The Quantum Harmonic Oscillator

The quantum mechanical harmonic oscillator is among our most important model systems.

The wavefunction is written here as a function of the (dimensionless) vibrational coordinate q (which is directly proportional to the real coordinate x). 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, H_v(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, the wavefunction is in black, the square of the wavefunction (the quantum probability distribution) is in red, and the classical turning points are located by blue ticks on the q axis. The dashed black 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, the wavefunction 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 as lines connecting the classical turning points at each energy. (The energy is expressed in multiples of the vibrational energy quantum.)