Classical and quantum mechanics of the damped harmonic oscillator


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harmonic oscillator

Dodonov and S. Mizrahi, Ann. Caldirola, Nuovo Cimento , 18 , No. CrossRef Google Scholar. Kanai, Progr.

L14.3 Algebraic solution of the harmonic oscillator.

Mancini, V. Man'ko, and P. Tombesi, Phys. Man'ko, J. Laser Res. Vogel and H. Risken, Phys. A 40 , Cahill and R. Glauber, Phys. Wigner, Phys. Sudarshan, Phys. Husimi, Proc. Japan , 23 , Kano, J. Bagrov, V. Belov, and A. Trifonov, Ann. Belov and V. Maslov, Dokl. Harmonic oscillator; Wave function; Hamiltonian; Eigen value.

The laws of nature discovered were all from the Newtonian mechanics. The classical physics was concerned with those aspect of nature for which question of the ultimate constitution of matter was not of immediate concern. Classical theories are phenomenological theories which attempt to describe and summarize experimental facts in limited domain of physics. But, in the case of microscopic world, it has been proved that classical theories were not of universal validity [ [1] - , [2] , [3] ].


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These laws describe behavior of mechanism of rigid bodies with respect to some material constants such as density, elasticity etc. But they are unable to explain why density has that value at certain specific physical condition, why wire breaks after exceeding limits, etc. It also fails to explain why metallic rod turns white from red when temperature is increased.

Harmonic oscillator

These problems bring revolution in physics through the discovery of quantum mechanics, which was a total surprise to the scientists. It describes the physical world in a way that was fundamentally new. In early stages, quantum theory seems to be a poor substitute for classical laws, but later from various experiments it was realized by the scientific community. In this paper special emphasis is given on the main building blocks of quantum mechanics which solve many problems, which in turn helps in developing various concepts in modern physics [ [4] - , [5] , [6] , [7] ]. After the discovery of radiators of energy bundles for black body radiations by Max Planck in , many quantum mechanical systems were evolved.

The harmonic oscillator is one of the most important quantum systems which can be solved more accurately and hence is of great interest. This concept had been in use since classical era to study vibrations of atoms or molecules in various conditions in solids. After the development of quantum theory in last century, people are now aware about the building block rules [ [8] - , [9] , [10] , [11] ]. But, the quantum behavior of free particles or particles constrained inside different potentials cannot be determined easily. Setting up such problem in terms of harmonic oscillator system helps to calculate required energy levels.

We are familiar that among all other branches in Physics, quantum mechanics is an essential branch which got enormous fame in first half of 20 th century, before which people were in belief for laws of classical mechanics.


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When students entered in undergraduate level of their studies, they were having a high impact of Newtonian mechanical laws. At higher studies, when these students came across concepts from quantum mechanics, it seems to be contradictory to the conventional theories. Hence many times various concepts of quantum theories become difficult to deal with.

There are hardly any efforts are taken to resolve issues while tackling large derivations particularly for a problem of harmonic oscillator [ [12] - , [13] , [14] , [15] , [16] , [17] ]. Many researchers attempted to bridge the gap and tried to solve problems in easier way, but with individual techniques. This pitfall itself is a motivation for present work. Thrust of this paper is to explore the theory of harmonic oscillator at a glance, which common readers seek to comprehend the mysterious world of quantum physics.

1. Introduction

The main advantage of this article is that the derivations are discussed in the light of implications for novices who have a keen interest to know more about the field. This paper summarizes derivations of 1-D harmonic oscillator with detailed description of equations used along with their significance and elaborating simple mathematical steps involved in deriving it.

It is an attempt to highlight basic concepts in quantum mechanics initializing from an empirical relation such as kinetic energy, potential functions, Hamiltonian, wave functions etc of the system and their applications in evaluating eigen energy values. Though a reader may find it much detailed, it is essential to present all the derivational steps to cater the needs of a beginner. Material and method. Harmonic oscillator performs undamped simple harmonic motion which is periodic with constant amplitude. The main features of harmonic oscillator; though they may be considered in various conditions, are its positive amplitude, periodicity, its phase determination and frequency, on the basis of which wave function can be assumed.

Harmonic oscillator is always defined in terms of second order differential equation whose solution is to be assumed to evaluate the energy levels. Classically, initial energy of the oscillator is due to its restoring force, which brings oscillator to the equilibrium position when displaced.


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  • But, quantum mechanically; the initial energy is determined in terms of a Hamiltonian, which consists of kinetic and potential energies of the system [ [18] - , [19] , [20] , [21] ]. The steps involved in determining energy eigen values for 1-D harmonic oscillators with different methods is highlighted in the flow chart shown in Fig. Figure 1. Flow chart for evaluating eigen values. Initially, the system of harmonic oscillator is described by considering various aspects. Under classical case, these conditions can be employed to determine potential of the system.

    However, a Hamiltonian is to be calculated if the system is in quantum state. The Hamiltonian is required if kinetic and potential energies are known. The assumed potential is utilized to setup a wave equation of second order. But, to solve such equations, it is a need to select some wave function which is based on the parameters required to evaluate. The result obtained by solving these equations is the eigen energy values of harmonic oscillator.

    These values are utilized for further solving the complex problems in which harmonic oscillators are to be used. The block diagram for various methods considered in this paper to solve harmonic oscillator problem in 1-D is shown in Fig.

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    Harmonic oscillator - Wikipedia

    Figure 2. Block diagram for different techniques to solve the Harmonic oscillator. Linear Harmonic oscillator by classical mechanics. Another, trivial consequence is that the decay rates of a quantum and a classical harmonic oscillator have to be the same. Note, however, that this is no longer the case if you add some sort of nonlinearity into the system. If you're able to measure the photon number in the real time, you will see photons entering and leaving the oscillator one by one, which is qualitatively very different from exponential decay.

    You'd have to observe many such trajectories starting from the same initial state and average them to see the exponential decay. And if you want to describe the decay of an atom, you have to use quantum mechanics. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Why does a damped quantum harmonic oscillator have the same decay rate as the equivalent classical system? Ask Question. Asked 4 years, 7 months ago.

    harmonic oscillator

    Active 2 years, 2 months ago. Viewed times. Example systems This system is realized many ways in nature. SRS 6, 4 4 gold badges 38 38 silver badges bronze badges. DanielSank DanielSank So it wasn't exactly how they behaved, but it was close enough.

    Classical and quantum mechanics of the damped harmonic oscillator Classical and quantum mechanics of the damped harmonic oscillator
    Classical and quantum mechanics of the damped harmonic oscillator Classical and quantum mechanics of the damped harmonic oscillator
    Classical and quantum mechanics of the damped harmonic oscillator Classical and quantum mechanics of the damped harmonic oscillator
    Classical and quantum mechanics of the damped harmonic oscillator Classical and quantum mechanics of the damped harmonic oscillator
    Classical and quantum mechanics of the damped harmonic oscillator Classical and quantum mechanics of the damped harmonic oscillator
    Classical and quantum mechanics of the damped harmonic oscillator Classical and quantum mechanics of the damped harmonic oscillator
    Classical and quantum mechanics of the damped harmonic oscillator Classical and quantum mechanics of the damped harmonic oscillator

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