Discrete time in quantum mechanics

Quantum Mechanics on Discrete Space and Time

We propose the assumption of quantum mechanics on a discrete space and time, which implies the modification of mathematical expressions for some postulates of quantum mechanics. In particular we have a Hilbert space where the vectors are complex functions of discrete variable. As a concrete example we develop a discrete analog of the one-dimensional quantum harmonic oscillator, using the depend...

Symmetries in discrete-time mechanics

Based on a general formulation for discrete-time quantum mechanics, introduced in 1], symmetries in discrete-time quantum mechanics are investigated. It is shown that any classical continuous symmetry leads to a conserved quantity in classical mechanics, as well as quantum mechanics. The transformed wave function, however, has the correct evolution if and only if the symmetry is nonanomalous.

Shape Invariant Potentials in “Discrete Quantum Mechanics”

Shape invariance is an important ingredient of many exactly solvable quantum mechanics. Several examples of shape invariant “discrete quantum mechanical systems” are introduced and discussed in some detail. They arise in the problem of describing the equilibrium positions of Ruijsenaars-Schneider type systems, which are “discrete” counterparts of Calogero and Sutherland systems, the celebrated ...

Arrival Time in Quantum Mechanics

TIME DEPENDENCE in QUANTUM MECHANICS

Time-dependent perturbation theory is formulated in a systematic way. The focus is shifted from the wavefunction to the unitary transformation that evolves the wavefunction from one slice of time to another. Perturbation theory is formulated in terms of a systematic iterative expansion of the unitary transformation in terms of the perturbing hamiltonian, expressed in the interaction representat...