Quantum mechanics on a circle: Husimi phase-space distributions and semiclassical coherent state propagators

Abstract

We discuss some basic tools for an analysis of one-dimensional quantum systems defined on a cyclic coordinate space. The basic features of the generalized coherent states, the complexifier coherent states, are reviewed. These states are then used to define the corresponding (quasi)densities in phase space. The properties of these generalized Husimi distributions are discussed, in particular their zeros. Furthermore, the use of the complexifier coherent states for a semiclassical analysis is demonstrated by deriving a semiclassical coherent state propagator in phase space.

Coherent states for the quantum mechanics on a torus

“…The expectation value of the angular momentum in the coherent state is l, α|J|l, α l, α|l, α ≈ l, (2.11) where the very good approximation of the relative error is [2]: ∆l/l ≈ 2π exp(−π 2 ) sin(2lπ)/l (see also the very recent paper [8]), so the maximal error arising in the case l → 0 is of order 0.1 per cent. We have remarkable exact equality for l integer or half-integer.…”

Quantum‐classical correspondence for a dc‐biased cavity resonator–Cooper‐pair transistor system

“…The parameter δ appearing in the Hamiltonian operator is a purely quantum signature of the nontrivial topology of the corresponding classical pendulum's configuration space S 1 . An interesting question concerns the particular value for δ that Nature chooses and why [67][68][69].…”

Coherent States on the Circle: Semiclassical Matrix Elements in the Context of Kummer Functions and the Zak transformation

“…In this paper, we discuss coherent states for a particle on a circle that have for instance been discussed in earlier work in [1][2][3][4][5] and references therein. In [1][2][3][4], coherent states in the Hilbert space L 2 (S 1 ) were constructed by means of the so-called Zak transformation [6], whereas in [5] complexifier coherent states [7] for the group U(1) were used, leading finally to the same kind of coherent states. These complexifier coherent states have been introduced in the framework of loop quantum gravity for the group SU (2) and their properties have been analysed in [8][9][10].…”

“…This relation, obtained in section VI, provides an alternative way for computing semiclassical matrix elements and expectation values respectively. It might also allow to reconsider the techniques from a different angle that have been used in the context of U(1) complexifier coherent states in [5,8,9,11,12] in order to estimate semiclassical expectation values and to obtain the classical limit. Although we will restrict to the one-dimensional case in this article, the Zak transformation, and thus also the results presented here, can be easily generalised to higher finite dimensional systems.…”

“…Next, we move on to quantum mechanics on a circle in section IV. Since, as in [5], we come from the complexifier coherent states, after a brief introduction to the Zak transformation, we apply it to the heat kernel, which is more convenient in this context, and not to the harmonic oscillator coherent states directly. For this purpose, we use former work from [19] and as expected we end up with the same coherent states for L 2 (S 1 ).…”