Consider a harmonic oscillator

$$H = \frac{1}{2}(p^2 + x^2)$$

We make a canonical transformation to $I = (p^2 + x^2)/2$ and $\varphi = \arctan(p/x)$. It is then easy to see that $\{I,\varphi \} = 1$ and the Hamiltonian reduces to

$$H = I$$

We now canonically quantize this Hamiltonian and we obtain that $\hat{I} = -i\hbar \partial_\varphi$. The stationary Schrödinger equation then simply reads

$$-i\hbar \partial_\varphi \psi = E\psi$$

with the obvious solution $\sim e^{i E_n \varphi/\hbar}, \,E_n = hn$. However, this is different from the result $E_n = h(n+1/2)$ we get by quantizing in the usual phase-space coordinates $p,x$.

This is a general pattern - it is very easy to find energy levels and other quantum numbers in action-angle coordinates but this will introduce a shift in the results as compared to the initial coordinate system.

This is probably a consequence of the nonlinear $\sim p^n x^k $ nature of the transformation and will thus introduce $\mathcal{O}(\hbar)$ differences due to the operator ordering ambiguity of canonical quantisation.This makes me believe that these shifts will always be only shifts by a constant.

It is not clear to me

- Whether the observable consequences (such as $E_n - E_m = h (n-m)$) are truly always independent of the phase-space coordinate system in which we execute canonical quantisation.
- How to compute the value of the shifts to the quantum numbers induced by the coordinate transform.

Does anyone know the answer to this?