= Solution
The <rotational invariance of a central-potential Hamiltonian> allows a simultaneous <eigenstate> of energy and the two-dimensional <orbital angular momentum> $L_z=-i\hbar\partial_\phi$. For a separated <wavefunction> $F(r)G(\phi)$, the <Laplacian in polar coordinates> gives
$$
\frac{r^2}{F}\left(F''+\frac1rF'\right)+\frac{G''}{G}+\frac{2m r^2}{\hbar^2}(E-V)=0.
$$
The $\phi$ term must be a constant; write $G''/G=-k^2$. The angular <eigenfunctions> can be chosen as $e^{ik\phi}$. Single-valuedness under $\phi\mapsto\phi+2\pi$ imposes $e^{2\pi i k}=1$, hence $k\in\mathbb Z$. The radial equation in the <separation of a two-dimensional central-potential eigenstate> is
$$
-\frac{\hbar^2}{2m}\left(f''+\frac1rf'-\frac{k^2}{r^2}f\right)+V(r)f=Ef.
$$
For a real <central potential> and the usual real self-adjoint radial boundary conditions, the radial equation admits a basis of real solutions: real and imaginary parts of a complex solution obey the same equation and boundary conditions. Choose a real normalized radial <eigenfunction>. Since the plane area element in <plane polar coordinates> is $r\,dr\,d\phi$, the normalization is
$$
\int_0^\infty r f(r)^2\,dr=1,\qquad
\boxed{\psi_k(r,\phi)=\frac{f(r)}{\sqrt{2\pi}}e^{ik\phi},\quad k\in\mathbb Z}.
$$
This establishes the intended separated simultaneous <eigenstate> form. \b[It is not the form of every stationary state.] The radial equation depends on $k^2$, so the $k$ and $-k$ sectors have the same energy. For $k\geq1$, their normalized superposition
$$
\psi_{\rm real}(r,\phi)=\frac{f(r)}{\sqrt\pi}\cos(k\phi)
$$
is a single-valued <stationary state> with that energy but is not a single angular exponential. <Quantum degeneracy> is precisely why <separation of variables> selects a convenient <eigenstate> basis rather than all vectors in an energy eigenspace. The printed assertion needs this qualification. The printed polar-coordinate aid also labels a gradient component tuple as a divergence; the <gradient> used below is the two-dimensional vector $\widehat{\mathbf r}\partial_r+\widehat{\boldsymbol\phi}r^{-1}\partial_\phi$.
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