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Dynamics

Coherent States of the Conformable Quantum Oscillator

Cresus F. L. Godinho, Claudio Maia Porto, Marcos Cardoso Rodriguez, Ion V. Vancea

Published Jul 4, 2025• 1 citation
Publisher

Abstract

Recent proposed conformable deformation of the quantum mechanics by a fractional parameter α∈(0,1] has been used to construct a conformable quantum harmonic oscillator which coincides with the standard quantum oscillator at α=1. We argue that there is a conformable generalization of the uncertainty principle and use it to define the coherent states along the general line of quantum mechanics. We focus on the coherent states of the conformable quantum harmonic oscillator in the xα-representation: we determine these states, compute their α-power of energy, their conformable time-dependent form and the conformable translation operator. Also, we show that the eigenvectors of the conformable annihilation operator are coherent states.

DOI
10.3390/dynamics5030026

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Recent proposed conformable deformation of the quantum mechanics by a fractional parameter α∈(0,1] has been used to construct a conformable quantum harmonic oscillator which coincides with the standard quantum oscillator at α=1. We argue that there is a conformable generalization of the uncertainty principle and use it to define the coherent states along the general line of quantum mechanics. We focus on the coherent states of the conformable quantum harmonic oscillator in the xα-representation: we determine these states, compute their α-power of energy, their conformable time-dependent form and the conformable translation operator. Also, we show that the eigenvectors of the conformable annihilation operator are coherent states.
This work addresses the study of the oscillator algebra, defined by four parameters ppp, qqq, ααα, and ννν. The time-independent Schrödinger equation for the induced deformed harmonic oscillator is solved; explicit analytic expressions of the energy spectrum are given. Deformed states are built and discussed with respect to the criteria of coherent state construction. Various commutators involving annihilation and creation operators and their combinatorics are computed and analyzed. Finally, the correlation functions of matrix elements of main normal and antinormal forms, pertinent for quantum optics analysis, are computed.
We construct representations of the Lie algebra using representations of the momentum and position operators satisfying the R -deformed Heisenberg relations, in which the fractional dimension d and angular momentum ℓ appear as parameters. The Bargmann index κ, which characterizes representations of the positive discrete series of , can take any positive value. We construct coherent states in fractional dimensions, in particular we extend the two well-known analytic representations of coherent states for , Perelomov and Barut–Girardello states, from dimension one to any dimension d . We generalize this construction to time-dependent coherent states by means of the symmetries of the quantum time-dependent harmonic oscillator in fractional dimensions. We investigate the uncertainty relations of the momentum and position operators with respect to these coherent states, and their dependence on the dimension.
In this Chapter we study the Weyl–Heisenberg group in the Schrödinger representation in arbitrary dimension n. One shows that it operates in the Hilbert space of quantum states (and on quantum operators) as a phase-space translation. Then applying it to a Schwartz class state of arbitrary profile we get a set of generalized coherent states. When we apply the Weyl–Heisenberg translation operator to the ground state of the n-dimensional Harmonic Oscillator, one gets the standard coherent states introduced by Schrödinger [233] (Naturwissenshaften 14:664–666, 1926) in the early days of quantum mechanics (1926). Later the coherent states have been extensively studied by Glauber [118, 119] (Phys. Rev. 131:2766–2788, 1963; Phys. Rev. 130:2529–2539, 1963) for the purpose of quantum optics and it seems that their name comes back to this work. The standard coherent states have been generalized by Perelomov [208] (Generalised Coherent States and Their Applications, 1986) to more general Lie groups than the Weyl–Heisenberg group. We also introduce the usual creation and annihilation operators in dimension n which are very convenient for the study of coherent states. We show that coherent states constitute a non-orthogonal over-complete system which yields a resolution of the identity operator in the Hilbert space and which allows a computation of the Hilbert–Schmidt norm and of the trace of respectively Hilbert–Schmidt class and trace-class operators. We study their time-evolution for the quantum Harmonic Oscillator hamiltonian and show that a time evolved coherent state located around phase-space point z is up to a phase a coherent state located around the phase-space point ztz_tzt​, where ztz_tzt​ is the phase-space point of the classical flow governed by the Harmonic Oscillator. This property was described by Schrödinger as the non-spreading of the time evolution of coherent states under the quantum Harmonic Oscillator dynamics. We also show how to go from the Schrödinger to the Fock–Bargmann representation using the standard coherent states.
In this paper, dynamics and quantum mechanical coherent states of a simple harmonic oscillator are considered in the framework of the Generalized Uncertainty Principle (GUP). Equations of motion for the simple harmonic oscillator are derived and some of their new implications are discussed. Then, coherent states of the harmonic oscillator in the case of the GUP are compared with the relative situation in ordinary quantum mechanics. It is shown that in the framework of GUP there is no considerable difference in definition of coherent states relative to ordinary quantum mechanics. But, considering expectation values and variances of some operators, based on quantum gravitational arguments, one concludes that although it is possible to have complete coherency and vanishing broadening in usual quantum mechanics, gravitational induced uncertainty destroys complete coherency in quantum gravity and it is not possible to have a monochromatic ray in principle.
This is a pedagogical paper where we present a physically motivated approach to introduce the coherent states of a harmonic oscillator from which it is simple to rigorously derive their mathematical definition. We do this in two different ways that turn out to be equivalent but emphasize two related but different aspects of classicality. First, we analyze which are the quantum states that are the closest one can get to a point in phase space and demonstrate the validity of the following theorem: (i) The product of the uncertainty in position and that of momentum saturates the bound imposed by Heisenberg uncertainty relations for all times if and only if the state is an eigenstate of the annihilation operator. Second, we analyze the way in which the difference between the expectation value of the energy and the energy associated with the expectation values of position and momentum depends on the state, and show the validity of the following theorem (ii) the difference between the expectation value of the energy and the energy associated with the expectation values is minimal if and only if the state is an eigenstate of the annihilation operator. We also show that the reason why coherent states are chosen as the most classical ones by the decoherence process induced by coupling the particle to an environment in the standard Quantum Brownian motion model, is precisely due to the validity of the two above theorems.