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

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The discussion confirms that unitary transformations in quantum mechanics (QM) correspond to canonical transformations in classical mechanics, as expressed by the equation \hat{U}\Psi[O] = \Psi[O']. The state \Psi[O] is a functional of the observable algebra, adhering to positivity and normalization conditions, while O' represents the canonically transformed observable. Reilly Atkinson references Dirac's Quantum Mechanics, specifically section 26, which elaborates on the analogy between unitary and canonical transformations, emphasizing that unitary transformations are more general than their canonical counterparts, highlighting the fundamental differences between QM and classical mechanics.

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

[tex]\hat{U}\Psi[O] = \Psi[O'][/tex]

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

what does it look like in the case of wave function [tex]\psi(x)[/tex] ?
 
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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
 

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