Wave packets in discrete quantum phase space

Abstract

The properties of quantum mechanics with a discrete phase space are studied. The minimum uncertainty states are found, and these states become the Gaussian wave packets in the continuum limit. With a suitably chosen Hamiltonian that gives free particle motion in the continuum limit, it is found that full or approximate periodic time evolution can result. This represents an example of revivals of wave packets that in the continuum limit is the familiar free particle motion on a line.Finally we examine the uncertainty principle for discrete phase space and obtain the correction terms to the continuum case.

Properties of finite Gaussians and the discrete-continuous transition

“…Note that in the continuum limit, when d tends to infinity, the periodicity of free evolution practically disappears, and one obtains the result known from continuousconfiguration quantum mechanics. A similar periodicity has been obtained in [3] for a free wavepacket moving in a discrete quantum phase space. However, in [3] the discrete-eigenvalues position and momentum operators were defined differently, with the result that the revivals appear for minimum uncertainty states but are only approximate for long time evolution in other cases.…”

“…A similar periodicity has been obtained in [3] for a free wavepacket moving in a discrete quantum phase space. However, in [3] the discrete-eigenvalues position and momentum operators were defined differently, with the result that the revivals appear for minimum uncertainty states but are only approximate for long time evolution in other cases.…”

“…except a few small values of d. Numerical results concerning the case κ = 1 are presented in table 3. Note that for different definitions of the position and momentum operators in a discrete quantum phase space [3], the minimum uncertainty of these operators is dependent on the discretization step (the exact formula is known as the generalized uncertainty principle) and approaches the result for the continuum case only for an infinitely fine discretization; the generalized uncertainty principle can be obtained from a quantum mechanical model with discrete eigenvalues for the coordinate operator [2]. On the contrary, in our case the uncertainty is approximately minimum for quite small d values.…”

Operator spreading in quantum maps

“…Several approaches to construct such sets of states for a particle on a finite-dimensional periodic lattice or for a quantum Hall system on a torus are known. [81][82][83] However, it is known to be impossible to obtain an N -dimensional orthogonal basis (for finite N ) of wavefunctions on a torus that is local in phase space, 84 although orthogonalization of coherent states can be implemented in the continuum as well as the N → ∞ limit. 80,85 This hinders an analytical exploration of the classical (N → ∞) and quantum (finite N ) dynamics in the same language in the Schrödinger picture, although phase-space representations of quantum mechanics 86 such as the Wigner quasi-probability representations of wavefunctions (density matrices) have been employed.…”

Properties of a discrete quantum field theory

“…One straightforward technique for producing a well-defined quantum mechanics on a discrete configuration space is to consider momentum space as being compact, in which case it is the position space that becomes discrete [2]. If one properly defines a Hamiltonian on this discrete space, one can represent various dynamics and one can produce in the continuum limit (where the discretization becomes small) various continuum Hamiltonians (for example the free particle) [3]. For wave packets localized in momentum space the topology has only a small effect on the dynamics and one obtains the usual time evolution in the continuum limit.…”