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Journal of Aba Teachers College

Two Examples of the Presentational Theory of Quantum Mechanics

LI Ji-jun

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

This paper systematically expounds the concrete forms and relevant problems of the coordinate functor and momentum functor in the coordinates and momentum presentations by way of the δ function and Dirac notation.

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The canonical commutation relation between the quantum mechanical operators of momentum and coordinate and the Ehrenfest theorem in the free particle case are derived solely from the rule of differentiation of the delta function [xδ′(x)=−δ(x)x\delta'(x) = - \delta(x)xδ′(x)=−δ(x)].
We begin with a presentation of a minimal set of von Neumann postulates while introducing language and notation to facilitate subsequent discussion of quantum calculations based in finite-dimensional Hilbert spaces. We also address two-state quantum systems, entanglement of multiple two-state systems, quantum angular momentum theory and quantum approaches to statistical mechanics. A concluding chapter gives an overview of issues associated with quantum mechanics in continuous infinite-dimensional Hilbert spaces.
This chapter introduces the representation of quantum states and operators in an orthonormal basis associated with a linear Hermitian operator, forming the foundation of matrix mechanics. The transformation between different representations is described, highlighting how operators and state vectors change under basis transformations. The projection operator is defined, and the closure (completeness) relation is expressed in terms of projection operators on the basis states. The Dirac bra–ket notation is introduced and thoroughly explained to unify representation theory. Detailed treatments of the coordinate and momentum representations are given, including the transformation between them via Fourier analysis. Finally, explicit forms of the coordinate and momentum operators acting on wavefunctions in the coordinate space are derived. Metadata/Keywords: matrix mechanics, representation theory, Hermitian operator, projection operator, closure relation, Dirac bra–ket notation, coordinate and momentum representations
In the Hamiltonian formulation, classical mechanics employs the commutative algebra of functions that are defined on phase space a point of which could be represented using Dirac delta distributions. In the absence of such a concrete existence of the notion of point in the quantum domain, we show how the quantum mechanical formalism emerges by replacing the commutative algebra by a noncommutative algebra of functions and introducing the quantum condition. The noncommutativity is achieved by deforming the product of two functions and by introducing the Moyal brackets, the basic Moyal brackets between two spatial coordinates and between position and momentum coordinates being noncommutative. The only other input we make use of is the quantum condition which is motivated from its classical analogue—the square of Dirac delta distributions.