Quantum-mechanical wave functions have a probabilistic
interpretation, where the probability density is the absolute square of the wave function *at a given time*,[i] *ρ*(*x*,*t*),
and can tell you the probability of finding a particle between *x* and *x* + Δ*x*, at that time *t*,
via the construction *ρ*(*x*,*t*)Δ*x*. In classical mechanics, however, the probability *
density* of a particle is either infinite or zero depending on whether the particle is in Δ*x* at time *t*,
or not (a classical particle has a definite location).

One must be careful to assert that there will be marked differences
when using the classical probability distribution to determine the quantum-mechanical probability density and
therefore the amplitude of the underlying energy eigenfunction. This use of a classical probability distribution
should only be used to give a rough idea (see the Building Energy Eigenfunctions section) of the *maximum*
amplitude of energy eigenfunctions vs. position. We must remember that we are comparing time-dependent classical
states to time-independent quantum states (energy eigenstates). Nonetheless, this comparison works rather well.[ii]

A classical probability distribution, via a time-spent diagram, can
inform you as to the maximum height of the peaks of the quantum-mechanical probability density for energy
eigenfunctions in the region that is classically allowed. For localized states to maintain a probabilistic
interpretation, the eigenfunction must go to zero as *x* → ± ∞ or at infinite walls.
French and Taylor conclude that the use of time-spent arguments in determining energy eigenfunction amplitudes “may not be generally appreciated, even by professionals,” which is as true now as it
was 35 years ago.

[i]
D. Styer, “Common Misconceptions Regarding Quantum Mechanics,” *Am. J. Phys.* **64**, 31-34 (1996).

[ii]
C. Leubner,
Margot Alber, and N. Schupfer, “Critique and Correction of the Textbook Comparison Between Classical and
Quantum Harmonic Oscillator Probability Densities,”* Am. J. Phys.* **56**, 1123-1129 (1988).