Universal Raising and Lowering Operators for a Discrete Energy Spectrum

S. A. Bugal'ter, G. M. Zhislin

Alberto Devoto, Bojan Pomorisac

Semjon A. Vugalter, G. M. Zhislin

A.H. Amer

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

Luis de la Peña, R. Montemayor

We show that it is possible to give to any (linear and Hermitian) operator with discrete spectrum, acting on a Hilbert space, a canonical form in terms of the commutator and anticommutator of a pair of adequately constructed raising and lowering operators. This construction enables one to investigate in a very compact way the (discrete) spectrum of the given operator, which becomes determined by a set of consistency conditions. We apply the method to several important problems of quantum mechanics.