Use the recurrence relation for the radial wave function to construct the $n=3$ radial states of hydrogen. Calculate the normalization constant for the $R_{32}(r)$ state.%(McIntyre 8.2)

By direct application of the differential operators, verify that the state $\vert 321\rangle\doteq \psi_{321}(r,\theta,\phi)$ is an eigenstate of $H,\, \mathbf{L}^2,$ and $L_z$ and determine the corresponding eigenvalues.%(McIntyre 8.5)

(McIntyre 8.6) Calculate the probability that the electron is measured to be within one Bohr radius of the nucleus for the $n=2$ states of hydrogen. Discuss the differences between the results for the $l=0$ and $l=1$ states.

Calculate the probability that the electron is measured to be in the classically forbidden region for each of the $n=2$ states of hydrogen. Discuss the differences between the results for the $\ell=0$ and $\ell=1$ states.

Consider a one-dimensional probability density $\mathcal{P}(z)$ along the $z$-axis obtained by integrating over a plane perpendicular to the $z$-axis, either in Cartesian coordinates $$\mathcal{P}(z)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\left|\psi_{n\ell m}(x,y,z)\right|^2 dx\,dy$$ or in cylindrical coordinates $$\mathcal{P}(z)=\int_{0}^{2\pi}\int_{-\infty}^{\infty}\left|\psi_{n\ell m}(\rho,\phi,z)\right|^2 \rho\, d\rho\,d\phi$$ Calculate this probability density for the superposition states $\ket{\psi_1}=\frac{1}{\sqrt{2}}\left(\ket{100}+\ket{210}\right)$ and $\ket{\psi_2}=\frac{1}{\sqrt{2}}\left(\ket{200}+\ket{210}\right)$. Use these probability densities to find the expectation value of the electric dipole moment $\mathbf{d}=q\mathbf{r}$ and verify that the moments for these two states are oppositely oriented. Plot and animate the probability densities to verify that one state is oscillating and one state is static.

A hydrogen atom is initially in the superposition state $$\ket{\psi(0)}=\frac{1}{\sqrt{14}}\ket{211}-\frac{2}{\sqrt{14}}\ket{3,2,-1}+\frac{3i}{\sqrt{14}}\ket{422}.$$

Consider the initial state ${1\over \sqrt{2}}\left(\vert 2,0,0\rangle +\vert 2,1,0\rangle\right)$ which is an $sp$ hybrid orbital which occurs in chemistry in the study of molecular bonding.