Page 105 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

96 General principles of quantum theory

and is an even function of each q k . If the potential energy V(q) is also an even
^
^
function of each q k , then we have H(q) ˆ H(ÿq) and
^
^ ^
^
^
^
^ ^
[H, Ð]f (q) ˆ H(q)Ðf (q) ÿ ÐH(q) f (q) ˆ H(q) f (ÿq) ÿ H(ÿq) f (ÿq) ˆ 0
^
^
Since the function f (q) is arbitrary, the commutator of H and Ð vanishes.
Thus, these operators have simultaneous eigenfunctions for systems with
V(q) ˆ V(ÿq).
If the potential energy of a system is an even function of the coordinates and
if ø(q) is a solution of the time-independent Schrodinger equation, then the
È
function ø(ÿq) is also a solution. When the eigenvalues of the Hamiltonian
operator are non-degenerate, these two solutions are not independent of each
other, but are proportional
ø(ÿq) ˆ cø(q)
These eigenfunctions are also eigenfunctions of the parity operator, leading to
the conclusion that c ˆÐ 1. Consequently, some eigenfunctions will be of even
parity while all the others will be of odd parity.
For a degenerate energy eigenvalue, the several corresponding eigenfunc-
^
tions of H may not initially have a de®nite parity. However, each eigenfunction
may be written as the sum of an even part ø e (q) and an odd part ø o (q)
ø(q) ˆ ø e (q) ‡ ø o (q)

where
1
ø e (q) ˆ [ø(q) ‡ ø(ÿq)] ˆ ø e (ÿq)
2
1
ø o (q) ˆ [ø(q) ÿ ø(ÿq)] ˆÿø o (ÿq)
2
È
Since any linear combination of ø(q) and ø(ÿq) satis®es Schrodinger's equa-
^
tion, the functions ø e (q) and ø o (q) are eigenfunctions of H. Furthermore, the
^
functions ø e (q) and ø o (q) are also eigenfunctions of the parity operator Ð, the
®rst with eigenvalue ‡1 and the second with eigenvalue ÿ1.

3.9 Hellmann±Feynman theorem
A useful expression for evaluating expectation values is known as the Hell-
mann±Feynman theorem. This theorem is based on the observation that the
Hamiltonian operator for a system depends on at least one parameter ë, which
can be considered for mathematical purposes to be a continuous variable. For
example, depending on the particular system, this parameter ë may be the mass
of an electron or a nucleus, the electronic charge, the nuclear charge parameter
Z, a constant in the potential energy, a quantum number, or even Planck's
^
constant. The eigenfunctions and eigenvalues of H(ë) also depend on this