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CRC Press eBooks

Matrix Aspects of Quantum Mechanics and Quantum Operators

F.J. Duarte

Published Jan 15, 2024• 0 citations
Publisher

Abstract

This chapter revises the fundamentals of matrix mechanics. The Pauli matrices are derived from a Hamiltonian perspective, and the tensor product of matrices is introduced as applied to Pauli matrices. Moreover, the density matrix is introduced from a Diracian perspective. Quantum operators, such as the position operator, the momentum operator, and the energy operator, are introduced, and examples are given on their use.

DOI
10.1201/9781003398707-11

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This chapter revises the fundamentals of matrix mechanics. The Pauli matrices are derived from a Hamiltonian perspective, and the tensor product of matrices is introduced as applied to Pauli matrices. Moreover, the density matrix is introduced from a Diracian perspective. Quantum operators, such as the position operator, the momentum operator, and the energy operator, are introduced, and examples are given on their use.
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The interpretation of quantum mechanics due to Lande' is applied to the connection between wave mechanics and matrix mechanics. The connection between the differential eigenvalue equation and the matrix eigenvalue equation for an operator is elucidated. In particular, we show that the elements of a matrix vector state are probability amplitudes with a structure rather than being mere constants. We obtain the most general expressions for the probability amplitudes for the description of spin-1/2 measurements. As a result, we derive spin-1/2 operators and vectors from first principles. The procedure used is analogous to that by which orbital angular momentum wavefunctions and operators are transformed to matrix mechanics vectors and matrices. The most generalized forms of the spin operators and their eigenvectors for spin-1/2 are derived and shown to reduce to the Pauli spin matrices and vectors in an appropriate limit.
The objective of this series of three papers is to axiomatically derive quantum mechanics from classical mechanics and two other basic axioms. In this first paper, Schreodinger's equation for the density matrix is fist obtained and from it Schroedinger's equation for the wave functions is derived. The momentum and position operators acting upon the density matrix are defined and it is then demonstrated that they commute. Pauli's equation for the density matrix is also obtained. A statistical potential formally identical to the quantum potential of Bohm's hidden variable theory is introduced, and this quantum potential is reinterpreted through the formalism here proposed. It is shown that, for dispersion free {\it ensembles% }, Schroedinger's equation for the density matrix is equivalent to Newton's equations. A general non-ambiguous procedure for the construction of operators which act upon the density matrix is presented. It is also shown how these operators can be reduced to those which act upon the wave functions.
The objective of this series of three papers is to axiomatically derive quantum mechanics from classical mechanics and two other basic axioms. In this first paper, Schroedinger's equation for the density matrix is fist obtained and from it Schroedinger's equation for the wave functions is derived. The momentum and position operators acting upon the density matrix are defined and it is then demonstrated that they commute. Pauli's equation for the density matrix is also obtained. A statistical potential formally identical to the quantum potential of Bohm's hidden variable theory is introduced, and this quantum potential is reinterpreted through the formalism here proposed. It is shown that, for dispersion free {\\it ensembles% }, Schroedinger's equation for the density matrix is equivalent to Newton's equations. A general non-ambiguous procedure for the construction of operators which act upon the density matrix is presented. It is also shown how these operators can be reduced to those which act upon the wave functions.
This chapter reminds, without entering into details, the main mathematical concepts and results relevant for finite system quantum mechanics. The basic postulates single out a Hilbert space of wave functions with self-adjoint linear operators corresponding to observables as was originally discovered by von Neumann. The chapter connects the contemporary terminology of linear Hilbert space operators with quantum physics. Important concepts like linear operators, measures, self-adjointness, spectral measures, density matrices, and tensor products are discussed and illustrated in the light of observables, probability for quantum systems and composite systems. A first example of a useful algebra of observables, the Weyl algebra, is described in detail and linked to the classical phase space of a point particle.
Abstract Previously, we viewed Hopfield nets from the point of view of statistical mechanics. In this chapter, we widen our perspective one again and view them from the point of view of quantum mechanics. This requires us to introduce completely different representations of the states and energy functions of Hopfield nets and relate them to the kind of tensor product spaces we studied earlier. At this point, this different perspective on Hopfield nets is of purely theoretical interest because it generally defies practical implementations on digital computers. However, it establishes a connection between Hopfield nets and important quantum mechanical concepts. In particular, we will get to know the notion of the Hamiltonian of a physical system, its eigenstates, and eigenvalues. Since the material in this chapter is more abstract and less intuitive than in previous chapters, we develop our ideas by looking at an exemplary simple Hopfield net and support our technical claims through practical computations and code examples.