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arXiv:1112.2418

The Berry Phase for Simple Harmonic Oscillators

Sergeĭ K. Suslov

Published Dec 12, 2011• 4 citations
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Abstract

We evaluate the Berry phase for a "missing" family of the square integrable wavefunctions for the linear harmonic oscillator, which cannot be derived by the separation of variables (in a natural way). Instead, it is obtained by the action of the maximal kinematical invariance group on the standard solutions. A simple closed formula for the phase (in terms of elementary functions) is found by integration with the help of a computer algebra system.

DOI
10.48550/arxiv.1112.2418

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We evaluate the Berry phase for a "missing" family of the square integrable wavefunctions for the linear harmonic oscillator, which cannot be derived by the separation of variables (in a natural way). Instead, it is obtained by the action of the maximal kinematical invariance group on the standard solutions. A simple closed formula for the phase (in terms of elementary functions) is found by integration with the help of a computer algebra system.
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We consider a six-parameter family of the square integrable wave functions for the simple harmonic oscillator, which cannot be obtained by the standard separation of variables. They are given by the action of the corresponding maximal kinematical invariance group on the standard solutions. In addition, the phase space oscillations of the electron position and linear momentum probability distributions are computer animated and some possible applications are briefly discussed. A visualization of the Heisenberg Uncertainty Principle is presented.
For a time-dependent $\ensuremath{\tau}$-periodic harmonic oscillator of two linearly independent homogeneous solutions of the classical equation of motion that are both bounded over time (stable), it is shown that there is a representation of states that are cyclic up to multiplicative constants under $\ensuremath{\tau}$ evolution or $2\ensuremath{\tau}$ evolution depending on the model. The set of the wave functions is complete. Berry's phase, which could depend on the choice of representation, can be defined under the $\ensuremath{\tau}$ or $2\ensuremath{\tau}$ evolution in this representation. If a homogeneous solution diverges as time goes to infinity, it is shown that such a Berry phase cannot be defined in any representation considered. Berry's phase for the driven harmonic oscillator is also considered. For the cases where Berry's phase can be defined, the phase is given in terms of solutions of the classical equation of motion.