Page 51 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

42 Schro Èdinger wave mechanics

1 1
2
hpiˆ pjA(p, t)j dp ˆ pA ( p, t)A(p, t)dp (2:15)
ÿ1 ÿ1
The expectation value hf ( p)i of any function f ( p)of p is given by an
expression analogous to equation (2.14)

1
2
hf ( p)iˆ f (p)jA(p, t)j dp (2:16)
ÿ1
In general, A(p, t) depends on the time, so that the expectation values hpi and
hf (p)i are also functions of time.
Both Ø(x, t) and A( p, t) contain the same information about the system,
making it possible to ®nd hpi using the coordinate-space wave function
Ø(x, t) in place of A( p, t). The result of establishing such a procedure will
prove useful when determining expectation values for functions of both
position and momentum. We begin by taking the complex conjugate of A( p, t)
in equation (2.8)

1 1
i( pxÿEt)="
A (p, t) ˆ p   Ø (x, t)e dx
2ð" ÿ1

Substitution of A (p, t) into the integral on the right-hand side of equation
(2.15) gives
1

1 i( pxÿEt)="
hpiˆ p   Ø (x, t)pA( p, t)e dx dp
2ð"
ÿ1

1 1 1
i( pxÿEt)="
ˆ Ø (x, t) p   pA( p, t)e dp dx (2:17)
ÿ1 2ð" ÿ1
In order to evaluate the integral over p, we observe that the derivative of
Ø(x, t) in equation (2.7), with respect to the position variable x,is

@Ø(x, t) 1 1 i i( pxÿEt)="
ˆ p   pA( p, t)e dp
@x 2ð" ÿ1 "
Substitution of this observation into equation (2.21) gives the ®nal result

1 " @

hpiˆ Ø (x, t) Ø(x, t)dx (2:18)
i @x
ÿ1
Thus, the expectation value of the momentum can be obtained by an integration
in coordinate space.
2
2
The expectation value of p is given by equation (2.16) with f ( p) ˆ p . The
expression analogous to (2.17) is

1 1 1
2 2 i( pxÿEt)="
hp iˆ Ø (x, t) p   p A( p, t)e dp dx
ÿ1 2ð" ÿ1
From equation (2.7) it can be seen that the quantity in square brackets equals