is clear that the form (\[sol\_Pi\_def\]) is valid not only under the assumption (\[sol\_Psi\_phi\]) but also in the general case when $ P_{3} = p_{z} \equiv -i {\partial}_{z} $; moreover $ \Pi $ is a gauge-invariant quantity (as it contains only the gauge-invariant canonical momentum $ {\bf P} $).

The wave functions corresponding to the spectrum (\[sol\_E\]) in the polar coordinate system $ (r, \varphi) $ are as follows: $$\label{sol_phi}
    \phi(r,\varphi) =
        {{\textstyle}\frac{1}{\sqrt{2\pi}}} \, e^{i (n - s - 1/2) \varphi}
        \begin{pmatrix}
            c_{1}   \, e^{-i \varphi / 2} \, I_{n-1,s}(\rho) \\
            i c_{2} \, e^{ i \varphi / 2} \, I_{n,s}  (\rho) \\
            c_{3}   \, e^{-i \varphi / 2} \, I_{n-1,s}(\rho) \\
            i c_{4} \, e^{ i \varphi / 2} \, I_{n,s}  (\rho) \\
        \end{pmatrix}
        \sqrt{eH}, \quad
    \rho = {{\textstyle}\frac{1}{2}} eHr^{2},$$ where $ I_{n,s}(\rho) $ are the Laguerre functions: $$I_{n,s}(\rho) =
        {{\textstyle}\sqrt{\frac{s!}{n!}}} \, e^{-\rho / 2} \rho^{(n - s)/2} L_{s}^{n-s}(\rho),$$ expressed by means of the generalized Laguerre polynomials $ L^{l}_{s}(\rho) $: $$L^{l}_{s}(\rho) =
        {{\textstyle}\frac{1}{s!}} \, e^{\rho} \rho^{-l}
        {{\textstyle}\frac{d^{s}}{d\rho^{s}}} \left( e^{-\rho} \rho^{s + l} \right);$$ $ s = 0, 1, \ldots $ is the radial quantum number; $ \{ c_{\mathfrak a} \} $ are constant coefficients depending on the particle state. 