Wave Function of a Harmonic Oscillator (Graph)
This program calculates the quantum mechanical wave function of a one-dimensional harmonic oscillator and displays a graph.
The unit of position is calculated as α = √(2πmω/h) = 1.
| \(\normalsize The\ wave\ function\ \psi(x)\\ \hspace{15px}of\ the\ quantum\ harmonic\ oscillator\\ (1)\ [-{\large\frac{\hbar^2}{2m}}{\large\frac{d^2}{dx^2}}\psi+{\large\frac{1}{2}}m\omega^2x]\psi=E\psi\\ \hspace{170px} E=\hbar\omega(n+{\large\frac{1}{2}})\\ (2)\ \psi_{n}(x)=\sqrt{\large\frac{\alpha}{2^nn!\sqrt{\normalsize\pi}}}H_n(\alpha x)e^{\small -\frac{1}{2}(\alpha x)^2}\\ \hspace{170px}\alpha=\sqrt{\large\frac{m\omega}{\hbar}}\\ \hspace{20px}{\large\int_{\small-\infty}^{\small\infty}}\psi_{n}(x)\psi_{n'}^*(x)dx=\delta_{nn'}\\\) |
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