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Energy eigenvalue spectroscopy
Author(s) -
Blinder S. M.
Publication year - 1967
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560010306
Subject(s) - eigenfunction , eigenvalues and eigenvectors , hamiltonian (control theory) , harmonic oscillator , wave function , moment (physics) , spectrum (functional analysis) , operator (biology) , quantum mechanics , mathematics , quantum , function (biology) , mathematical analysis , physics , chemistry , mathematical optimization , biochemistry , repressor , transcription factor , gene , evolutionary biology , biology
A wave function which is other than an exact eigenfunction, if it obeys appropriate analytical conditions, can be considered to represent the initial configuration of a nonstationary state. In the course of its subsequent time development the quantum system exhibits implicitly its entire eigenvalue spectrum. A method based, in principle, on Fourier analysis of the evolving quantum system is applied to the direct calculation of the energy eigenvalue spectrum. The spectral function is expressed as a moment expansion, in terms of expectation values of powers of the Hamiltonian. When the expansion is truncated, as it must be in any practical application, the corresponding spectral function represents a smeared‐out eigenvalue spectrum. An alternative approximation leads to the quantum‐mechanical method of moments. As the number of terms is increased, the computed spectrum becomes sharper and more accurate. In certain cases the moment expansion can be circumvented, if the action of the evolution operator can be evaluated in closed form. This is equivalent to finding some solution of the time‐dependent Schrödinger equation. The various methods of eigenvalue spectroscopy are applied to the harmonic oscillator.