Probability and Energy-eigenfunction Amplitude

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 + Δx, at that time t, via the construction ρ(x,tx.  In classical mechanics, however, the probability density of a particle is either infinite or zero depending on whether the particle is in Δx 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).