Energies for a Heaviside-Lambda Potential Well

V

0

6

a

2

This Demonstration calculates the bound energy levels of a particle in an inverted Heaviside-lambda (vee-shaped) potential well of depth

V

0

and width

2a

, using the semiclassical Wentzel–Kramers–Brillouin (WKB) method. The numerical results are within 1% of the values that would be obtained from the exact solutions of the corresponding Schrödinger equation. The energies are determined by the Sommerfeld–Wilson quantization conditions

∮

2m[E-V(x)]

dq=n+

1

2

h

. With

ℏ=m=1

, the integral reduces to

4

a1+E

V

0



∫

0

E+

V

0

-

V

0

a

x

dx

, noting that

x=±a1+

E

V

0

are the classical turning points. This can be solved for the energy levels:

E

n

=-

V

0

+

1/3

2

4

2/3

3π

V

0

a

2/3

n+

1

2

,

n=0,1,2,…,

n

max

. The highest bound state is given by

n

max

=-

1

2

+

4a

2

V

0

3π

, where

⌊⌋

is the floor, which for positive numbers is simply the integer part.