Recursive Diagonalization of Quantum Hamiltonians to all order in $\hbar$

TL;DR

This paper introduces a recursive method to diagonalize quantum Hamiltonians to all orders in , incorporating Berry phase corrections and quantum effects for particles and electrons, advancing semiclassical analysis.

Contribution

It develops a differential equation approach for recursive diagonalization of Hamiltonians in powers of , extending semiclassical methods with higher-order quantum corrections.

Findings

Quantum corrections of order ^2 for energy and velocity of massless particles.

Formal all-order  expansion for Bloch electron spectra and dynamics.

Enhanced understanding of Berry phase effects in quantum Hamiltonian diagonalization.

Abstract

We present a diagonalization method for generic matrix valued Hamiltonians based on a formal expansion in power of $\hbar $. Considering $\hbar $ as a running parameter, a differential equation connecting two diagonalization processes for two very close values of $\hbar $ is derived. The integration of this differential equation allows the recursive determination of the series expansion in powers of $\hbar $ for the diagonalized Hamiltonian. This approach results in effective Hamiltonians with Berry phase corrections of higher order in $\hbar $, and deepens previous works on the semiclassical diagonalization of quantum Hamiltonians which led notably to the discovery of the intrinsic spin Hall effect. As physical applications we consider spinning massless particles in isotropic inhomogeneous media and show that both the energy and the velocity get quantum corrections of order $\hbar ^{2}$. We also derive formally to all order in $\hbar $ the energy spectrum and the equations of motion of Bloch electrons in an external electric field.