where it moves slowly, and least near the equilibrium position (x0), where it moves
rapidly.
Exactly the opposite behavior occurs when a quantum-mechanical oscillator is
in its lowest energy state of n0. As shown, the probability density 0 ^2 has its
maximum value at x0 and drops off on either side of this position. However,
this disagreement becomes less and less marked with increasing n.The lower graph
of Fig. 5.13 corresponds to n10, and it is clear that 10 ^2 when averaged over
xhas approximately the general character of the classical probability P. This is
another example of the correspondence principle mentioned in Chap. 4: In the limit
of large quantum numbers, quantum physics yields the same results as classical
physics.
It might be objected that although 10 ^2 does indeed approach Pwhen smoothed
out, nevertheless 10 ^2 fluctuates rapidly with xwhereas Pdoes not. However, this
objection has meaning only if the fluctuations are observable, and the smaller the spac-
ing of the peaks and hollows, the more difficult it is to detect them experimentally.
The exponential “tails” of 10 ^2 beyond x Aalso decrease in magnitude with
increasing n.Thus the classical and quantum pictures begin to resemble each other
more and more the larger the value of n, in agreement with the correspondence prin-
ciple, although they are very different for small n.Quantum Mechanics 191
x = –Ax = +AP| 10 |^2x = –Ax = +AP| 0 |^2Figure 5.13Probability densities for the n0 and n10 states of a quantum-mechanical harmonic
oscillator. The probability densities for classical harmonic oscillators with the same energies are shown
in white. In the n10 state, the wavelength is shortest at x0 and longest at xA.x = –Ax = +A(^1)
x = –Ax = +A
(^2)
x = –Ax = +A
(^3)
x = –Ax = +A
(^4)
x = –Ax = +A
(^0)
x = –Ax = +A
(^5)
Figure 5.12The first six harmonic-
oscillator wave functions. The ver-
tical lines show the limitsAand
Abetween which a classical os-
cillator with the same energy
would vibrate.
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