# Well energy eigenvalues difference (graphical description)

Give a qualitative graphical argument that the difference in energy eigenvalues between the finite and infinite square wells is larger for higher energy states.

The energy of a particle in a well (finite or infinite) can be written as

$E_n=(\hbar^2/2m)*k_n^2=(\hbar^2*4\pi^2)/2m*1/\lambda_n^2$

where $k_n$ is the wave vector (thus $k=2\pi/\lambda$)

Since between two consecutive energy levels there is a difference of just half wavelength $\lambda/2$ added to the wave for each energy level (and all the standing wave NEED to be accommodated by the same well width), it results that the difference between $\lambda_n$  and $\lambda_{n+1}$ decreases with $n$ (the energy level).

Therefore

$1/\lambda_{n+1} -1/\lambda_n$   increases with $n$

and so does the energy difference between two consecutive energy levels

$E_{n+1}-E_n=C*(1/\lambda_{n+1}^2 -1/\lambda_n^2)$  increases with $n$

The only difference between infinite and finite wells is that in the latter case the standing wave describing the particle extends a bit also into the walls.