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- Copyright 2002 John S. Winn. All rights reserved. Last updated September 18, 2002.
- Ahead to the spherical harmonic functions.
The quantum mechanical harmonic oscillator is among our most important model systems. This page supplements pages 403 through 408 and Figure 12.8 on page 408 in the text.
The wavefunction Y is a function of the vibrational coordinate q. It is indexed by the quantum number v (where v = 0, 1, 2, ...), and it is a product of a normalization factor, a Gaussian function exp(-q2/2), and the Hermite polynomial of order v, Hv(q). Below, you can change the v quantum number (keep it less than 11) and watch the wavefunction and its associated energy change.
In the graph on the top, Y is in black, Y2 (the quantum probability distribution) is in red, and the classical turning points are located by blue ticks on the q axis. The dashed line is the classical probability distribution.
The probability of finding the oscillator with a coordinate q in the range qmin to qmax is given by the integral over q of the square of the wavefunction between the limits qmin to qmax. You can change the values of these integral limits below and see how the probability changes for various integration regions. The graph on the left shows you the integration range in dark green.
In the graph on the bottom, Y and its square are shown superimposed on a plot of the potential energy function with the first 11 (v = 0 to 10) allowed energies drawn in as lines connecting the classical turning points at each energy. (The energy is expressed in multiples of the vibrational energy quantum.)
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