# 416: Numerical Solutions for a Modified Harmonic Potential

This tutorial deals with the following potential function:

$V(x, d) = \bigg| _{ \infty ~otherwise}^{ \frac{1}{2} k(x-d)^2~if~x \geq 0 + d \leq 0}$

If d = 0 we have the harmonic oscillator on the half-line with eigenvalues 1.5, 3.5, 5.5, ... for k = $$\mu$$ = 1. For large values of d we have the full harmonic oscillator problem displaced in the x-direction by d with eigenvalues 0.5, 1.5, 2.5, ... for k = $$\mu$$ = 1. For small to intermediate values of d the potential can be used to model the interaction of an atom or molecule with a surface.

Integration limit: xmax = 10

Effective mass: $$\mu$$ = 1

Force constant: k = 1

Potential energy minimum: d = 5

Potential energy:

$V(x,d) = \frac{k}{2} (x-d)^2$

Integration algorithm:

Given



Normalize wavefunction:

$\psi (x) = \frac{ \psi (x)}{ \sqrt{ \int_{0}^{x_{max}} \psi (x)^2 dx}}$

Energy guess: E = 0.5

Calculate average position:

$X_{avg} = \int_{0}^{x_{max}} \psi (x) x \psi (x) dx = 5$

Calculate potential and kinetic energy:

$V_{avg} = \int_{0}^{x_{max}} \psi (x) V(x,d) \psi (x) dx = 0.25$

$T_{avg} = E - V_{avg} = 0.25$

Exercises:

• For d = 0, k = $$\mu$$ = 1 confirm that the first three energy eigenvalues are 1.5, 3.5 and 5.5 Eh. Start with xmax = 5, but be prepared to adjust to larger values if necessary. xmax is effectively infinity.
• For d = 5, k = $$\mu$$ = 1 confirm that the first three energy eigenvalues are 0.5, 1.5 and 2.5 Eh. Start with xmax = 10, but be prepared to adjust to larger values if necessary.
• Determine and compare the virial theorem for the exercises above.
• Calculate the probability that tunneling is occurring for the ground state for the first two exercises. (Answers: 0.112, 0.157)