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Separation of variables in the central Coulomb potential gives ${\psi }_{nlm}=R_{nl}(r)Y_l^m(\theta ,\varphi )$. The radial wavefunction depends on $n$ and $l$, independently of $m$. For the hydrogen radial wavefunction in the first excited shell, put $\rho =r/a_0$. The Bohr radius is $a_0=4\pi {ϵ}_0{\hslash }^2/(\mu e^2)$, where $\mu$ is the electron–proton reduced mass. Choose the real phasesNear the origin, $R_{20}\to 1/(\sqrt{2}{a}_0^{3/2})$, whereas $R_{21}\sim r/(\sqrt{24}{a}_0^{5/2})$. More generally, regular hydrogen radial wavefunctions behave as $r^l$. At large $r$, both displayed functions have a polynomial factor proportional to $r$ multiplying $e^{-r/(2a_0)}$; $R_{20}$ approaches zero from below and $R_{21}$ from above with these phase choices.

The $2s$ radial wavefunction crosses zero once, at $r=2a_0$, and has its negative minimum at $4a_0$. The $2p$ radial wavefunction has no radial node at positive $r$, and reaches its maximum at $2a_0$. The count is $n-l-1$; neither the origin nor the limit at infinity is counted as a radial node. These are sketches of $R$, rather than of $rR$ or the radial probability density $r^2|R{|}^2$.

Both $z$ and $p_z$ commute with $L_z$ and are odd under parity. Their angular selection rules therefore require the same magnetic quantum number and opposite orbital parity. Within the nonrelativistic point-Coulomb $n=2$ hydrogen atom eigenspace, the only angularly permitted off-diagonal pair is $|2s⟩$ and $|2p,m=0⟩$.

Those necessary angular conditions do not guarantee a nonzero matrix element. For the Coulomb potential Hamiltonian $H={\mathbf{p}}^2/(2m)+V(r)$,All states within this exact $n=2$ eigenspace have the same energy, so every projected $p_z$ matrix element vanishes. In contrast, the position matrix element is nonzero: with conventional real spherical harmonics and hydrogen radial wavefunction in the first excited shell, $⟨2s|z|2p,0⟩=-3a_0$. Its sign depends on basis phases, while its magnitude does not.

Thus the electric-dipole selection rules for hydrogen allow a linear position perturbation to mix these degenerate states, but a momentum perturbation cannot be assigned the same matrix merely from parity. The conclusion concerns the unperturbed Coulomb eigenspace; degeneracy-breaking interactions require their own energy eigenstates and matrix elements.