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Mean and Variance in Quantum Theory


Calculation of the mean of an observable in quantum mechanics is typically assumed to require that the state vector be in the domain of the corresponding self-adjoint operator or for a mixed state that the operator times the density matrix be in the trace class. We remind the reader that these assumptions are unnecessary. We state what is actually needed to calculate the mean of an observable as well as its variance.

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  1. See [5, 9], or [10] for an account of Borel sets and projection-valued measures.

  2. The letter \(\lambda \) is used for the variable in Eq. (3) and elsewhere since it is commonly used for eigenvalues. The support of \(\mu _{\psi }\) is the spectrum of \(A\).

  3. An observant referee has pointed out that the right side of (4) is used in Prugovečki’s book [9], p. 263] as the definition of mean value.

  4. This is equivalent to saying that the identity function \(x \mapsto x\) is in \(L^1(\mathcal {R}, \mu _{\psi })\) where \(\mu _{\psi }\) is defined by (11).

  5. See [6], pp. 64–66].


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Correspondence to Andrew Vogt.

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Gray, J.E., Vogt, A. Mean and Variance in Quantum Theory. Found Phys 45, 883–888 (2015).

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  • Projection-valued measure
  • Pure state
  • Mixed state
  • Trace

Mathematics Subject Classification

  • 81Q10
  • 47B25
  • 62P35