Page 68 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

2.7 Particles in three dimensions 59

where the gradient operator = is de®ned as
@ @ @
= i ‡ j ‡ k
@x @ y @z
Using these relations, we may express the Hamiltonian operator in three
dimensions as
ÿ" 2
^ 2
H ˆ = ‡ V(r)
2m
2
where the laplacian operator = is de®ned by
@ 2 @ 2 @ 2
2 .
= = = ˆ ‡ ‡
@x 2 @ y 2 @z 2
È
The time-dependent Schrodinger equation is
@Ø(r, t)
^
i" ˆ HØ(r, t)
@t
ÿ" 2
2
ˆ = Ø(r, t) ‡ V(r)Ø(r, t) (2:68)
2m
The stationary-state solutions to this differential equation are
Ø n (r, t) ˆ ø n (r)e ÿiE n t=" (2:69)

where the spatial functions ø n (r) are solutions of the time-independent
È
Schrodinger equation
ÿ" 2
2
= ø n (r) ‡ V(r)ø n (r) ˆ E n ø n (r) (2:70)
2m
The most general solution to equation (2.68) is

X ÿiE n t="
Ø(r, t) ˆ c n ø n (r)e (2:71)
n
where c n are arbitrary complex constants.
The expectation value of a function f (r, p) of position and momentum is
given by

"

hf (r, p)iˆ Ø (r, t) f r, = Ø(r, t)dr (2:72)
i
Equivalently, expectation values of three-dimensional dynamical quantities
may be evaluated for each dimension and then combined, if appropriate, into
vector notation. For example, the two Ehrenfest theorems in three dimensions
are