CRC Press eBooks

Linear Harmonic Oscillator in the Energy Representation

A.H. Amer

Published Nov 5, 2025• 0 citations

Abstract

This chapter examines the linear harmonic oscillator in the energy (Fock) representation, emphasizing its algebraic structure and spectral properties. It is shown that the energy spectrum is discrete and bounded from below. The energy raising and lowering (ladder) operators are introduced, and the energy eigenvalues are derived. The eigenstates are constructed algebraically from the ground state by repeated application of the raising operator, forming the Fock basis. These energy eigenstates are then transformed into the coordinate representation, where the corresponding wavefunctions are expressed in terms of Hermite polynomials. This establishes a connection between the algebraic and coordinate formulations of the quantum harmonic oscillator. Metadata/Keywords: linear harmonic oscillator, Fock representation, ladder operators, energy spectrum, energy eigenstates, ground state, Hermite polynomials, raising and lowering operators

DOI: 10.1201/9781003677109-10

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