Suppose to have a Hamiltonian $H(q,p)$ defined on a phase space with (q,p) and suppose that exist a canonical transformation $$(Q,P)=X(q,p)$$ such that the classical dynamics is equivalent using $H'(Q,P)$ and (Q,P). Now, what happens if I quantize the system using (q,p) or (Q,P)? I would say that I get two Hilbert spaces E and E' that are not isomorphic in general because a canonical transformations could be non linear. Is it true? If so, why we are used to quantize a system in a specific (q,p) phase space? I observed this ambiguity in the quantization of the hydrogen atom: the system is quantized in the phase space $(x,y,z,p_{x},p_{y},p_{z})$ and only in the quantum world we shift to polar coordinates: what would happen if this change is made at a classical level and then we quantize the phase space (r,$\theta$,$\phi$,p$_{r}$,p$_{\theta}$,p$_{\phi}$)?
1 Answers
Within your question lies an assumption that is important to state explicitly: you have a semiclassical picture in mind where you have a physically meaningful classical Hamilton function $H(q,p)$ on phase space (or some other classical observable) and you would like to promote it to a physically meaningful quantum Hamiltonian $\hat{H} = H(x,-\mathrm{i} \nabla - A)$. The fact that both need to be physically meaningful places restrictions on your quantization procedure.
A quantization of observables (as opposed to states, which is also an option) is a systematic association of suitable functions on phase space with operators on a Hilbert space. In general, quantizations are not unique, for example you can choose your favorite operator ordering in a variation Weyl calculus or choose a gauge in magnetic Weyl calculus.
However, choosing a non-symmetric operator ordering leads to mathematically well-defined operators that are usually not physically meaningful. On the other hand, the gauge freedom in choosing a vector potential is different in that respect: if $A' = A + \nabla \chi$ are two equivalent gauges, then $f \bigl ( x , - \mathrm{i} \hbar \nabla - A' \bigr )$ and $f \bigl ( x , -\mathrm{i} \hbar \nabla - A , x \bigr )$ defined via magnetic Weyl calculus are unitarily equivalent operators related by $\mathrm{e}^{+ \mathrm{i} \chi}$.
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