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AIP conference proceedings

Study of damped harmonic oscillator in view of quantum calculus

Pintu Bhattachary, Ravi Ranjan

Published Jan 1, 2024• 0 citations
Publisher

Abstract

We display some important definitions of q-calculus along with some essential formulae of q-Laplace transform. Then, we apply them to realize the locomotion of a damped harmonic oscillator for over-damped, under damped, and critically damped situation. Thus, the mathematical statement of motion of a damped harmonic oscillator (DHO) is revisited and reviewed in view of quantum calculus. The dimensionality of all the parameters associated to the problem are discussed to find exact solutions. It is also found that, as q tends to 1, the solution takes the form as it is in classical Leibnitz calculus.

DOI
10.1063/5.0224388

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We display some important definitions of q-calculus along with some essential formulae of q-Laplace transform. Then, we apply them to realize the locomotion of a damped harmonic oscillator for over-damped, under damped, and critically damped situation. Thus, the mathematical statement of motion of a damped harmonic oscillator (DHO) is revisited and reviewed in view of quantum calculus. The dimensionality of all the parameters associated to the problem are discussed to find exact solutions. It is also found that, as q tends to 1, the solution takes the form as it is in classical Leibnitz calculus.
Using the invariant operator method and the unitary transformation method together, we obtained discrete and continuous solutions of the quantum damped driven harmonic oscillator. The wave function of the underdamped harmonic oscillator is expressed in terms of the Hermite polynomial while that of the overdamped harmonic oscillator is expressed in terms of the parabolic cylinder function. The eigenvalues of the underdamped harmonic oscillator are discrete while that of the critically damped and the overdamped harmonic oscillators are continuous. We derived the exact phases of the wave function for the underdamped, critically damped and overdamped driven harmonic oscillator. They are described in terms of the particular solutions of the classical equation of motion.
In this paper we use the algebraic and the invariant method to study the time-dependent damped harmonic oscillator from classical and quantum points of view. The solution of the classical equation of motion and the wavefunction solving the time-dependent Schrödinger equation are found explicitly. We show that the original time-dependent quantum-mechanical problem is completely related to the well known time-independent harmonic oscillator. In addition, we elucidate the intimate connection between the damped harmonic oscillator (DHO) and the generalized harmonic oscillator (GHO). More importantly, the evolution of the states of the DHO cannot be cyclic, in contradistinction with the states of the GHO. Explicit expressions for both the dynamical and the geometric angles and phases are deduced in the adiabatic limit. The coherent states describing the invariant-angle variables of the classical DHO are constructed; they allow us to recover the classical evolution and invariant from the quantum evolution.
An approach to quantization of the damped harmonic oscillator (DHO) is developed on the basis of a modified Bateman Lagrangian (MBL); thereby some quantum mechanical aspects of the DHO are clarified. We treat the energy operator for the DHO, in addition to the Hamiltonian operator that is determined from the MBL and corresponds to the total energy of the system. It is demonstrated that the energy eigenvalues of the DHO exponentially decrease with time and that transitions between the energy eigenstates occur in accordance with the Schr\"odinger equation. Also, it is pointed out that a new critical parameter discriminates different behaviors of transition probabilities.
In this short paper, a very brief introduction of the quantum harmonic oscillator has been provided. The understanding of the transition from a classically damped oscillator and the quantum mechanical model of the latter is included via a clear mathematics involving simple calculus. Only the one dimensional time dependent Schrodinger’s equation is being studied. This paper is intended to simply provide an analytical description of the subject, there are no advance calculations implying further suggestions in quantum theory.
Using the equation of motion for a linearly damped harmonic oscillator as fundamental equation, a condition has been obtained that must be satisfied by its Lagrangian and a number of functional forms has been critically investigated. The Lagrangians that successfully describe a classical damped oscillator have been quantized using Feynman’s elegant path integral method. However, it is found that the expectation value of energy of the damped oscillator in all the cases becomes infinitely large for t→∞, ruling out the possibility of treating the damped quantum system as a one-body problem.
The present review is devoted to the study of certain aspects of anharmonic, time-dependent and damped oscillator(s) system using different theoretical techniques. A theoretical understanding of these systems is important for application in many problems in physics, mechanics and other fields. We discuss in detail the difficulties in the theoretical analysis of the problem. In particular we discuss here the regular, well-behaved perturbative solution, the large quantum number behaviour of anharmonic oscillator(s) using the technique of coherent states, exact solution of quantum anharmonic oscillators, the electromagnetic radiation emitted by a charged particle executing damped anharmonic oscillator motion using Krylov-Bogoliubov approximation method, use of invariants to obtain solution and coherent states of time-dependent oscillator(s), the derivation of perturbative frequencies of a damped coupled anharmonic oscillators system using suitable canonical transformation in the framework of Hamilton-Jacobi formalism and the quantisation and construction of coherent states of a damped oscillator using time-dependent operators.