Bound States of a Semi-Infinite Potential Well

well depth

V

0

0.1

0.5

1.

1.5

This Demonstration shows the bound state energy levels and eigenfunctions for a semi-infinite potential well defined by

V(x)=

∞,x<0

0,0<x<a

V

0

,x>a

. The solutions are obtained by solving the time-independent Schrödinger equation in each region, and requiring continuity of both the wavefunction and its first derivative. This leads to a transcendental equation of the form

tan(α

E

)=-

E/(

V

0

-E)

, where

α≡

2m

2

a

/

2

ℏ

, (

α=10

is shown). The graphical solutions to this equation give the bound state energy levels, and are shown in the left panel. The right panel shows the energy levels and the corresponding eigenfunctions for varying well depth.