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WOLFRAM|DEMONSTRATIONS PROJECT

Bound-State Solutions of the Schrödinger Equation by Numerical Integration

​
ψ'(0.2)
0
energy
1
potential
V(x) =
2
x
/2
V(x) = |x|
This Demonstration shows the mathematical solution of the time-independent Schrödinger equation for two potentials, the harmonic oscillator
V(x)=
2
x
/2
and an anharmonic oscillator with
V(x)=|x|
. The wavefunction
ψ(x)
is fixed at
ψ(0.2)=2
. You can obtain linearly independent solutions by numerical integration for different values of the derivative
ψ'(0.2)
and the energy level
E
. The vertical dashed lines indicate the locations of the classical turning points.
For any value of
E
, it is always possible to tune in a value of
ψ'(0.2)
such that
ψ(x)
goes to zero either in the limit as
x∞
or in the limit as
x-∞
. The goal is to find the discrete values of
E
(the eigenvalues) such that an acceptable solution
ψ(x)
(the eigenfunction) exists that goes to zero for both limits
x∞
and
x-∞
.
The eigenvalues of the harmonic oscillator are
E
n
=n+
1
2
,
n=0
,
1
,
2
, …. The first five eigenvalues of the anharmonic oscillator are
E
0
=0.8086
,
E
1
=1.8558
,
E
2
=2.5781
,
E
3
=3.2446
, and
E
4
=3.8257
.
Units are such that
ℏ=1
,
m=1
, and the proportionality coefficients in the potential functions are as indicated.
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