Page 53 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

44 Schro Èdinger wave mechanics

dhxi d 1 2 1 @ 2
ˆ xjØ(x, t)j dx ˆ x jØ(x, t)j dx
dt dt @t
ÿ1 ÿ1

i" 1 @ @Ø @Ø
ˆ x Ø ÿ Ø dx
2m @x @x @x
ÿ1
where equation (2.12) has been used. Integration by parts of the last integral
gives
1
dhxi i" @Ø @Ø i" 1 @Ø @Ø
ˆ x Ø ÿ Ø ÿ Ø ÿ Ø dx
dt 2m @x @x ÿ1 2m ÿ1 @x @x
The integrated part vanishes because Ø(x, t) goes to zero as x approaches ( )
in®nity. Another integration by parts of the last term on the right-hand side
yields

dhxi 1 1 " @
ˆ Ø Ø dx
dt m i @x
ÿ1
According to equation (2.18), the integral on the right-hand side of this
equation is the expectation value of the momentum, so that we have
dhxi
hpiˆ m (2:23)
dt
Equation (2.23) is the quantum-mechanical analog of the classical de®nition of
momentum, p ˆ mv ˆ m(dx=dt). This derivation also shows that the associa-
tion in quantum mechanics of the operator ("=i)(@=@x) with the momentum is
consistent with the correspondence principle.
The second relationship is obtained from the time derivative of the expecta-
tion value of the momentum hpi in equation (2.18),

dhpi d 1 " @Ø " 1 @Ø @Ø @ @Ø
ˆ Ø dx ˆ ‡ Ø dx
dt dt i @x i @t @x @x @t
ÿ1 ÿ1

We next substitute equations (2.11) for the time derivatives of Ø and Ø and
obtain
" #

1 2 2 2 2
dhpi ÿ" @ Ø @Ø @ " @ Ø
ˆ ‡ VØ ‡ Ø ÿ VØ dx
dt 2m @x 2 @x @x 2m @x 2
ÿ1
3
2
ÿ" 2 1 @ Ø @Ø " 2 1 @ Ø 1 dV

ˆ dx ‡ Ø dx ÿ Ø Ø dx
2m @x 2 @x 2m @x 3 dx
ÿ1 ÿ1 ÿ1
(2:24)
where the terms in V cancel. The ®rst integral on the right-hand side of
equation (2.24) may be integrated by parts twice to give