Page 117 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

108 Harmonic oscillator
r 
1 2E
A ˆ
ù m
Thus, equation (4.3) takes the form
r 
1 2E
x ˆ sin(ùt ‡ b) (4:7)
ù m
By de®ning a reduced distance y as
r 
m
y ù x (4:8)
2E
so that the particle oscillates between y ˆÿ1 and y ˆ 1, we may express the
equation of motion (4.7) in a universal form that is independent of the total
energy E
y(t) ˆ sin(ùt ‡ b) (4:9)
As the particle oscillates back and forth between y ˆÿ1 and y ˆ 1, the
probability that it will be observed between some value y and y ‡ dy is
P(y)dy, where P(y) is the probability density. Since the probability of ®nding
the particle within the range ÿ1 < y < 1 is unity (the particle must be some-
where in that range), the probability density is normalized

1
P(y)dy ˆ 1
ÿ1
The probability of ®nding the particle within the interval dy at a given distance
y is proportional to the time dt spent in that interval
dt
P(y)dy ˆ c dt ˆ c dy
dy
so that
dt
P(y) ˆ c
dy
where c is the proportionality constant. To ®nd P(y), we solve equation (4.9)
for t
1 ÿ1
t(y) ˆ [sin (y) ÿ b]
ù
and then take the derivative to give
c 1
2 ÿ1=2
2 ÿ1=2
P(y) ˆ (1 ÿ y ) ˆ (1 ÿ y ) (4:10)
ù ð
where c was determined by the normalization requirement. The probability
density P(y) for the oscillating particle is shown in Figure 4.1.