• Corpus ID: 233231607

A simple proof of Floquet's theorem for the case of unitary evolution in quantum mechanics

@inproceedings{Martin2021ASP,
  title={A simple proof of Floquet's theorem for the case of unitary evolution in quantum mechanics},
  author={J. D. D. Martin and A. N. Poertner},
  year={2021}
}

• J. D. Martin, A. Poertner
• Published 1 March 2021
• Physics

where E is a diagonal matrix containing the ǫj ’s, and Φ(t) is periodic; i.e., Φ(t) = Φ(t+ T ) for all t. Despite the relative simplicity of this result, it is not standard quantum mechanics textbook fare (with at least one exception [4]). But due to the fundamental nature of this “time version” of Bloch’s theorem and recent activity in time-periodic quantum-mechanical Floquet systems (see, for example, Ref. [5]), it may be desirable to study it in quantum mechanics courses. The purpose of this short note is to present a simple, constructive proof of Floquet’s theorem in the special case of quantum-mechanical unitary time evolution (Eq. 5), appropriate for physics undergraduate students. This proof arises naturally from consideration of how one might numerically compute time-evolution in an efficient manner. The steps in the derivation are easily remembered, and involve only the standard results of basic quantum mechanics, as found in typical textbooks [6–8]. First note that for any positive integerN , the evolution from t = NT to t = (N + 1)T as given by U((N + 1)T,NT ) is the same as for from t = 0 to t = T as given by U(T, 0). This symmetry can be seen by replacing t by t−T in the TDSE, since H(t+T ) = H(t) for all t. Since t in U(t, 0) is potentially much greater than T , it makes sense to compute U(T, 0), then raise it to the largest non-negative integer power N such that NT ≤ t. Then finally, compute evolution over the “left-over” interval from NT to t (of duration less than T ). In other words, we consider the factorization: