Any permutation corresponding to one of the operators P áâã other than P 123
is equivalent to one or two pairwise exchanges. Accordingly, we introduce the
^
^
^
linear hermitian exchange operators P 12 , P 23 , and P 31 with the properties
^ 9
P 12 Ø(i, j, k) ˆ Ø(j, i, k) =
^ i 6ˆ j 6ˆ k ˆ 1, 2, 3 (8:20)
P 23 Ø(i, j, k) ˆ Ø(i, k, j)
^ ;
P 31 Ø(i, j, k) ˆ Ø(k, j, i)
^
The exchange operator P áâ interchanges the particles with coordinates q á and
^
q â . It is obvious that the order of the subscripts in P áâ is immaterial, so that
^ ^ ^ ^ ^
P á⠈ P âá . The permutations from P 213 , P 132 , and P 321 are the same as those
^
^
^
from P 12 , P 23 , and P 31 , respectively, giving
^ ^ ^ ^ ^ ^
P 213 ˆ P 12 ,
P 132 ˆ P 23 ,
P 321 ˆ P 31
^
The permutation from P 231 may also be obtained by ®rst applying the exchange
^
^
operator P 12 and then the operator P 23 . Alternatively, the same result may be
^
^
^
obtained by ®rst applying P 23 followed by P 31 or by ®rst applying P 31 followed
^
by P 12 . This observation leads to the identities
^ ^ ^ ^ ^ ^ ^ (8:21)
P 231 ˆ P 23 P 12 ˆ P 31 P 23 ˆ P 12 P 31
A similar argument yields
^ ^ ^ ^ ^ ^ ^
P 312 ˆ P 31 P 12 ˆ P 23 P 31 ˆ P 12 P 23 (8:22)
These permutations of the three particles are expressed in terms of the
minimum number of pairwise exchange operators. Less ef®cient routes can
^
^
also be visualized. For example, the permutation operators P 132 and P 231 may
also be expressed as