Is it true that unitary transform in QM corresponds to canonical transform

[kakarukeys]

Is it true that unitary transform in QM corresponds to canonical transformation in classical mechanics in this way:

$$\hat{U}\Psi[O] = \Psi[O']$$

state $$\Psi[O]$$ is a functional of the observable algebra, satisfying positivity and normalization conditions. $$O'$$ is the canonically transformed observable.

what does it look like in the case of wave function $$\psi(x)$$ ?

 

[reilly]

See Dirac's Quantum Mechanics, section 26, for a very thorough discussion of this matter. In short, Dirac notes that while unitary and contact/canonical transformations are analogous, unitary xforms are far more general than canonical ones -- QM and classical mechanics are very different.
Regards,
Reilly Atkinson