Energies for Particle in a Gaussian Potential Well

​
V
0
4
σ
1.5
α
0.78
basis functions
1
2
3
4
5
6
7
8
The Schrödinger equation for a particle in a one-dimensional Gaussian potential well
V(x)=-
V
0
-
2
x
/2
2
σ
e
, given by
-
2
ℏ
2m
ψ''
(x)
+V(x)ψ(x)=ϵψ(x)
, has never been solved analytically. This Demonstration derives an approximation for the first few bound-state energies,
ϵ
n
<0
, using the linear variational method. The wavefunction is approximated by a linear combination
ψ(x)=
N-1
∑
n=0
c
n
ϕ
n
(x)
. It is convenient to take the basis functions
ϕ
n
(x)
as the corresponding orthonormalized eigenfunction of the linear harmonic oscillator:
ϕ
n
(x)=
1
n
2
n!
1/4
α
π
-α
2
x
/2
e
H
n
(
α
x)
, where
H
n
is the
th
n
Hermite polynomial and
α
is a scaling constant to be determined variationally. After evaluating the matrix elements
H
nm
=
∞
∫
-∞
ϕ
n
(x)-
1
2
2
d
d
2
x
+V(x)
ϕ
m
(x)dx
​
over the selected set of
N
basis functions, Mathematica can calculate the
N
eigenvalues in a single step, from which we select only those with negative values. For convenience, we set
ℏ=m=1
, so that all distances are expressed in bohrs (Bohr radii) and energy quantities in hartrees.
The graphic shows the computed energy levels for selected values of
V
0
,
σ
,
α
, and
N
, superposed on the potential energy function. By an estimation based on the WKB method, the number of bound states is approximated by
floor4σ
V
0
2π
+
1
2

.

References

D.A. McQuarrie, Quantum Chemistry, Sausalito, CA: University Science Books, 1983, pp. 266-275. Or numerous other texts on quantum mechanics or quantum chemistry.

External Links

Quantum Mechanics (ScienceWorld)
Interleaving Theorems for the Rayleigh-Ritz Method in Quantum Mechanics

Permanent Citation

S. M. Blinder
​
​"Energies for Particle in a Gaussian Potential Well"​
​http://demonstrations.wolfram.com/EnergiesForParticleInAGaussianPotentialWell/​
​Wolfram Demonstrations Project​
​Published: August 14, 2012