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Exponential Transformations for the Harmonic Chain with a Molecular Defect
Author(s) -
Brühl S.,
Sigmund E.,
Wagner M.
Publication year - 1977
Publication title -
annalen der physik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.009
H-Index - 68
eISSN - 1521-3889
pISSN - 0003-3804
DOI - 10.1002/andp.19774890310
Subject(s) - formalism (music) , eigenvalues and eigenvectors , exponential function , hamiltonian (control theory) , physics , lattice (music) , transformation matrix , mathematical physics , quantum mechanics , mathematical analysis , mathematics , art , musical , mathematical optimization , kinematics , acoustics , visual arts
The exponential transformation, developed in an earlier paper [1], is applied to the Hamiltonian of a linear harmonic chain with a molecular defect. The resulting eigenvalue equation is solved for the localized frequency. A discussion of the renormalized in‐band frequencies shows that in good approximation the entire Hamiltonian is diagonalized by a single transformation. This is of great advantage, since in the classical Lifshitz formalism each single frequency has to be evaluated separately. Furthermore, a simpler transformation is discussed, which is derived from an U ‐matrix formalism. Numerical results of the two transformations are given for a chain with 999 lattice points and compared with the exact values from the classical Lifshitz formalism.