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Oxford University Press eBooks

The foundations of quantum mechanics

Peter Joshua Atkins, Ronald Friedman

Published Nov 11, 2010• 7 citations
Publisher

Abstract

Operators in quantum mechanics Linear operators Eigenfunctions and eigenvalues Representations Commutation and non-commutation The construction of operators Integrals over operators Dirac bracket and matrix notation Hermitian operators The postulates of quantum mechanics States and wavefunctions
DOI
10.1093/hesc/9780199541423.003.0001

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The Postulates and Principles of Quantum Mechanics Commutability and Compatibility of Quantum Observables Non-Commuting Operators and Uncertainty Principle Quantum Dynamics and Schrodinger Equation Time Evolution Operator and Propagator Symmetry, Transformation and Continuous Groups Lie Groups and Lie Algebras O(3) Group, SU(2) Group and Angular Momentum Dynamical Symmetry and Hydrogen Atom Lorentz Group and Group Representation Chiral Transformation and Spinor Algebra Gamma Matrix Algebra and Discrete Symmetry in Dirac Spinor System.
Abstract Mathematical formalism of quantum mechanics is closely connected with the theory of linear operators. A key quantum-mechanical principle is that physical quantities (observables) are represented by Hermitian linear operators that act on state vectors that belong to a Hilbert vector space. This chapter analyzes problems that deal with the basic concepts of the theory of linear operators; eigenfunctions, eigenvalues, mean values; the projection operators; quantum-mechanical representations of operators and wave-functions; and unitary operators.
An overview of main operators in quantum mechanics is presented. Operators for position, linear momentum, kinetic energy, potential energy, total energy, and angular momentum are introduced. Properties of linear operators are described. Commutative property of operators has profound implications because a pair of non-commutative operators cannot be measured accurately at the same time. Eigenfunctions and eigenvalues of an operator are defined, and operator form of Schrodinger’s equation is given. Most operators in quantum mechanics are of the Hermitian category because they always have real eigenvalues. The condition to test Hermitian nature of operators is specified. Postulates of quantum mechanics are stated to serve as a convenient starting point.
This text presents a rigorous mathematical account of the principles of quantum mechanics, in particular as applied to chemistry and chemical physics. Applications are used as illustrations of the basic theory. The first two chapters serve as an introduction to quantum theory, although it is assumed that the reader has been exposed to elementary quantum mechanics as part of an undergraduate physical chemistry or atomic physics course. Following a discussion of wave motion leading to Schrödinger's wave mechanics, the postulates of quantum mechanics are presented along with essential mathematical concepts and techniques. The postulates are rigorously applied to the harmonic oscillator, angular momentum, the hydrogen atom, the variation method, perturbation theory, and nuclear motion. Modern theoretical concepts such as hermitian operators, Hilbert space, Dirac notation, and ladder operators are introduced and used throughout. This text is appropriate for beginning graduate students in chemistry, chemical physics, molecular physics and materials science.