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arXiv:quant-ph/0602100 · Foundations of Physics Letters

Energy Translation and Proper-Time Eigenstates

Zhiyong Wang, Cai-Dong Xiong, Ailin Zhang

Published Dec 1, 2006• 3 citations
arXivHTMLPDF
DOI
10.1007/s10702-006-1061-y

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The space-translation method for treating the interaction of electromagnetic radiation and matter in the electric dipole approximation is investigated from the standpoint of gauge invariance. Unless the wavefunction of the space-translation method is properly transformed, the probability of finding the system in an energy eigenstate is not in general correct. In the electric dipole approximation the probability of finding the system in an energy eigenstate is most conveniently calculated in the electric field gauge where the interaction is of the length form. The length form of the interaction is not in general equal to the velocity form or the acceleration form which comes from the space-translation method. The charged harmonic oscillator is used as an example.
In analogy to spontaneous breaking of continuous space translation symmetry in the process of space crystal formation, it was proposed that spontaneous breaking of continuous time translation symmetry could lead to time crystal formation. In other words, a time-independent system prepared in the energy ground state is expected to reveal periodic motion under infinitely weak perturbation. In the case of the system proposed originally by Wilczek, spontaneous breaking of time translation symmetry cannot be observed if one starts with the ground state. We point out that the symmetry breaking can take place if the system is prepared in an excited eigenstate. The latter can be realized experimentally in ultracold atomic gases. We simulate the process of the spontaneous symmetry breaking due to measurements of particle positions and analyze the lifetime of the resulting symmetry broken state.
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.