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On formulas in closed form for Van Vleck expansions
Author(s) -
Gora Edwin K.
Publication year - 1972
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.560060412
Subject(s) - mathematics , operator (biology) , canonical form , canonical transformation , matrix (chemical analysis) , rank (graph theory) , transformation (genetics) , bilinear form , phase space , pure mathematics , type (biology) , physics , quantum mechanics , combinatorics , ecology , biochemistry , chemistry , repressor , biology , transcription factor , quantum , gene , materials science , composite material
Canonical transformations have been widely used to simplify Hamiltonians and other operators. In molecular and in solid state theory, the so‐called Van Vieck expansion is usually employed for this purpose while in theories of particles interacting with fields a combination of canonical transformations in closed form with Van Vleck type expansions has been found effective. For some of the transformations used in applications formulas in closed form are well known. It will be shown here that such formulas can be derived whenever the transformation function is bilinear in the canonical variables, and further that the use of matrix operators makes it possible to simplify these derivations substantially. The Cayley‐Hamilton theorem is then used to express the expansions for the matrix operators in closed form. The number of separate operator terms appearing in the formulas thus obtained is the same as the rank of the matrices used. To calculate the coefficients of these operator terms a new type of special functions is introduced. The resulting linear canonical transformations include generalized rotations in both ordinary and phase‐space. Explicit results have been obtained for several two‐ to four‐dimensional problems.