© Daniel Kleppner.
The n = 40 wavefunction for the harmonic oscillator, found by solving Schrödinger's equation, is shown on the left. The corresponding probability distribution, which oscillates rapidly in space, is shown on the right. As the quantum number n increases, the oscillation rate increases, and the probability of finding the particle in a given region approximates a smooth curve at the average height of each oscillation. In classical physics, the probability of finding the particle at a particular position is proportional to the fraction of time the particle spends there, which is inversely proportional to the particle's velocity. The particle therefore is most likely to be found at a turning point, and spends much less time near the equilibrium position. The classical and the quantum predictions get closer and closer as n increases, with one important exception: The classical curve diverges at the turning point where the particle has zero velocity, while the quantum curve is smoothed over. This is a characteristic difference between the quantum and classical pictures. (Unit: 5)