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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• Berry Curvature in Graphene: A New Approach
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• General relativity effects in precision spin experimental tests of fundamental symmetries
• Effective Quantum Theories for Transport in Inhomogeneous Systems with Non-trivial Band Structure
• Semiclassical quantization of electrons in magnetic fields: the generalized Peierls substitution
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• Validation of the Eriksen Method of the Exact Foldy-Wouthuysen Representation
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• Dirac Particles in a Gravitational Field
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