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Theory of Quantization of Temperature in Linear Harmonic Oscillator

Poora Ram

Published Jan 1, 2016• 0 citations
Publisher

Abstract

Here a theoretically relation is presented between the frequency and the temperature of a linear harmonic oscillator. A mathematical derivation is proved here, that shows how frequency of linear harmonic oscillator varies with the temperature. This is also shown that, at given frequency of oscillator only a minimum value of temperature can be found out till at particular limit. This theory also gives a new concept of quantization of temperature for harmonic oscillator or all the bodies whose are constructed of such type of oscillator. Term of uncertainty in temperature is also explained in this theory.

DOI
10.13140/rg.2.2.36670.31046

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Here a theoretically relation is presented between the frequency and the temperature of a linear harmonic oscillator. A mathematical derivation is proved here, that shows how frequency of linear harmonic oscillator varies with the temperature. This is also shown that, at given frequency of oscillator only a minimum value of temperature can be found out till at particular limit. This theory also gives a new concept of quantization of temperature for harmonic oscillator or all the bodies whose are constructed of such type of oscillator. Term of uncertainty in temperature is also explained in this theory.
As known all physical properties of solids are described well by the system of quantum linear harmonic oscillators. It is shown in the present paper that the system consisting of classical linear harmonic oscillators having temperature dependent masses or (and) frequencies has the same partition function as the system consisting of quantum linear harmonic oscillators having temperature independent masses and frequencies while the means of the square displacements of the positions of the oscillators from their mean positions for the system consisting of classical linear harmonic oscillators having: the temperature dependent masses; temperature dependent frequencies; and temperature dependent masses and frequencies differ from each other and from that of the system consisting of quantum linear harmonic oscillators, and hence, the system consisting of classical linear harmonic oscillators describes well the thermodynamic properties of the system consisting of quantum linear harmonic oscillators and solids.
The Gaussian noise model for a harmonic oscillator subjected to random linear fluctuations (e.g., thermal noise) is used to derive generalized Heisenberg and entropic uncertainty relations for the position-momentum and number-phase observables. The lower bounds are partitioned into pure quantum and pure noise terms, and the minimum-uncertainty states are determined. Formally similar results of Abe and Suzuki [Phys. Rev. A 41, 4608 (1990)] for the position-momentum case, derived using the thermofield formalism, are shown to correspond to amplification noise rather than thermal noise.
The harmonic oscillator is defined by a quadratic potential. Many important problems in physics reduce to that of a quantum harmonic oscillator because the leading term of a potential near its minimum is quadratic. It is a general and noteworthy feature of quantum systems that the lowest energy is not zero. Calculate the probability that the quantum oscillator is found beyond the classical limit. Agreement with the uncertainty principle may be checked by calculating the expectation values of x2 and p2 and then taking the square root of the product. For large quantum numbers, the solution must approach that of the classical harmonic oscillator.
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 concept of harmonic oscillator particularly one dimensional (1-D) is mentioned in literature repeatedly and is explained in more complex manner by using various methods. This creates difficulties in understanding the description of the concept for new learners. The purpose of this article was to enlighten different methods to formulate harmonic oscillator in improving knowledge about detailed steps to derive eigen energy values in more comprehensible manner for the beginners. The energy values are derived by using classical method, quantum mechanically, Schrodinger time independent equation, perturbation technique, variation method, WKB approximation etc. A coherent way of derivation of eigen values using various approaches makes this article as unique.