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arXiv:1510.04056

Calculation of quantum eigens with geometrical algebra rotors

A. Dargys, A. Acus

Published Oct 14, 2015• 0 citations
arXivHTMLPDF

Abstract

A practical computation method to find the eigenvalues and eigenspinors of quantum mechanical Hamiltonian is presented. The method is based on reduction of the eigenvalue equation to well known geometric algebra rotor equation and, therefore, allows to replace the usual det(H-E)=0 quantization condition by much simple vector norm preserving requirement. In order to show how it works in practice a number of examples are worked out in Cl_{3,0} (monolayer graphene and spin in the quantum well) and in Cl_{3,1} (two coupled two-level atoms and bilayer graphene) algebras.

DOI
10.48550/arxiv.1510.04056

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A practical computation method to find the eigenvalues and eigenspinors of quantum mechanical Hamiltonian is presented. The method is based on reduction of the eigenvalue equation to well known geometric algebra rotor equation and, therefore, allows to replace the usual det(H-E)=0 quantization condition by much simple vector norm preserving requirement. In order to show how it works in practice a number of examples are worked out in Cl_{3,0} (monolayer graphene and spin in the quantum well) and in Cl_{3,1} (two coupled two-level atoms and bilayer graphene) algebras.
Abstract We present a simple algebraic procedure that can be applied to solve a range of quantum eigenvalue problems without the need to know the solution of the Schrödinger equation. The procedure, presented with a pedagogical purpose, is based on algebraic consistency conditions that must be satisfied by the eigenvalues of a couple of operators proper of the problem. These operators can be either bilinear forms of the raising and lowering operators appropriate to the problem, or else auxiliary operators constructed by resorting to the factorization of the Hamiltonian. Different examples of important quantum-mechanical textbook problems are worked out to exhibit the clarity and simplicity of the algebraic procedure for determining the spectrum of eigenvalues without knowing the eigenfunctions. For this reason the material presented may be particularly useful for undergraduate students or young physicists.
The number of exactly solvable eigenvalue problems has been rapidly increasing since the advent of quantum mechanics and there are many efficient ways to treat them. As a general rule, it is possible to write exactly solvable abstract eigenvalue equations in terms of the generators of finite-dimensional Lie algebras and solve them by means of purely algebraic methods. Matrix elements are commonly necessary for the application of perturbation theory and variational principles, among other approximate methods. The chapter discusses examples of three-dimensional Lie algebras with well-known quantum-mechanical realizations. Typically, exactly solvable quantum-mechanical problems are either onedimensional models or lead to separable equations in many dimensions. In the coordinate representation both cases reduce to ordinary differential equations of second order that determine the eigenvalues and separation constants compatible with the physical boundary conditions. In its purest form, a Lie algebraic method is representation-independent. Extensive application of algebraic methods to selected physical problems is already available elsewhere.
In the present thesis, the properties of the entanglement spectra of the ground state of graphene bilayers Bernal-stacked is analytically investigated. In addition, the entanglement spectra of the ground state of the honeycomb lattice in the presence of superconductivity instabilities are analytically studied. We consider not only the eigenvalues of the reduced density matrix that form the entanglement spectrum, but also, its eigenstates. From these eigenstates, we construct topological quantities, such as the Berry phase and the Chern number, in order to investigate the topological properties of the entanglement spectrum. In the first part of this thesis, we present an analytical study of the graphene bilayers, mainly focusing on the effects of the trigonal warping. When the term causing the trigonal warping is neglected, the entanglement Hamiltonian obtained by tracing out one layer shows a proportionality with the energetic Hamiltonian of the remaining monolayer graphene, in the limit of strongly coupled layers. We demonstrate that this proportionality leads to an agreement of the topological quantities of these Hamiltonians. Moreover, we demonstrate that the entanglement spectrum of graphene bilayers with a trigonal warping spectrum clearly differs in the geometric shape from the energy spectrum of the remaining monolayer graphene. However, there is an agreement of the topological quantities such as Berry phase contributions to Chern numbers. In the second part of this thesis, we give a detailed study of graphene in the presence of superconductivity instabilities, mainly considering the s-wave and the chiral dx2−y2 +idxy state.We investigate the relationship between the entanglement and energy spectrum, making use of the concepts of the Chern number constructed from eigenstates of the entanglement Hamiltonian. We demonstrate that the entanglement and remaining subsystem Hamiltonians can have different topologies. These findings are illustrated by considering the entanglement Hamiltonian of the ground state of graphene with dx2−y2 + idxy superconductivity obtained by tracing out one spin direction. Our investigations are based on closed analytical expressions for the full eigensystem in the entire Brillouin zone of bilayer graphene with a trigonally warped spectrum and graphene with superconductivity instabilities.