Wavefunction and Notations in Quantum Mechanics

Luca Salasnich

Peter Joshua Atkins, Ronald Friedman

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

Elizabeth Gire, Edward Price

This paper is part of the Focused Collection on Upper Division Physics Courses.]The formalism of quantum mechanics includes a rich collection of representations for describing quantum systems, including functions, graphs, matrices, histograms of probabilities, and Dirac notation.The varied features of these representations affect how computations are performed.For example, identifying probabilities of measurement outcomes for a state described in Dirac notation may involve identifying expansion coefficients by inspection, but if the state is described as a function, identifying those expansion coefficients often involves performing integrals.In this study, we focus on three notational systems: Dirac notation, algebraic wavefunction notation, and matrix notation.These quantum notations must include information about basis states and their associated complex probability amplitudes.In this theory paper, we identify four structural features of quantum notations, which we term individuation, degree of externalization, compactness, and symbolic support for computational rules.We illustrate how student reasoning interacts with these structural features with episodes from interviews with advanced undergraduate physics majors reasoning about a superposition state of an infinite square well system.We find evidence of the students coordinating different notations through the use of Dirac notation, using an expression in Dirac notation to guide their work in another notation.These uses are supported by the high degree of individuation, compactness, and symbolic support for computation and the moderate degree of externalization provided by Dirac notation.

Brian P. Anderson

Notation : constants, symbols, and operators -- Linear algebra -- Quantum mechanics formalism -- Postulates -- Bases and representations -- Operators -- Wavefunctions -- Time dependence, frames, and "pictures" -- Exactly solvable problems -- Angular momentum -- Two-level systems and spin 1=2 -- Net angular momentum -- Approximation methods -- Hydrogen and atomic structure -- Advanced topics -- Appendix.