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Journal of Chemical Education

A note on the Schrödinger representation of the momentum and energy operators

Tai L. Chow

Published Jul 1, 1992• 2 citations
Publisher

Abstract

Observables in quantum mechanics can not be regarded as ordinary algebraic variables as in classical mechanics; instead, every observable can be represented by an operator.
DOI
10.1021/ed069p537

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Physical observables which are measured in a particular state of a dynamical system can be conveniently represented in classical mechanics by mathematical variables, on the grounds that all have definite values in any classical state. In contrast, the statistical interpretation of wave functions describing the quantum states implies that there are uncertainties in the measurement of any two observables which can be represented by canonically conjugate variables, such as position q and momentum p q . This requires any measured dynamical observable A in a quantum state to be associated with an expectation value 〈 A 〉.
Abstract Quantum mechanics is written in the language of linear algebra. On the Schrodinger picture the theory represents quantum-mechanical states using the elements of a Hilbert space and represents observable physical properties and the standard dynamics using the linear operators on the state space. We consider the mathematical notions for understanding and working with the standard formulation of quantum mechanics. Each mathematical notion is characterized geometrically, algebraically, and physically. The mathematical representation of quantum-mechanical superpositions is discussed.
As a very important example for dynamical symmetries in the context of q‐generalized quantum mechanics the algebra aa† − q−2a†a = 1 is investigated. It represents the oscillator symmetry SUq(1, 1) and is regarded as a commutation phenomenon of the q‐Heisenberg algebra which provides a discrete spectrum of momentum and space, i.e., a discrete Hilbert space structure. Generalized q‐Hermite functions and systems of creation and annihilation operators are derived. The classical limit q → 1 is investigated. Finally the SUq(1, 1) algebra is represented by the dynamical variables of the q‐Heisenberg algebra.