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