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

Abstract

We present a diagonalization method for generic matrix valued Hamiltonians based on a formal expansion in power of $\hslash$. Considering $\hslash$ as a running parameter, a differential equation connecting two diagonalization processes for two very close values of $\hslash$ is derived. The integration of this differential equation allows the recursive determination of the series expansion in powers of $\hslash$ for the diagonalized Hamiltonian. This approach results in effective Hamiltonians with Berry phase corrections of higher order in $\hslash$, 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 ${\hslash }^2$. We also derive formally to all order in $\hslash$ the energy spectrum and the equations of motion of Bloch electrons in an external electric field.

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• Dissipationless Quantum Spin Current at Room Temperature
• Conservation of Angular Momentum, Transverse Shift, and Spin Hall Effect in Reflection and Refraction of Electromagnetic Wave Packet
• Modified geometrical optics of a smoothly inhomogeneous isotropic medium: the anisotropy, Berry phase, and the optical Magnus effect
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Cited by in corpus (3)

• Berry Curvature in Graphene: A New Approach
• Semiclassical quantization of electrons in magnetic fields: the generalized Peierls substitution
• Diagonal Representation for a Generic Matrix Valued Quantum Hamiltonian