Energy values in infinite potential well

[phys2]

I was wondering why the spacing between energy values keep increasing for the infinite potential well?

 

[Leeondamiky]

The energy equation for a particle inside an infinite square well is

$E_{n}$ = $\frac{\hbar^2 \pi^2 n^2}{8 m a^2}$

So $E_{n} \propto n^2$

So the relationship between energy and increasing n is non-linear.

 

[tom.stoer]

I was wondering why the spacing between energy values keep increasing for the infinite potential well?

The solution of the Schrödinger equation inside the square well is a plane wave identical to a free particle, but due to the boundary conditions only discrete levels n are allowed. The momentum p scales with n, i.e. p~n; the energy scales with p², so E=p²/2m~n².