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John von Neumann's spectral theorem for self-adjoint operators is acornerstone of quantum mechanics. Among other things, it also provides aconnection between expectation values of self-adjoint operators and expectedvalues of real-valued random variables. This paper presents a plain-spokenformulation of this theorem in terms of Dirac's bra and ket notation, which isbased on physical intuition and provides techniques that are important forperforming actual calculations. The goal is to engage students in aconstructive discussion about similarities and differences in the use of randomvariables in classical and quantum mechanics. Special emphasis is given onoperators that are simple functions of noncommuting self-adjoint operators. Thepresentation is self-contained and includes detailed calculations for the mostrelevant results.

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