I am exploring the quantum mechanical description of a free particle in 2 dimensions and seeking clarification on the equivalence of formulations in Cartesian and polar coordinates. The core of my inquiry revolves around whether these two coordinate systems yield the same physical predictions, particularly in terms of probabilities for particle measurements.
Initial Formulations:
Cartesian Coordinates (x, y): The wave function is $\Psi(x,y,t)$, and the Hamiltonian is represented as:
$$ \hat{H} = -\frac{\hbar^2}{2m} \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right) $$
The Schrödinger equation is:
$$ i\hbar \frac{\partial \Psi}{\partial t} = \hat{H} \Psi $$
Polar Coordinates (r, θ): The wave function is $\tilde{\Psi}(r,\theta,t)$, and the Hamiltonian in polar coordinates is:
$$ \hat{H} = -\frac{\hbar^2}{2m} \left( \frac{\partial^2}{\partial r^2} + \frac{1}{r}\frac{\partial}{\partial r} + \frac{1}{r^2}\frac{\partial^2}{\partial \theta^2} \right) $$
Corresponding Schrödinger equation:
$$ i\hbar \frac{\partial \tilde{\Psi}}{\partial t} = \hat{H} \tilde{\Psi} $$
Given an initial condition $\Psi(x,y,0) = f(x,y) = \tilde{f}(r,\theta) = \tilde{\Psi}(r,\theta,0)$, I am interested in understanding whether the evolution of these wave functions in their respective coordinate systems leads to identical physical predictions.
Question 1.
Specifically, for a given region $R$ in Cartesian coordinates, corresponding to $\tilde{R}$ in polar coordinates, will the probabilities of measuring the particle at time $t$ in $R$ or $\tilde{R}$ be the same when calculated using $\Psi(x,y,t)$ and $\tilde{\Psi}(r,\theta,t)$, respectively?
Question 2.
In the formulation above, we have quantized the free particle in Cartesian coordinates, and then we have transformed the Hamiltonian to polar coordinates. What if I do the Hamiltonian formulation of the classical free particle in polar coordinates, and then quantize promoting $\theta$, $r$ and their conjugate momenta to operators? Would I obtain the same system?
