Physical Chemistry , 1st ed.

potential energy. That is, wavefunctions are nonzero and therefore the oscillator

can exist beyondits classical turning point. This suggests the paradoxical conclu-

sion that the oscillator must have negative kinetic energy! Actually, the “paradox”

aspect is based on classical expectations. This is not the first example of quantum

mechanics proposing something that goes against classical expectations. Tunneling

of a particle through a finite barrier is another, and the wavefunction’s existence

beyond the classical turning point is similar to tunneling. In this case, the “wall”

is a curved potential energy surface, not a straight up-and-down barrier.

Recall that the particle’s probability of existing anywhere along its one-

dimensional space is proportional to ^2. Several plots of^2 are shown in

Figure 11.5. The top plot has a high quantum number, and its shape is start-

ing to mimic the behavior of a classical harmonic oscillator: it moves very

quickly near x0 (and has a lower probability of existence there), but pauses

near the turning point and has a higher probability of being found there. This

is another example of the correspondence principle: for high quantum num-

bers (and therefore high energies), quantum mechanics approaches the expec-

tations of classical mechanics.

Example 11.7

Evaluate the average value of the momentum (in the xdirection) for  1 of a

harmonic oscillator.

Solution

Using the definition of the momentum operator, we need to evaluate

pxN^2 





[H 1 (1/2x) ex

(^2) /2
]* i




x

[H 1 (1/2x) ex

(^2) /2
)] dx
It would be easier to simply use the form of the Hermite polynomial in terms
of1/2xinstead of(although it can be done either way; use your judgment
regarding which you prefer). From Table 11.1:
pxN^2 


(21/2x ex
(^2) /2
)* i




x

(21/2x ex

(^2) /2
) dx
The complex conjugate does not change anything. Evaluating the derivative
in the right-hand part of the expression, and bringing the constants outside
the integral:
px 4 iN^2 


x ex
(^2) /2
(ex
(^2) /2
x^2 ex
(^2) /2
) dx
which simplifies to
px 4 iN^2 


(xex
2
x^3 ex
2
) dx
Both terms inside the parentheses are odd over the range of integration, over-
all. Therefore, their integrals are exactly zero. So
px 0
Given that momentum is a vector quantity and that the mass is traveling back
and forth in both directions, it should make sense that the average value of
the momentum is zero.
11.4 The Harmonic Oscillator Wavefunctions 329
n  0
n  1
n  2
n  3
n  4
x
x  0
V ^12 kx^2
Figure 11.5 Plots of the first five ^2 wave-
functions, superimposed on the potential energy
diagram. As the quantum number increases, the
probability that the particle is in the center of the
potential energy well decreases and the probabil-
ity of its being at the sides of the potential well
increases. At high quantum numbers, quantum
mechanics is mimicking classical mechanics.
This is another example of the correspondence
principle.