Quantum Mechanics - Examples - Harmonic Oscillator

Harmonic Oscillator

Main article: Quantum harmonic oscillator

As in the classical case, the potential for the quantum harmonic oscillator is given by:

This problem can be solved either by solving the Schrödinger equation directly, which is not trivial, or by using the more elegant "ladder method", first proposed by Paul Dirac. The eigenstates are given by:

\psi_n(x) = \sqrt{\frac{1}{2^n\,n!}} \cdot \left(\frac{m\omega}{\pi \hbar}\right)^{1/4} \cdot e^{ - \frac{m\omega x^2}{2 \hbar}} \cdot H_n\left(\sqrt{\frac{m\omega}{\hbar}} x \right), \qquad n = 0,1,2,\ldots.

where Hn are the Hermite polynomials:

and the corresponding energy levels are

.

This is another example which illustrates the quantization of energy for bound states.

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