Quantum index theory: Relations between quantum phase and quantum number

Ciann Dong Yang, Teng Yi Chang

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


This paper extends the index theory originally valid only for classical nonlinear systems to quantum mechanical systems. Based on the Hamilton quantum mechanics, probabilistic behavior described by a wavefunction ψ(x) can be represented equivalently by nonlinear, complex-valued Hamilton equations of motion, from which we can identify fixed points, evaluate their indices and derive quantum index theory. Three quantum indices are discovered for quantum systems. The momentum index np counting the net change of the momentum phase is the counterpart of the classical index and is an indication of whether a quantum trajectory is closed. The wavefunction index n ψ counting the net change of the wavefunction phase, called Berry's geometrical phase, is responsible for the Aharonov-Bohm effect and for the Wilson-Sommerfeld quantization law. The combined index nψ= np + nψ counts the net phase change of the wavefunction derivative ψ'. Apart from their geometric meanings, it is pointed out here that the three indices, np, nψ and nψ, act as quantum numbers to represent quantization levels, respectively, for the kinetic energy Ek, the quantum potential energy Q and the combined energy Ek +Q.

Original languageEnglish
Pages (from-to)907-925
Number of pages19
JournalInternational Journal of Nonlinear Sciences and Numerical Simulation
Issue number7
Publication statusPublished - 2009

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Computational Mechanics
  • Modelling and Simulation
  • Engineering (miscellaneous)
  • Mechanics of Materials
  • Physics and Astronomy(all)
  • Applied Mathematics


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