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 as a 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 of 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. @S1050-2947~97!06606-7# The infinite square-well potential is an important model in quantum mechanics, and insights into the dynamics exhibited in this model speak immediately to a wide range of physical systems. It is a good description for onedimensional bound-state problems, whenever the confining potential is steeper than that of the parabolically curved harmonic-oscillator potential. It is also an important first approximation to the confining potential seen in semiconductor quantum wells. It is perhaps surprising that there is new physics in the time evolution of wave functions in a square well. The energy eigenvalues and eigenstates found from solving Schro ¨dinger’s equation have simple analytic expressions, and one can readily describe the time evolution of a wave function in this system via its eigenstate expansion. But if we look ahead to Fig. 2, for example, we see that time-evolved wave functions can appear to ‘‘clone’’ the initial wave function throughout the well, a phenomenon that cannot be anticipated from such an eigenstate expansion. The primary motivation for this work is to provide a formalism, a foundation, for understanding this behavior. In this paper, we describe the evolution of a wave function in the infinite square well in terms of full and fractional revivals. We will show that this system is ideal for understanding fractional revival phenomena: the fractional revivals in the infinite square well occur to all orders and for any initial wave function, with no stipulations on its initial localization or its energy bandwidth. Fractional revivals are usually thought of as isolated incidents in a quantum system’s time evolution, a few special moments in the long-term time dynamics of a carefully excited wave packet. In the infinite square well they allow for a full description of time evolution.