Page 50 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

2.3 Expectation values of dynamical quantities 41

time t. By Parseval's theorem (equation (B.28)), if Ø(x, t) is normalized, then
its Fourier transform A( p, t) is normalized,

1 1
2 2
jØ(x, t)j dx ˆ jA(p, t)j dp ˆ 1
ÿ1 ÿ1
The transform A( p, t) is called the momentum-space wave function, while
Ø(x, t) is more accurately known as the coordinate-space wave function.
When there is no confusion, however, Ø(x, t) is usually simply referred to as
the wave function.

2.3 Expectation values of dynamical quantities
Suppose we wish to measure the position of a particle whose wave function is
2
Ø(x, t). The Born interpretation of jØ(x, t)j as the probability density for
®nding the associated particle at position x at time t implies that such a
measurement will not yield a unique result. If we have a large number of
particles, each of which is in state Ø(x, t) and we measure the position of each
of these particles in separate experiments all at some time t, then we will obtain
a multitude of different results. We may then calculate the average or mean
value hxi of these measurements. In quantum mechanics, average values of
dynamical quantities are called expectation values. This name is somewhat
misleading, because in an experimental measurement one does not expect to
obtain the expectation value.
By de®nition, the average or expectation value of x is just the sum over all
possible values of x of the product of x and the probability of obtaining that
value. Since x is a continuous variable, we replace the probability by the
probability density and the sum by an integral to obtain

1
2
hxiˆ xjØ(x, t)j dx (2:13)
ÿ1
More generally, the expectation value hf (x)i of any function f (x) of the
variable x is given by

1
2
hf (x)iˆ f (x)jØ(x, t)j dx (2:14)
ÿ1
Since Ø(x, t) depends on the time t, the expectation values hxi and hf (x)i in
equations (2.13) and (2.14) are functions of t.
The expectation value hpi of the momentum p may be obtained using the
momentum-space wave function A( p, t) in the same way that hxi was obtained
from Ø(x, t). The appropriate expression is