Page 107 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

98 General principles of quantum theory
* + * + * +
^
dhAi d ^ @Ø @Ø @A
A Ø
^
ˆ hØjAjØiˆ ^ ‡ Ø A ‡ Ø Ø
dt dt @t @t @t
Equation (3.55) may be substituted for the time derivatives of the wave function
to give
* +

dhAi i ^ ^ i ^^ @A ^
ˆ hHØjAjØiÿ hØjAHjØi‡ Ø Ø
dt " " @t
* +

i i @A ^
^^
^ ^
ˆ hØjHAjØiÿ hØjAHjØi‡ Ø Ø
" " @t
* +

i @A ^
^
^
ˆ hØj[H, A]jØi‡ Ø Ø
" @t

i @A ^
^
^
ˆ h[H, A]i‡
" @t
^
where the hermiticity of H and the de®nition (equation (3.3)) of the commu-
^
tator have been used. If the operator A is not an explicit function of time, then
the last term on the right-hand side vanishes and we have
dhAi i ^ ^
ˆ h[H, A]i (3:72)
dt "
^
^
^
If we set A equal to unity, then the commutator [H, A] vanishes and equation
(3.72) becomes
dhAi
ˆ 0
dt
or
d d
^
hØjAjØiˆ hØjØiˆ 0
dt dt
We thereby obtain the result in Section 2.2 that if Ø is normalized, it remains
normalized as time progresses.
^
^
If the operator A in equation (3.72) is set equal to H, then again the
commutator vanishes and we have
dhAi dhHi dE
ˆ ˆ ˆ 0
dt dt dt
Thus, the energy E of the system, which is equal to the expectation value of the
Hamiltonian, is conserved if the Hamiltonian does not depend explicitly on
time.
^
By setting the operator A in equation (3.72) equal ®rst to the position
variable x, then the variable y, and ®nally the variable z, we can show that