bei48482_FM

Example 5.7

Find the expectation value x for the first two states of a harmonic oscillator.

Solution

The general formula for x is

x 





x ^2 dx

In calculations such as this it is easier to begin with yin place of xand afterward use Eq. (5.67)

to change to x.From Eq. (5.72) and Table 5.2,

0     

1  4

ey

(^2)  2
1 

1  4


1  2
(2y)ey
(^2)  2
The values of x for n0 and n1 will respectively be proportional to the integrals
n0:


y 0 ^2 dy


yey
2
dy ey
2



 0
n1:


y 1 ^2 dy


y^3 ey
2
dy
  ey
2



 0
The expectation value x is therefore 0 in both cases. In fact, x 0 for allstates of a harmonic
oscillator, which could be predicted since x0 is the equilibrium position of the oscillator
where its potential energy is a minimum.
y^2
2
1
4
1
2
1
2
2 m
2 m
192 Chapter Five
bei48482_ch05.qxd 1/17/02 12:17 AM Page 192