Phase space quantization and the uncertainty principle

Michael J. W. Hall, Marcel Reginatto

Б. И. Садовников, E. E. Perepelkin, Наталья Германовна Иноземцева

Jianlan Wu, Jianshu Cao

The algebraic structure of the quantum Morse oscillator is explored to formulate the coherent state, the phase-space representations of the annihilation and creation operators, and their classical limits. The formulation allows us to calculate the linear and nonlinear quantum response functions for microcanonical Morse systems and to demonstrate the linear divergence in the corresponding classical response function. On the basis of the uncertainty principle, the classical divergence is removed by phase-space averaging around the microcanonical energy surface. For the Morse oscillator, the classical response function averaged over quantized phase space agrees exactly with the quantum response function for a given eigenstate. Thus, phase-space averaging and quantization provide a useful way to establish the classical-quantum correspondence of anharmonic systems.

Murray Gerstenhaber

Deformation quantization generally produces families of cohomologically equivalent quantizations of a single physical system. We conjecture that the physically meaningful ones (i) allow enough observable energy distributions, i.e., ones for which no pure quantum state has negative probability, and (ii) reduce the uncertainty in the probability distribution of the resulting quantum states. For the simple harmonic oscillator this principle selects the classic Groenewold-Moyal (or Weyl) product on phase space while for the free particle, in which there is no real quantization, all cohomologically equivalent quantizations are equally good.

Carlos Alberto Alcalde, Kanishka Tyagi