Question

A particle is trapped in an infinite potential well. Describe what happens to the particle’s ground-state energy and wave function as the potential walls become finite and get lower and lower until they finally reach zero (U = 0 everywhere).

Answer 1

ground state energy of a particle in an infinite potential well is

when the potential well becomes finite with potential U the ground state energy is

where

is the penetration depth

The wave function extends to a distance of outside the well and decays exponentially. The wave becomes essentially 0 beyond .

as U becomes less and less penetration of the wave beyond the walls increases. The case where U=0 becomes imaginary , this the case where there are no potential walls , the potential well is non-existing and particle is free . This is the case the particle can be anywhere from - to +