Energy levels in finite 1d well

[stone]

could anyone suggest the methods for solving the energy eigenvalues in a 1d finite potential well. are there any websites where we can directly get these instead of writing programs for rootfinding

 

[G01]

In the process of finding the energy levels for the finite square well, you end up with a transcendental equation of the form:

$$\tan z=\sqrt{\frac{z_o}{z}-1}$$

where:

$$z=\frac{a}{\hbar}\sqrt{2m(E+V_o)$$

$$z_o=\frac{a}{\hbar}\sqrt{2mV_o}$$

a= 1/2 the width of the well.

Here, z depends on the energy of the particle trapped, and z_o depends on the potential. It is impossible to solve this equation for z, so we can't analytically find E. So, we're stuck with numerical and graphical approaches in this case.

Reference for the info in this post:

Griffith's "Introduction to Quantum Mechanics"