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

 

[denian]

The relationship between unitary transforms in quantum mechanics and canonical transformations in classical mechanics is a complex and nuanced one. While there are some similarities between the two, it is not accurate to say that a unitary transform in QM always corresponds to a canonical transform in classical mechanics.

Firstly, it is important to clarify that the term "canonical transform" in classical mechanics refers to a specific type of transformation that preserves the form of Hamilton's equations. These equations describe the evolution of a system over time and are based on the Hamiltonian, which is a function of the system's position and momentum variables. In contrast, the term "unitary transform" in quantum mechanics refers to a type of transformation that preserves the inner product between two quantum states.

In certain cases, a unitary transform in quantum mechanics can be related to a canonical transform in classical mechanics. This is known as the Wigner-Weyl correspondence and it is based on the fact that the quantum mechanical operators representing observables can be mapped onto classical phase space functions. However, this correspondence is not always straightforward and it depends on the specific observables and transformations involved.

In the case of the wave function \psi(x), the unitary transform \hat{U}\psi(x) would result in a new wave function \psi'(x) that is related to the original one by a phase factor. This is because the unitary transform only acts on the quantum state, not on the underlying classical variables. On the other hand, a canonical transform in classical mechanics would result in a new set of classical variables, such as position and momentum, that are related to the original ones by a specific transformation. Therefore, it is not accurate to say that the unitary transform corresponds to the canonical transform in this case.

In summary, while there are some connections between unitary transforms in quantum mechanics and canonical transformations in classical mechanics, they are not equivalent and cannot be simply interchanged. They represent different mathematical concepts and should be understood and applied accordingly.