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The Quantum Harmonic Oscillator

Alexander Eichler, Oded Zilberberg

Published Sep 26, 2023• 0 citations
Publisher

Abstract

Abstract In this chapter, we enter the quantum regime. Following a succinct introduction to the topic, we present the main properties of the quantum harmonic oscillator, such as the ground state energy caused by quantum fluctuations, the harmonic energy spectrum of Fock states, coherent states, the density matrix of mixed states and its time evolution calculated with the von Neumann master equation, and various quantum state representations.

DOI
10.1093/oso/9780192862709.003.0009

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Abstract In this chapter, we enter the quantum regime. Following a succinct introduction to the topic, we present the main properties of the quantum harmonic oscillator, such as the ground state energy caused by quantum fluctuations, the harmonic energy spectrum of Fock states, coherent states, the density matrix of mixed states and its time evolution calculated with the von Neumann master equation, and various quantum state representations.
The time evolution of a randomly modulated quantum harmonic oscillator is studied by introducing the master equation for the reduced density operator s(t)s(t)s(t). The antinormally ordered representation is adopted for s(t)s(t)s(t), and one is thus confronted with the problem of solving the partial differential equation obeyed by the reduced matrix element s‾\overline{s}s of s(t)s(t)s(t). The two limiting cases of short- and long-range correlations of the frequency fluctuations are treated in detail.
Abstract The equilibrium properties of an open harmonic oscillator are considered in three steps: first the creation and destruction operators are generalized for open dynamics and the creation operator is used to construct coherent states. The second step consists of the introduction of the Heisenberg representation where the dynamical decoherence is identified. Finally it is pointed out that the quantum fluctuations generate non-continuous limit for infinitesimal system–environment interactions and at the border of the under- and over-damped oscillator.
Abstract The ground state of the linear harmonic oscillator is the solution to a minimization problem, and this opens up two new ways to completely solve Schrödinger’s time-independent equation. The first exploits systematic patterns that emerge from a formal relationship between two different integral representations of the energy. A remarkable invariant form of the ground-state probability-density equation under specific conditions is the leading principle of the second procedure, where a uniform notation for all mathematical object involved facilitates calculations and generates useful schemes to quickly deduce the main functional relationships of the problem. We believe these procedures may represent useful educational alternatives to standard treatments that address this fundamental problem.
Operator algebra techniques are employed to derive the quantum evolution operator for the harmonic oscillator. The derivation begins with the construction of the annihilation and creation operators and the determination of the wave function for the coherent state as well as its time-dependent evolution, and ends with the transformation of the propagator in a mixed position-coherent-state representation to the desired one in configuration space. Throughout the entire procedure, besides elementary operator manipulations, it is only necessary to solve linear differential equations and to calculate Gaussian integrals.
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 work is devoted to the investigation of the influence of a heat bath on the physical processes in a quantum system. We use the density matrix theory as one of the most powerfool tool for investigation of quantum relaxation. In the beginning of the work (chapter 2) we mention and recall the most important steps of derivation of the equation of motion for the reduced density matrix (master equation) for an arbitrary quantum system in diabatic representation interacting with the environment modeled by a set of independent harmonic oscillators. Chapter 3 deals with the question of the border between classical and quantum effects and reports on a study of the environmental influence on the time evolution of a coherent state or the superposition of two coherent states of a harmonic oscillator as a simple system displaying the peculiarities of the transition from quantum to classical regime. Chapters 4 and 5 concern the electron transfer (ET) problem, namely the mathematical description of the ET in molecular zinc-porphyrin-quinone complexes modeling artificial photosynthesis (chapter 4) and photoinduced processes in the porphyrin triad (chapter 5). Each chapter starts with an introduction and ends with a brief summary. The main achievements of the present work are summarized in the Conclusions.