Fractional wave-function revivals in the infinite square well

We describe the time evolution of a wave function in the infinite square well using a fractional revival formalism, and show that at all times the wave function can be described asa superposition of translated copies of the initial wave function. Using the model of a wave form propagating on a dispersionless string from classical mechanics to describe these translations, we connect the reflection symmetry of the square-well potential to a reflection symmetry in the locations of these translated copies, and show that they occur in a ''parity-conserving'' form. The relative phases of the translated copies are shown to depend quadratically on the translation distance along the classical path. We conclude that the time-evolved wave function in the infinite square well can be described in terms of translations pf the initial wave-function shape, without approximation and without any reference to its energy eigenstate expansion. That is, the set of translated initial wave functions forms a Hilbert space basis for the time-evolved wave functions.