Time-Dependent Superposition of Harmonic Oscillator Eigenstates

Consider a time-dependent superposition of quantum harmonic oscillator eigenstates, , where the eigenfunctions and eigenvalues are given by and , respectively. Here is the Hermite polynomial. The Hamiltonian for this system is and its energy expectation value is given by .

Choosing "energy levels" shows the complex wavefunction in the upper panel, where the shape is its modulus and the coloring represents its argument, the range to corresponding to colors from red to magenta. The potential energy curve is drawn for visualization purposes. The lower panel shows the eigenvalues in blue and the energy of the superposition state in red.

Choosing the view "expectation value" shows the same superposition in both the position and momentum representations, where the wavefunctions are connected via . The two left panels show the position space probability density and position expectation value , while the right panels show the momentum space probability density and momentum expectation value . The lower panel shows a parametric plot of the expectation values which, in the very special case of a harmonic potential, resembles classical phase space.