Schrodinger by Numerical Integration
CO22 Solutions of Schrödinger’s equation by numerical integration
Abstract
The aim of this program is to solve Schrödinger’s equation via a numerical method and to compare our results with the analytical result. The harmonic oscillator potential was taken as the potential in the equation. This was used because the solutions to this one dimensional system are well known, and an analytic solution is relatively easy to program. The idea is that once we have verified that our numerical approach works for this system it can be extended to work for any one dimensional potential that we want, even if the analytic solutions are not known.
Introduction
The time independent Schrödinger’s equation in one dimension is:
\* MERGEFORMAT
We are looking at the simple harmonic oscillator for which , where k is the spring constant. To make the equation easier to solve, we use non dimensional variable:
and
Therefore
Equation (1) reduces to:
\* MERGEFORMAT
Analytic solutions exist for equation (2), that take the form:
Where are Hermite polynomials for n positive integers and zero given by the relation:
With corresponding eigenvalues:
With this I could then write a program to show the analytic solution of this problem. For the numerical solution I used Numerov’s method. This says that a differential of the form:
Has a solution, for a uniformly spaced set of points of difference δ the value of ψ at point v+1 is approximately related to the points v and v-1 by the equation:
Where and are the values of and at . Due to the symmetry of the potential there will be both even and odd solutions. The boundary conditions will be:
Even:
Odd:
A second point can be calculated via a Taylor
Abstract
The aim of this program is to solve Schrödinger’s equation via a numerical method and to compare our results with the analytical result. The harmonic oscillator potential was taken as the potential in the equation. This was used because the solutions to this one dimensional system are well known, and an analytic solution is relatively easy to program. The idea is that once we have verified that our numerical approach works for this system it can be extended to work for any one dimensional potential that we want, even if the analytic solutions are not known.
Introduction
The time independent Schrödinger’s equation in one dimension is:
\* MERGEFORMAT
We are looking at the simple harmonic oscillator for which , where k is the spring constant. To make the equation easier to solve, we use non dimensional variable:
and
Therefore
Equation (1) reduces to:
\* MERGEFORMAT
Analytic solutions exist for equation (2), that take the form:
Where are Hermite polynomials for n positive integers and zero given by the relation:
With corresponding eigenvalues:
With this I could then write a program to show the analytic solution of this problem. For the numerical solution I used Numerov’s method. This says that a differential of the form:
Has a solution, for a uniformly spaced set of points of difference δ the value of ψ at point v+1 is approximately related to the points v and v-1 by the equation:
Where and are the values of and at . Due to the symmetry of the potential there will be both even and odd solutions. The boundary conditions will be:
Even:
Odd:
A second point can be calculated via a Taylor