# 9.22: Numerical Solutions for a Modified Harmonic Potential

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

This tutorial deals with the following potential function:

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

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:

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

Integration algorithm:

Given

$\nonumber$

Normalize wavefunction:

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

Energy guess: E = 0.5

Calculate average position:

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

Calculate potential and kinetic energy:

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

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

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)

This page titled 9.22: Numerical Solutions for a Modified Harmonic Potential is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Frank Rioux via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.