Diagonal Representation for a Generic Matrix Valued Quantum Hamiltonian

Pierre Gosselin (1) , Herve Mohrbach (2, 3)

1 IF - Institut Fourier
2 FCN - Fédération de Chimie de Nancy
3 LPMC - Laboratoire de physique moléculaire et des collisions

A general method to derive the diagonal representation for a generic matrix valued quantum Hamiltonian is proposed. In this approach new mathematical objects like non-commuting operators evolving with the Planck constant promoted as a running variable are introduced. This method leads to a formal compact expression for the diagonal Hamiltonian which can be expanded in a power series of the Planck constant. In particular, we provide an explicit expression for the diagonal representation of a generic Hamiltonian to the second order in the Planck constant. This last result is applied, as a physical illustration, to Dirac electrons and neutrinos in external fields.

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]; Autre [cond-mat.other]; Physique des Hautes Energies - Théorie [hep-th]; Physique Quantique [quant-ph]

Pierre Gosselin, Herve Mohrbach. Diagonal Representation for a Generic Matrix Valued Quantum Hamiltonian. 2009. ⟨hal-00202491v5⟩