Quantum mechanics has a well-known procedure for evaluating the(adsbygoogle = window.adsbygoogle || []).push({}); expectation valueof an observable quantity in a given quantum state. First one must obtain the quantum operator O that is associated with the observable quantity. Then the rule for computing the expectation value is: Apply O to the wave function psi, multiply the result with psi*, and finally integrate over spatial variables.

The procedure described above can be extended to evaluate thevarianceof an observable quantity. Now the rule is: apply the operator O twice in succession to the wave function psi, multiply with psi* and integrate over space. From this result, subtract the square of the expectation value.

All this is in clear analogy with the standard rules of statistics. However, in QM things are less clear-cut. For example, it can be demonstrated that the variance may acquire a negative value!

My question is: How should one interpret a negative value for the variance of an observable quantity?

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# Negative variance of an observable quantity

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