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Connections between perturbation theory and the Van Vleck transformation: Illustrative calculations on the perturbed harmonic oscillator
Author(s) -
Westhaus Paul
Publication year - 1982
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.560210309
Subject(s) - unitary transformation , unitary state , harmonic oscillator , perturbation (astronomy) , eigenvalues and eigenvectors , stationary state , poincaré–lindstedt method , ground state , transition of state , quantum mechanics , perturbation theory (quantum mechanics) , formalism (music) , physics , anharmonicity , mathematical physics , classical mechanics , coherent states , quantum , political science , law , visual arts , art , musical
We continue to examine the connection between perturbation theory and the Van Vleck unitary transformation. Here we illustrate the formalism derived earlier by applying it to compute the stationary states of the perturbed harmonic oscillator. We find that each solution of the traditional Brillouin‐Wigner perturbation theory equations gives rise to a different unitary transformation which, when operating on the unperturbed ground state, produces one or the other of the perturbed eigenstates. With any of the perturbed states able to be reached by a unitary transformation on the unperturbed ground state, we advise caution in using approximate solutions of the perturbation equations in general cases, lest an unexpected stationary state be obtained.