The corresponding Schrödinger equation can be solved with the substitutions and (which are the normal mode coordinates), which reduces the problem to a two-dimensional harmonic oscillator. The energy eigenstates are then and and the wavefunction is .

This Demonstration plots (substituting back the regular displacements) and its modulus squared (which is the PDF of the displacements) for states to and their linear combinations, in which you can vary both magnitudes and phases . Some plots are significantly slower to display, especially the ones with more complex wavefunctions.