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Hello,
suppose one has a classical canonical transformation between two sets of canonical variables such that the new (primed) positions and momenta can be written as functions of the old (unprimed) ones.
[tex]{\cal K}: x_i \to x_i^\prime(x); \quad p_i \to p_i^\prime(p)[/tex]
Using these relations one can check immediately that the Poisson brackets remain unchanged.
This is translated to quantum mechanical language as follows: the positions and momenta are replaced by operators, the Poisson brackets are translated into canonical commutation relations, which remain unchanged as well:
[tex][x_i,p_k] = [x_i^\prime,p_k^\prime] = i\delta_{ik}[/tex]
The canonical transformation itself is replaced by an operator acting on the positions and momenta
[tex]{\cal K}: x_i \to x_i^\prime = U_{\cal K}\, x_i\, U^\dagger_{\cal K}; \quad p_i \to p_i^\prime = U_{\cal K}\, p_i\, U^\dagger_{\cal K}[/tex]
Questions:
Examples are rotations, translations, transformation to center-of-mass frame, ...
suppose one has a classical canonical transformation between two sets of canonical variables such that the new (primed) positions and momenta can be written as functions of the old (unprimed) ones.
[tex]{\cal K}: x_i \to x_i^\prime(x); \quad p_i \to p_i^\prime(p)[/tex]
Using these relations one can check immediately that the Poisson brackets remain unchanged.
This is translated to quantum mechanical language as follows: the positions and momenta are replaced by operators, the Poisson brackets are translated into canonical commutation relations, which remain unchanged as well:
[tex][x_i,p_k] = [x_i^\prime,p_k^\prime] = i\delta_{ik}[/tex]
The canonical transformation itself is replaced by an operator acting on the positions and momenta
[tex]{\cal K}: x_i \to x_i^\prime = U_{\cal K}\, x_i\, U^\dagger_{\cal K}; \quad p_i \to p_i^\prime = U_{\cal K}\, p_i\, U^\dagger_{\cal K}[/tex]
Questions:
- are there additional conditions for the canonical transformations [tex]{\cal K}[/tex] in order to translate them into quantum mechanical operators [tex]U_{\cal K}[/tex]?
- given a specific canonical transformations [tex]{\cal K}[/tex], that means the functional relationship between new and old position and momenta, how does one construct the unitary operator [tex]U_{\cal K}[/tex] generating the quantum mechanical transformation?
Examples are rotations, translations, transformation to center-of-mass frame, ...
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