Shouldn't the operator be applied to both the wf anD its modulus?

[jshrager]

Why is it:

$$ \langle A \rangle = \int dV ~ \psi^* (\hat{A} \psi) $$

As opposed to:

$$ \langle A \rangle = \int dV ~ (\hat{A} \psi^*) (\hat{A} \psi) $$

The op can substantially change the wf, so it would seem to make more mathematical sense (at least from a linear algebra pov) to operate on both the wf and its mod, otherwise, as we do it now (first expr) you're getting a very odd product. Is this is a property of the way that wfs are formulated? Are they a special case where this isn't required? Or maybe it's a magical property of hermetian operators? Or maybe it just works and I shouldn't think too hard about it.

 

[mfb]

1) it works
2) the second formula is not linear
3) There is probably some deeper reason why this works

 

[kith]

If you start with the standard definition of the expectation value of a random variable <A> = Ʃ_i p_ia_i -where the a_is are the possible outcomes and p_i is the probability to obtain the i-th- your first formula follows from the QM postulates about these quantities. I don't know if this is satifying to you. ;-)