Page 103 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

94 General principles of quantum theory

answer that question, we assume that Ø(q, t) is any arbitrary solution of the
parital differential equation (3.55). We suppose further that the set of functions
ø n (q) which satisfy the eigenvalue equation (3.58) is complete. Then we can,
in general, expand Ø(q, t) in terms of the complete set ø n (q) and obtain
X
Ø(q, t) ˆ a n (t)ø n (q) (3:60)
n
The coef®cients a n (t) in the expansion are given by
a n (t) ˆhø n (q)jØ(q, t)i (3:61)
and are functions of the time t, but not of the coordinates q. We substitute the
expansion (3.60) into the differential equation (3.55) to obtain

X " @a n (t) X " @a n (t)
^
‡ a n (t)H ø n (q) ˆ ‡ E n a n (t) ø n (q) ˆ 0(3:62)
i @t i @t
n n
^
where we have also noted that the functions ø n (q) are eigenfunctions of H in

accordance with equation (3.58). We next multiply equation (3.62) by ø (q),
k
the complex conjugate of one of the eigenfunctions of the orthogonal set, and
integrate over the spatial variables

X " @a n (t)
‡ E n a n (t) hø k (q)jø n (q)i
i @t
n

X " @a n (t) " @a k (t)
ˆ ‡ E n a n (t) ä kn ˆ ‡ E k a k (t) ˆ 0
i @t i @t
n
Replacing the dummy index k by n, we obtain the result
a n (t) ˆ c n e ÿiE n t=" (3:63)
where c n is a constant independent of both q and t. Substitution of equation
(3.63) into (3.60) gives equation (3.59), showing that equation (3.59) is indeed
the most general form for a solution of the time-dependent Schrodinger
È
equation. All solutions may be expressed as the sum over stationary states.

3.8 Parity operator
^
The parity operator Ð is de®ned by the relation
^
Ðø(q) ˆ ø(ÿq) (3:64)
Thus, the parity operator reverses the sign of each cartesian coordinate. This
operator is equivalent to an inversion of the coordinate system through the
origin. In one and three dimensions, equation (3.64) takes the form
^
^
^
^
Ðø(x) ˆ ø(ÿx), Ðø(r) ˆ Ðø(x, y, z) ˆ Ðø(ÿx, ÿy, ÿz) ˆ ø(ÿr)
^ 2
The operator Ð is equal to unity since