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Overview on the Hilbert Space Harmonics Oscillator

Rinto Agustino, Saiman Saiman

Published Oct 1, 2014• 0 citations
OpenAlex

Abstract

Abstract. Have been calculated harmonics oscillator with Hilbert Space review. Study of harmonics oscillator in classical mechanics usually uses Newtonian and Lagrange mechanicswhile in quantum mechanics using Hamiltonian. Keywords: Hilbert Space, harmonics oscillator

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Abstract In both classical and quantum physics the harmonic oscillator is one of the most important physical systems [1–4]. Furthermore, because of the simplicity and symmetry of the Hamiltonian operator, the quantum mechanical oscillator is particularly suited to analysis by abstract operator methods. For this reason the study of the one dimensional oscillator is a useful introduction to the application of operator methods in quantum mechanics. In Sections 2.1–2.4 we use these methods to determine energy eigenvalues and eigenkets, and matrix elements; the analysis is extended to isotropic multidimensional oscillators in Section 2.10. Quasiclassical states of the oscillator are discussed in Section 2.5. In Section 2.6 shift operators are used to calculate wave functions. In Sections 2.7–2.11 we discuss some applications of the theory of the oscillator.
Starting with a real abstract algebra which encapsulates the algebraic structures of both classical and quantum mechanics, this paper presents a self–contained real-ization of the latter in phase space. Having both mechanics formulated in the same space opens two new windows into the comparative study of foundations. This stems from the fact that the same physical problem, as defined by a given Hamil-tonian, can be solved in several independent ways. The exact solutions can then be compared. Thus, comparison of the classical and quantum solutions in phase space offers new epistemological insights into Bohr’s correspondence principle, while com-parison of the quantum solutions in the different formalisms of Hilbert space and phase space yields new physical insights. These general ideas are then tested on the harmonic oscillator. The analytic ground work is presented in Part I, the exact solutions will be derived in Part II. 1.
A simple harmonic oscillator, or oscillator in short, is one of the most important systems in physics as well as in quantum mechanics. The path integral and Hamiltonian of a simple harmonic oscillator are leading exemplars for the description of a wide class of physical systems. An oscillator is a theoretical model of great utility, primarily due to its simplicity that allows for the exact derivation of many results. The simple harmonic oscillator can be generalized to the case of infinitely many oscillators, and is also the starting point for analysis of many nonlinear quantum systems. Path integrals for the simple harmonic oscillator are all based on Gaussian integration, which has been discussed in Section 7.2. The harmonic oscillator is described by Gaussian path integrals , and is a bedrock of the general theory of path integration. Moreover, all perturbation expansions of nonlinear path integrals about the oscillator path integral, which includes the semi-classical expansion, are based on results of Gaussian integration. The properties of the harmonic oscillator are analyzed in the coordinate space representation, and its path integral as well as its correlation functions are analyzed. All formulas and derivations are for Euclidean time. In Sections 11.1 and 11.2 the Hamiltonian and state space of the oscillator are studied and the correlator is derived using state space methods. In Section 11.2 the infinite time oscillator is introduced and in Sections 11.3–11.5 the path integral for the oscillator is evaluated.