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Institution of Engineering and Technology eBooks

Quantum harmonic oscillator and allied topics

Alireza R. Baghai-Wadji

Published Dec 18, 2023• 0 citations
Publisher

Abstract

The discussion in this chapter centers around the ideal and perturbed Quantum Harmonic Oscillator (QHO) and allied topics. The QHO is the quantum mechanical counterpart of the classical harmonic oscillator. QHO is a fundamental quantum system with distinctively rich properties. The Hamiltonian operator describing the QHO is Hermitian and consists of a second-order differential operator term and a quadratic term in the position operator.

DOI
10.1049/sbew569g_ch1

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The discussion in this chapter centers around the ideal and perturbed Quantum Harmonic Oscillator (QHO) and allied topics. The QHO is the quantum mechanical counterpart of the classical harmonic oscillator. QHO is a fundamental quantum system with distinctively rich properties. The Hamiltonian operator describing the QHO is Hermitian and consists of a second-order differential operator term and a quadratic term in the position operator.
Quantum harmonic oscillator (QHO) involves square law potential (x2) in the Schrodinger equation and is a fundamental problem in quantum mechanics. It can be solved by various conventional methods such as (i) analytical methods where Hermite polynomials are involved, (ii) algebraic methods where ladder operators are involved, and (iii) approximation methods where perturbation, variational, semiclassical, etc. techniques are involved. Here we present the general outcomes of the two conventional semiclassical approximation methods: the JWKB method (named after Jeffreys, Wentzel, Kramers, and Brillouin) and the MAF method (abbreviated for “modified Airy functions”) to solve the QHO in a very good precision. Although JWKB is an approximation method, it interestingly gives the exact solution for the QHO except for the classical turning points (CTPs) where it diverges as typical to the JWKB. As the MAF method, it enables very approximate wave functions to be written in terms of Airy functions without any discontinuity in the entire domain, though, it needs careful treatment since Airy functions exhibit too much oscillatory behavior. Here, we make use of the parity conditions of the QHO to find the exact JWKB and approximate MAF solutions of the QHO within the capability of these methods.
This thesis investigates the quantum harmonic oscillator (QHO) through the path integral formulation. It begins with a review of classical dynamics and the standard operator approach to establish a foundational framework. The core of the work involves the systematic construction and evaluation of the QHO propagator, demonstrating how the transition amplitude naturally separates into a classical contribution and quantum fluctuations. Finally, the discussion extends to systems with infinite degrees of freedom, illustrating how the oscillator structure serves as a bridge to quantum field theory and the emergence of second quantization.
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.
The purpose of this work is to introduce, in a simple, intuitive way, the coherent and squeezed states of the quantum harmonic oscillator (QHO), through a series of exercises, which are solved in detail. Starting from the application of a spatial translation to the ground state of the QHO, we introduce the spatial and momentum translations, focusing on their application to the QHO, which leads us to the displacement operator. Next, we introduce the coherent states and examine their basic aspects. We then proceed to give a simple and purely intuitive introduction to the squeezed states and we conclude by identifying the coherent states as states of minimum energy expectation value compared to the respective squeezed states. The reader is assumed to have a basic knowledge of the postulates and the mathematical formalism of quantum mechanics, including the Dirac notation and the ladder operator method of the QHO.
This chapter begins discussing the classical harmonic oscillator and its solution as a second-order differential equation of the general type. Next, the one-particle quantum harmonic oscillator is presented, its corresponding Hamiltonian functionand wave equation followed by the undimensionalization of this wave equation yielding a second-order differential equation of the general type which leads to the general Hermite equation and Hermite polinomials as partial solution of the wave function. The energy for each wave function is obtained from the recursion relation of the Hermite equation. Afterwards, the numerical analysis of the one-particle harmonic oscillator is given by using the Numerov numerical method. Finally, a corresponding Fortran code for this problem is presented in order to plot the wave function according to a pre-determined quantum number. Plots of the wave function obtained from the Fortran code are also given. In the end, some exercises are given to include other information in the Fortran code.(153words).