@obastani : very good comment.
Aside: If |Psi> is a state with wavefunction <x|Psi>, then is there a linear operator A such that <x|A|Psi> = log(<x|Psi>)?
Note: Logarithm of a complex function.
Aside: If |Psi> is a state with wavefunction <x|Psi>, then is there a linear operator A such that <x|A|Psi> = log(<x|Psi>)?
Note: Logarithm of a complex function.
<x|A(|Psi> + |Chi>) = log(<x|(|Psi> + |Chi>))
= log(<x|Psi> + <x|Chi>)
= log(<x|Psi>) log(<x|Chi>)
/= <x|A|Psi> + <x|A|Chi> log(a + b) = log(a) log(b)
log(a + b) /= log(a) + log(b)
The idea behind my question: does the Shannon entropic integral correspond to the L2 length of some projected state? Leading to a prescription to prepare a state with minimum uncertainty product of canonically conjugate physical quantities.
Afaik, in classical statistical physics the log() shows up when the N! in a binomial probability distribution limits to large N via the Stirling approximation. It would be interesting to find a different route for log() to enter the picture from a quantum standpoint.
All admittedly vague and hand waving speculation.