Question

Explain why in the case of the quantum harmonic oscillator the wave function can cross the potential barrier and why does the same not happen in the case of the infinite potential well?

Explain in detail

Answer 1

To answer this question we must understand the phenomenon of quantum tunneling.

Let's start with classical mechanics, if we have a particle with energy E moving towards a wall of potential V (let's say a hill). If E is more than V the particle will cross the hill otherwise no, there is no doubt in classical mechanics. Probability of crossing or not crossing the hill is very clear 1 no matter what happens.

Here come quantum mechanics, for the same particle, it says there is always some probability of finding the particle beyond the hill even if its energy E is less than the potential V. The probability of finding the particle beyond the wall is zero only if potential is infinitely large( Infinite well potential).

So only in case of infinite well potential the probability of finding the particle beyond wall is zero. It means wave function can't cross the wall. While the harmonic oscillator is simple case of a finite potential wall, and we know there is always some probability that particle crosses the wall or the wave function can cross the wall.

The same thing can be verified by the solution of schoringer's equation for both finite and infinite well potentials methamatically.