Particle in Infinite Square Well

[AgingStudent49]

TL;DR Summary

Comparison of Free vs. Fixed end boundary conditions for a particle in an infinite square well

In thinking about the particle in an infinite square well, it the commonly espoused boundary conditions of ψ(0) = 0 and ψ(L) = 0 seem somewhat arbitrary. What in nature forces the wave function to vanish at the boundaries? If the particle can't escape and there is no energy loss, why not impose free end boundary conditions where the wave is reflected from each boundary? In working through this, the time-independent wave function with free-end boundary conditions is:

$ψ(x) = sqrt(2/L) cos(2nπx/L)$​

and the energy levels are:

$En = 4 (n^2 π^2 ħ^2)/(2m L^2)$​

(Notice that the energy levels are 4 times greater that those resulting from the fixed-end boundary conditions.).

Comments are encouraged.

 

[PeroK]

TL;DR Summary: Comparison of Free vs. Fixed end boundary conditions for a particle in an infinite square well

In thinking about the particle in an infinite square well, it the commonly espoused boundary conditions of ψ(0) = 0 and ψ(L) = 0 seem somewhat arbitrary. What in nature forces the wave function to vanish at the boundaries?

It's not so much arbitrary as idealized. The infinite square well can be seen as the limit of a finite square well with a large potential boundary. In that case, continuity of the wave function imposes the boundary conditions.