Spectral theorem for dummies: A pedagogical discussion on quantum probability and random variable theory

Abstract

John von Neumann's spectral theorem for self-adjoint operators is a cornerstone of quantum mechanics. Among other things, it also provides a connection between expectation values of self-adjoint operators and expected values of real-valued random variables. This paper presents a plain-spoken formulation of this theorem in terms of Dirac's bra and ket notation, which is based on physical intuition and provides techniques that are important for performing actual calculations. The goal is to engage students in a constructive discussion about similarities and differences in the use of random variables in classical and quantum mechanics. Special emphasis is given on operators that are simple functions of noncommuting self-adjoint operators. The presentation is self-contained and includes detailed calculations for the most relevant results.