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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 Schrodinger’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...
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