Quantization of the damped harmonic oscillator is taken as leitmotiv to gently introduce elements of quantum probability theory for physicists. To this end, we take (graduate) students in physics as entry level and explain the physical intuition and motivation behind the, sometimes overwhelming, math machinery of quantum probability theory. The main text starts with the quantization of the (undamped) harmonic oscillator from the Heisenberg and Schroedinger point of view. We show how both treatments are special instances of a quantum probabilistic quantization procedure: the second quantization functor. We then apply the second quantization functor to the damped harmonic oscillator and interpret the quantum dynamics of the position and energy operator as stochastic processes.
Quantization of the damped harmonic oscillator is taken as leitmotiv to gently introduce elements of quantum probability theory for physicists. To this end, we take (graduate) students in physics as entry level and explain the physical intuition and motivation behind the, sometimes overwhelming, math machinery of quantum probability theory. The main text starts with the quantization of the (undamped) harmonic oscillator from the Heisenberg and Schroedinger point of view. We show how both treatments are special instances of a quantum probabilistic quantization procedure: the second quantization functor. We then apply the second quantization functor to the damped harmonic oscillator and interpret the quantum dynamics of the position and energy operator as stochastic processes.
By the time a student of physics is ready to tackle quantum mechanics he has become familiar with the classical harmonic vibrator through many examples, and knows the crucial role it played in the development of Planck's ideas. It is natural then to concentrate on the mathematical aspects of the harmonic oscillator equation in quantum mechanics, the solution of Schrödinger's equation, normalization of the wave-function, calculation of mean values and of matrix elements, leaving the physics to look after itself. At a more advanced level the harmonic vibrator provides the entry into field theories, second quantization etc., and as a general rule it tends to be viewed more as a vehicle for instruction in more interesting matters than as a physical system having its own considerable interest and importance. Here we shall seek to redress the balance and study the vibrator as a thing in itself, without losing sight of the variety of physical problems to which the results can be applied. One especially important set of applications, however, will get little attention at this stage – the vibrations of compound bodies and of extended physical systems provide such wealth of interest as to justify a volume to themselves; and it is to such matters that the whole of the third part of this work will be devoted. As soon as one begins to contemplate the harmonic vibrator one becomes aware of the exceptional nature of its behaviour, in that it conforms more closely than any other system to the classical rules.
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.
Written for a two-semester graduate course in Quantum Mechanics, this comprehensive text helps develop the tools and formalism of Quantum Mechanics and its applications to physical systems. It suits students who have taken some introductory Quantum Mechanics and Modern Physics courses at undergraduate level, but it is self-contained and does not assume any specific background knowledge beyond appropriate fluency in mathematics. The text takes a modern logical approach rather than a historical one and it covers standard material, such as the hydrogen atom and the harmonic oscillator, the WKB approximations and Bohr-Sommerfeld quantization. Important modern topics and examples are also described, including Berry phase, quantum information, complexity and chaos, decoherence and thermalization, nonstandard statistics, as well as more advanced material such as path integrals, scattering theory, multiparticles and Fock space. Readers will gain a broad overview of Quantum Mechanics, as solid preparation for further study or research.
Second quantization is a powerful technique for describing quantum mechanical processes in which the number of excitations of a single particle is not conserved. A textbook example of second quantization is the presentation of the simple harmonic oscillator in terms of creation and annihilation operators, which, respectively, represent addition or removal of quanta of energy from the oscillator. Our aim in this article is to bolster this textbook example. Accordingly, we explore the physics of coupled second-quantized oscillators. These explorations are phrased as exactly solvable eigenvalue problems, the mathematical structure providing a framework for the physical understanding. The examples we present can be used to enhance the discussion of second-quantized harmonic oscillators in the classroom, to make a connection to the classical physics of coupled oscillators, and to acquaint students with systems employed at the frontiers of contemporary physics research.
A series of successive quantizations is considered, starting with the quantization of a non-relativistic or relativistic point particle: (1) quantization of a particle’s position, (2) quantization of wave function, (3) quantization of wave functional. The latter step implies that the wave packet profiles forming the states of quantum field theory are themselves quantized, which gives new physical states that are configurations of configurations. In the procedure of quantization, instead of the Schrödinger first-order equation in time derivative for complex wave function (or functional), the equivalent second-order equation for its real part was used. In such a way, at each level of quantization, the equation a quantum state satisfies is just like that of a harmonic oscillator, and wave function(al) is composed in terms of the pair of its canonically conjugated variables.
Abstract By selecting a right generalized coordinate X, which contains the general solutions of the classical motion equation of a forced damped harmonic oscillator, we obtain a simple Hamiltonian which does not contain time for the oscillator such that Schrödinger equation and its solutions can be directly written out in X representation. The wave functin in x representation are also given with the help of the eigenfunctions of the operator X^ in x representation. The evolution of ⟨x^⟩ is the same as in the classical mechanics, and the uncertainty in position is independent of an external influence; one part of energy mean is quantized and attenuated, and the other is equal to the classical energy.