Electro and Mechanical Oscillators

Stanislav Ordin

Published Oct 24, 2024• 0 citations

Abstract

In order to determine the specifics of the quantization of motion of charged particles, a comparative analysis of the mechanical oscillator with the electro-mechanical and purely electrical oscillators was carried out. It has been shown that the use of the standard harmonic oscillator equation for quantization does not take into account the effect of electric charge inertia, which is taken into account when using the Heaviside impedance. The formal addition to the equation of the harmonic oscillator of a term determined by the inertia of the charge leads to contradictions. So, in order to UNDERSTAND quantization, a more physical refinement of the Schrödinger equation, embedded in the fundamentals of quantization, is required.

DOI: 10.1201/9781003570615-17

Related papers

Physical model of Schrodinger electron. Feynman convenient way in mathematical description of its quantum behaviour

Josiph Mladenov Rangelov

The physical model of a nonrelativistic quantized Schrodinger's electron (SE) is offered. The behaviour of the SE well spread elementary electric charge had been understood by means of two independent and different in a frequency and size motions. The description of this resultant motion may be done by substitution of the classical Wiener continuous integral with the quantized Feynmam continuous integral. There are possibility to show by means of the existent not only formal but substantial analogy between the quadratic differential wave equation in partial derivatives of Schrodinger and quadratic differential particle equation in partial derivatives of Hamilton-Jacoby that the addition of a kinetic energy of the stochastic harmonic oscillation of some quantized micro particles to the kinetic energy of classical motion of the same micro particles formally determines their wave behaviour.It turns out the stochastic motion of the quantized micro particles powerfully to break up the smooth thin line of the classical motion of the same micro particle in many broad cylindrically spread path. The SE participate in stochastically roughly determined circumferences within different flats and with different radii, with centres which are successively arranjed over short and very disorderly orientated lines. Therefore the quantized motion of some micro particle cannot be descripted by smooth thin well contured (focused) line, describing the classical motion of the macro particle.

arXiv:quant-ph/0001078Jan 21, 20000 citations

To what extent do the classical equations of motion determine the quantization scheme?

J. Cislo, J. Lopuszański

A simple example of one particle moving in a (1+1) space–time is considered. As an example we take the harmonic oscillator. We confirm the statement that the classical equations of motion do not determine at all the quantization scheme. To this aim we use two inequivalent Lagrange functions, yielding Euler–Lagrange equations, having the same set of solutions. We present in detail the calculations of both cases to emphasize the differences occuring between them.

arXiv:math-ph/0012005Journal of Mathematical PhysicsNov 1, 200112 citations

Quantum Harmonic Oscillator

Z. Ficek

The study of quantum properties of the harmonic oscillator is important in physics as many real-world systems oscillate harmonically. Motion of systems in a confined space is often modeled as being a quantized harmonic motion or, in first instance, is approximated by a harmonic motion. This chapter helps the reader to solve the equation employing an algebraic operator technique, which is based on the Dirac notation. This approach has several definite advantages and exploits the commutation relations among the operators involved and their properties. The chapter transforms the stationary Schrodinger equation into a second-order differential equation and finds the solution to the equation with the aid of a more advanced mathematical technique that involves special functions. The energy of a harmonic oscillator is quantized, with the sequence of values. The difference in energy between adjacent energy levels is equal to the energy of a single photon, ħω.

Mar 3, 20171 citation

Quantized Energy Distributions for the Harmonic Oscillator

Bayard Rankin

An earlier study of quantum-mechanical time has suggested that, from the quantum-theoretic point of view, time is a random variable. The present work explores how such a description of time can be used to bring greater detail into the analysis of the one-dimensional harmonic oscillator. Specifically, the classical equations for harmonic motion are used as a point transformation between phase space and time-energy space, permitting probability distributions in the two spaces to be different representations of the same thing. The familiar energy-quantization hypothesis is supplemented with the hypothesis that quantized energy values are means (averages) of energy distributions characteristic of each energy state. With the probability distribution of time determined from the previously cited study, the desired quantized energy distributions follow from the requirements: (1) that they should be normalized, non-negative, and with means equal to the quantized energy values; (2) that the equivalent representation of distribution in position-momentum space should have marginal densities that are proportional (except near the origin) to the densities in position and momentum, ${|\ensuremath{\phi}(q)|}^{2}$ and ${|\ensuremath{\psi}(p)|}^{2}$, which are postulated in quantum theory. Requirement (2) is a weakening of the Born postulate. In the time-energy space, the probability distributions exclude energies smaller than the mode (energy value with maximum probability). In position-momentum space this exclusion requirement becomes the dynamical restriction: $\frac{{q}^{2}m{\ensuremath{\omega}}^{2}}{2}+\frac{{p}^{2}}{2m}\ensuremath{\ge}\frac{{\ensuremath{\Lambda}}_{n}\ensuremath{\hbar}\ensuremath{\omega}}{2}$, where, for the first three energy states, ${\ensuremath{\Lambda}}_{0}=0$, ${\ensuremath{\Lambda}}_{1}=\frac{3}{2}$, ${\ensuremath{\Lambda}}_{2}=2+\ensuremath{\surd}\frac{3}{2}$. In general terms it is thus shown that the quantized harmonic oscillator can be described as a statistical ensemble of classical oscillators. Some comparisons with work on phase-space formulations of quantum mechanics (especially with the work of Wigner) are made and some aspects of the theory that should submit to experimental verification are pointed out.

Physical ReviewJan 28, 19662 citations

Langevin Quantization of the Time-Dependent Harmonic Oscillator

Antonio Defendi, Marco Roncadelli

As an application of the recently proposed Langevin formulation of quantum dynamics, we quantize the time-dependent harmonic oscillator. In this way, we are able to explicitly compute the quantum-mechanical propagator. At variance with previous results, the expression found in this letter contains no auxiliary function to be determined by solving (as yet unsolved) non-linear differential equations, but it is written in terms of two series which arise quite naturally within the present formalism. Our strategy is compared with ordinary time-dependent perturbation theory.

Europhysics Letters (EPL)Jul 20, 19933 citations

Canonical quantization of fields

Yisong Yang

Abstract This chapter begins its exploration of field quantization with an illustration of how a harmonic oscillator may be quantized, which motivates and preludes the subsequent discussion on canonical quantization, also known as the second quantization. Then it uses the formalism to quantize the Klein-Gordon and Schrodinger equations. Afterwards, goes on to present the quantization of the Maxwell equations, which establishes the celebrated photoelectric effect through a derivation of the Planck-Einstein relation and Compton-Debye formula. Finally, the chapter concludes with a comparison study of the thermodynamic properties of a harmonic oscillator with respect to its classical and quantum-mechanical descriptions.

Jul 27, 20230 citations