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The Harmonic Oscillator II

We discuss our results for the eigenfunctions and energy eigenvalues of the harmonic oscillator,

$\displaystyle \psi_n(x)$	$\displaystyle =$	$\displaystyle \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} \frac{1}{\sqrt{n! 2^n}}H_n\left(\sqrt{\frac{m\omega}{\hbar}}x\right) e^{-\frac{m\omega}{2\hbar}x^2}$
$\displaystyle E_n$	$\displaystyle =$	$\displaystyle \hbar \omega \left( n+\frac{1}{2}\right),\quad n=0,1,2,3,...$	(247)

It is very instructive to compare it with our results for the eigenfunctions and eigenenergies of the infinite well potential,

$\displaystyle \phi_n(x)$

$\displaystyle =$

$\displaystyle \sqrt{\frac{2}{L}}\sin \left(\frac{n\pi x}{L}\right ),\quad \varepsilon_n= \frac{n^2 \hbar^2 \pi^2}{2mL^2},\quad n=1,2,3,...$

(248)

Tobias Brandes 2004-02-04