This simulation animates harmonic oscillator wavefunctions that are built from arbitrary superpositions of the lowest eight definite-energy wavefunctions. The “clock faces” show phasor diagrams for the complex amplitudes of these eight basis functions, going from the ground state at the left to the seventh excited state at the right, with the outside of each “clock” corresponding to a magnitude of 1. The current wavefunction is then built by summing the eight basis functions, multiplied by their corresponding complex amplitudes. As time passes, each basis amplitude rotates in the complex plane at a frequency proportional to the corresponding energy.
You can plot either the real and imaginary parts of the wavefunction (shown in orange and blue, respectively), or the probability density and phase, with the phase represented by hues going from red (pure real and positive) to light green (pure imaginary and positive) to cyan (pure real and negative) to purple (pure imaginary and negative) and finally back to red.
Click on any clock face to change the corresponding amplitude. To see an individual basis function, click “zero” and then click on the corresponding clock face. You can also create a coherent state (or an approximation thereof), which oscillates back and forth somewhat like a classical particle would. The parameter α determines the amplitude of this oscillation, with α2 equal to the average number of energy units above the ground state. For large values of α, however, the true coherent state is not well approximated using only the lowest eight basis states.
By Daniel V. Schroeder, Physics Dept., Weber State University