The Hydrogen Atom

Students should be able to:

1. (Recurrence)

   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)

2. (HydrogenVerify)

   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)

3. (BohrRadius)

   (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.

4. (Forbidden)

   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.

5. (DipoleMoment)

   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.

6. (Hydrogen)

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

   1. What are the possible results of a measurement of the energy and with what probabilities would they occur? Plot a histogram of the measurement results. Calculate the expectation value of the energy.

   2. What are the possible results of a measurement of the angular momentum operator $\mathbf{L}^2$ and with what probabilities would they occur? Plot a histogram of the measurement results. Calculate the expectation value of $\mathbf{L}^2$.

   3. What are the possible results of a measurement of the angular momentum component operator $L_z$ and with what probabilities would they occur? Plot a histogram of the measurement results. Calculate the expectation value of $L_z$.

   4. How do the answers to (a), (b), and (c) depend upon time?

7. (SPHybrid)

   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.

   1. If you measure the energy of this state, what possible values could you obtain?

   2. What is this state as a function of time?

   3. Calculate the expectation value $\langle\hat L^2\rangle$ in this state, as a function of time. Did you expect this answer? Comment.

   4. Write the time-dependent state in wave function notation.

   5. Calculate the expectation value $\langle \hat z \rangle$ as a function of time. Do you expect this answer?

8. (SPnotHybrid)

   Consider the initial state ${1\over \sqrt{2}}\left(\vert 1,0,0\rangle +\vert 2,1,0\rangle\right)$. Note, this is not an $sp$ hybrid orbital such as occurs in chemistry in the study of molecular bonding.

   1. If you measure the energy of this state, what possible values could you obtain?

   2. What is this state as a function of time?

   3. Calculate the expectation value $\langle\hat L^2\rangle$ in this state, as a function of time. Did you expect this answer? Comment.

   4. Write the time-dependent state in wave function notation.

   5. Calculate the expectation value $\langle \hat z \rangle$ as a function of time. Do you expect this answer?

9. (HydrogenTable) Complete the table (EigenTableHydrogen_Empty.pdf) which summarizes much of what you've learned about the eigenstates of the hydrogen atom by filling in all the empty boxes.
10. (McIntyre807)

    (McIntyre 8.7) Calculate the probability that the electron is measured to be in the classically forbidden region for the $n=2$ states of hydrogen. Discuss the differences between the results for the $l=0$ and $l=1$ states.

11. (McIntyre814)

    McIntyre 8.14

    A hydrogen atom is initially in the superposition state

    $\vert \psi(0) \rangle = \frac{1}{\sqrt{14}}\vert 211\rangle - \frac{2}{\sqrt{14}}\vert 32,-1\rangle + \frac{3}{\sqrt{14}}\vert 422\rangle$.

    1. What are the possible results of a measurement of the energy and with what probabilities would they occur? Plot a histogram of the measurement results. Calculate the expectation value of the energy.

    2. What are the possible results of a measurement of the angular momentum operator $L^2$ and with what probabilities would they occur? Plot a histogram of the measurement results. Calculate the expectation value of $L^2$.

    3. What are the possible results of a measurement of the angular momentum component operator $L_z$ and with what probabilities would they occur? Plot a histogram of the measurement results. Calculate the expectation value of $L_z$.