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.

Details

For a discussion of the WKB method, see the Demonstration "WKB Computations on Morse Potential".

External Links

WKB Computations on Morse Potential

Permanent Citation

S. M. Blinder
​
​"Energies for a Heaviside-Lambda Potential Well"​
​http://demonstrations.wolfram.com/EnergiesForAHeavisideLambdaPotentialWell/​
​Wolfram Demonstrations Project​
​Published: January 6, 2011