Page 218 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

8.1 Permutations of identical particles 209

of mass m. If we label one of the particles as particle 1 and the other as particle
^
2, then the Hamiltonian operator H(1, 2) for the system is
^ p 2 ^ p 2
^ 1 2
H(1, 2) ˆ ‡ ‡ V(q 1 , q 2 ) (8:1)
2m 2m
where q i (i ˆ 1, 2) represents the three-dimensional (continuous) spatial
coordinates r i and the (discrete) spin coordinate ó i of particle i. In order for
these two identical particles to be indistinguishable from each other, the
Hamiltonian operator must be symmetric with respect to particle interchange,
i.e., if the coordinates (both spatial and spin) of the particles are interchanged,
^
H(1, 2) must remain invariant
^
^
H(1, 2) ˆ H(2, 1)
^
^
If H(1, 2) and H(2, 1) were to differ, then the corresponding Schrodinger
È
equations and their solutions would also differ and this difference could be
used to distinguish between the two particles.
È
The time-independent Schrodinger equation for the two-particle system is
^
H(1, 2)Ø í (1, 2) ˆ E í Ø í (1, 2) (8:2)
where í delineates the various states. The notation Ø í (1, 2) indicates that the
®rst particle has coordinates q 1 and the second particle has coordinates q 2 .If
we exchange the two particles so that particles 1 and 2 now have coordinates
q 2 and q 1 , respectively, then the SchroÈdinger equation (8.2) becomes
^
^
H(2, 1)Ø í (2, 1) ˆ H(1, 2)Ø í (2, 1) ˆ E í Ø í (2, 1) (8:3)
^
where we have noted that H(1, 2) is symmetric. Equation (8.3) shows that
^
Ø í (2, 1) is also an eigenfunction of H(1, 2) belonging to the same eigenvalue
E í . Thus, any linear combination of Ø í (1, 2) and Ø í (2, 1) is also an eigen-
^
function of H(1, 2) with eigenvalue E í . For simplicity of notation in the
following presentation, we omit the index í when it is clear that we are
referring to a single quantum state.
The eigenfunction Ø(1, 2) has the form of a wave in six-dimensional space.

The quantity Ø (1, 2)Ø(1, 2) dr 1 dr 2 is the probability that particle 1 with
spin function ÷ 1 is in the volume element dr 1 centered at r 1 and simultaneously
particle 2 with spin function ÷ 2 is in the volume element dr 2 at r 2 . The product

Ø (1, 2)Ø(1, 2) is, then, the probability density. The eigenfunction Ø(2, 1)

also has the form of a six-dimensional wave. The quantity Ø (2, 1)Ø(2, 1) is
the probability density for particle 2 being at r 1 with spin function ÷ 1 and
simultaneously particle 1 being at r 2 with spin function ÷ 2 . In general, the two
eigenfunctions Ø(1, 2) and Ø(2, 1) are not identical. As an example, if
Ø(1, 2) is