Abstract
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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Notes
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 \(A\).
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).
See [6], pp. 64–66].
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Gray, J.E., Vogt, A. Mean and Variance in Quantum Theory. Found Phys 45, 883–888 (2015). https://doi.org/10.1007/s10701-015-9898-1
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DOI: https://doi.org/10.1007/s10701-015-9898-1