Chapter 5 Exercises: The Hydrogen Atom

Section A: Spherical Coordinates and the 3D Schrodinger Equation (Problems 1--6)

Problem 1 (Basic)

Convert the following points from Cartesian to spherical coordinates: (a) $(x, y, z) = (1, 1, \sqrt{2})$ (b) $(x, y, z) = (0, 0, -3)$ (c) $(x, y, z) = (2, 0, 0)$

For each point, verify your answer by converting back to Cartesian.

Problem 2 (Basic)

The volume element in spherical coordinates is $d^3\mathbf{r} = r^2\sin\theta\;dr\;d\theta\;d\phi$.

(a) Verify that the total integral $\int_0^\infty\int_0^\pi\int_0^{2\pi} r^2\sin\theta\;dr\;d\theta\;d\phi$ diverges, and explain why this is expected for the volume of all space.

(b) Compute the volume of a sphere of radius $R$ using the spherical volume element.

(c) Compute the surface area of a sphere of radius $R$ by integrating over the angular variables alone.

Problem 3 (Basic)

A free particle in three dimensions has the wavefunction $\psi = Ae^{i\mathbf{k}\cdot\mathbf{r}}$ with $\mathbf{k} = (k_x, k_y, k_z)$.

(a) Show that this satisfies $-(\hbar^2/2m)\nabla^2\psi = E\psi$ and find $E$ in terms of $|\mathbf{k}|$.

(b) What is the degeneracy of the energy level $E$? (Hint: how many distinct $\mathbf{k}$ vectors satisfy $|\mathbf{k}|^2 = 2mE/\hbar^2$?)

Problem 4 (Intermediate)

Verify by direct computation that the Laplacian of $f(r) = e^{-r/a_0}/r$ in spherical coordinates gives:

$$\nabla^2\left(\frac{e^{-r/a_0}}{r}\right) = \frac{1}{a_0^2}\frac{e^{-r/a_0}}{r} - 4\pi\delta^3(\mathbf{r})e^{-r/a_0}$$

Hint: Treat $r > 0$ and $r = 0$ separately. For $r > 0$, use the radial Laplacian directly. The delta function arises from $\nabla^2(1/r) = -4\pi\delta^3(\mathbf{r})$.

Problem 5 (Intermediate)

Consider a particle in a 3D isotropic harmonic oscillator potential $V(r) = \frac{1}{2}m\omega^2 r^2$.

(a) Write the 3D Schrodinger equation in spherical coordinates.

(b) By separating variables $\psi(r,\theta,\phi) = R(r)Y_l^m(\theta,\phi)$, derive the radial equation.

(c) Without solving the radial equation, argue on physical grounds why the energy levels of the 3D isotropic oscillator are $E_N = (N + 3/2)\hbar\omega$ where $N = 2n_r + l$ and $n_r = 0, 1, 2, \ldots$ is the number of radial nodes.

Problem 6 (Intermediate)

The 3D infinite spherical well has $V(r) = 0$ for $r < a$ and $V(r) = \infty$ for $r > a$.

(a) Write the radial equation for this potential (for $r < a$).

(b) For $l = 0$, show that the solutions are $R(r) = A\sin(kr)/r$ and find the allowed values of $k$ from the boundary condition $R(a) = 0$.

(c) Compute the first three energy levels for $l = 0$ and compare with the 1D infinite well.

Section B: Spherical Harmonics (Problems 7--12)

Problem 7 (Basic)

(a) Verify that $Y_0^0 = 1/\sqrt{4\pi}$ is normalized:

$$\int_0^{2\pi}\int_0^\pi |Y_0^0|^2\sin\theta\;d\theta\;d\phi = 1$$

(b) Verify the orthogonality of $Y_1^0$ and $Y_1^1$:

$$\int_0^{2\pi}\int_0^\pi Y_1^{0*} Y_1^1\sin\theta\;d\theta\;d\phi = 0$$

Problem 8 (Basic)

Write out explicitly the real combinations of spherical harmonics that correspond to the $p_x$, $p_y$, and $p_z$ orbitals. Show that they are proportional to $x/r$, $y/r$, and $z/r$ respectively.

Problem 9 (Intermediate)

(a) Show that $Y_l^m(\theta,\phi)$ is an eigenfunction of $\hat{L}_z = -i\hbar\partial/\partial\phi$ with eigenvalue $m\hbar$.

(b) Given that $\hat{L}^2 = -\hbar^2\hat{\Lambda}^2$, verify that $Y_1^0 = \sqrt{3/4\pi}\cos\theta$ is an eigenfunction of $\hat{L}^2$ with eigenvalue $2\hbar^2$.

Problem 10 (Intermediate)

The addition theorem for spherical harmonics states:

$$P_l(\cos\gamma) = \frac{4\pi}{2l+1}\sum_{m=-l}^{l} Y_l^{m*}(\theta',\phi')Y_l^m(\theta,\phi)$$

where $\gamma$ is the angle between the directions $(\theta,\phi)$ and $(\theta',\phi')$.

(a) For $l = 0$, verify this formula trivially.

(b) For $l = 1$, show that the formula gives $\cos\gamma = \cos\theta\cos\theta' + \sin\theta\sin\theta'\cos(\phi - \phi')$, which is the standard result from spherical trigonometry.

Problem 11 (Advanced)

Compute the matrix elements $\langle l', m'|\cos\theta|l, m\rangle$ for the transition $l = 1 \to l' = 0$ (all allowed $m$ values). Use the identity:

$$\cos\theta = \sqrt{\frac{4\pi}{3}}Y_1^0(\theta,\phi)$$

and the orthogonality of spherical harmonics. These matrix elements determine the relative intensities of spectral lines.

Problem 12 (Advanced)

(a) Expand the function $f(\theta,\phi) = \sin^2\theta\cos(2\phi)$ in terms of spherical harmonics.

Hint: Note that $\sin^2\theta\cos(2\phi) = \sin^2\theta\frac{1}{2}(e^{2i\phi} + e^{-2i\phi})$.

(b) What values of $l$ and $m$ appear in the expansion? What does this tell you about the angular momentum content of this function?

Section C: Radial Wavefunctions and Energy Levels (Problems 13--20)

Problem 13 (Basic)

(a) Compute the first four hydrogen energy levels $E_1, E_2, E_3, E_4$ in eV.

(b) What is the energy required to ionize hydrogen from the ground state? From the $n = 2$ state?

(c) What is the wavelength of the photon emitted in the transition $n = 3 \to n = 2$ (Balmer-$\alpha$)?

Problem 14 (Basic)

(a) Verify that the ground state wavefunction $\psi_{100} = (1/\sqrt{\pi})(1/a_0)^{3/2}e^{-r/a_0}$ is normalized.

Hint: $\int_0^\infty r^2 e^{-2r/a_0}dr = a_0^3/4$.

(b) What is the probability of finding the electron inside a sphere of radius $a_0$? Evaluate the integral numerically.

Problem 15 (Intermediate)

For the hydrogen ground state $\psi_{100}$:

(a) Compute $\langle r \rangle$ and $\langle r^2 \rangle$. Verify that $\langle r \rangle = 3a_0/2$ and $\langle r^2\rangle = 3a_0^2$.

(b) Compute the uncertainty $\Delta r = \sqrt{\langle r^2\rangle - \langle r\rangle^2}$.

(c) Compute $\langle 1/r \rangle$ and $\langle 1/r^2 \rangle$. Verify the general formulas given in Section 5.7.

(d) Compute the expectation value of the kinetic energy $\langle T \rangle$ using $\langle T \rangle = -\langle V \rangle/2$ (the virial theorem) and verify that $\langle T \rangle + \langle V \rangle = E_1$.

Problem 16 (Intermediate)

The $2s$ radial wavefunction is $R_{20}(r) = \frac{1}{2\sqrt{2}}(1/a_0)^{3/2}(2 - r/a_0)e^{-r/2a_0}$.

(a) Find the radial node (the value of $r$ where $R_{20} = 0$, excluding $r = 0$ and $r = \infty$).

(b) Compute the radial probability density $P_{20}(r) = r^2|R_{20}|^2$ and find its maxima.

(c) Sketch $P_{20}(r)$. At what radius is the electron most likely to be found?

Problem 17 (Intermediate)

(a) How many radial nodes does $R_{42}(r)$ have?

(b) How many angular nodes does $\psi_{42m}$ have?

(c) At the nucleus ($r = 0$), what is $R_{nl}(0)$ for $l = 0$? For $l \geq 1$? Explain physically.

(d) Show that $|R_{n0}(0)|^2 = 1/(\pi n^3 a_0^3)$ for all $n$ (for the s-states).

Problem 18 (Advanced)

The radial equation for hydrogen with substitution $u(r) = rR(r)$ reads:

$$-\frac{\hbar^2}{2m_e}\frac{d^2u}{dr^2} + V_{\text{eff}}(r)u = Eu$$

(a) Compute $V_{\text{eff}}(r)$ for $l = 0, 1, 2$ and sketch all three on the same plot. Identify the classical turning points for $E = E_2$.

(b) For each $l$ value, determine the number of bound states with $n = 2$. Explain why there is no $2d$ state.

Problem 19 (Advanced)

Virial theorem for the Coulomb potential. For a Hamiltonian $\hat{H} = \hat{T} + \hat{V}$ with $V(r) \propto r^k$:

$$2\langle T \rangle = k\langle V \rangle$$

(a) For the Coulomb potential ($k = -1$), show that $\langle T \rangle = -\frac{1}{2}\langle V \rangle$ and $\langle T \rangle = -E_n$.

(b) Use this to compute $\langle T \rangle$ and $\langle V \rangle$ for the ground state of hydrogen without performing any integrals.

(c) Why is $\langle T \rangle > 0$ even though $E_n < 0$?

Problem 20 (Advanced)

Stark effect preview. When an external electric field $\mathcal{E}$ is applied along the $z$-axis, the perturbation is $H' = e\mathcal{E}z = e\mathcal{E}r\cos\theta$.

(a) Compute the matrix element $\langle 2,1,0|r\cos\theta|2,0,0\rangle$ explicitly. You will need the radial wavefunctions $R_{20}$ and $R_{21}$ and the angular integral involving $Y_1^0$ and $Y_0^0$.

(b) Show that $\langle 2,0,0|r\cos\theta|2,0,0\rangle = 0$. What symmetry is responsible?

(c) The non-vanishing matrix element in (a) means the $n = 2$ level splits linearly in $\mathcal{E}$. This is the linear Stark effect, which we will study in Chapter 18.

Section D: Spectral Calculations and Selection Rules (Problems 21--25)

Problem 21 (Basic)

(a) Compute the wavelengths of the first four lines of the Lyman series ($n_f = 1$, $n_i = 2, 3, 4, 5$).

(b) What is the series limit (the shortest wavelength in the Lyman series)? What physical process does this correspond to?

(c) Repeat for the first four lines of the Balmer series ($n_f = 2$).

Problem 22 (Intermediate)

Determine which of the following transitions are allowed by electric dipole selection rules ($\Delta l = \pm 1$, $\Delta m = 0, \pm 1$), and state the spectral series for the allowed ones:

(a) $3p \to 1s$ (b) $3d \to 1s$ (c) $4f \to 3d$ (d) $4s \to 3s$ (e) $5d \to 3p$ (f) $3s \to 2p$

Problem 23 (Intermediate)

A hydrogen atom is in the $n = 4$ level.

(a) List all possible transitions to lower levels that are allowed by selection rules.

(b) For each allowed transition, compute the photon wavelength and identify the spectral series.

(c) Draw an energy level diagram showing all allowed transitions from $n = 4$.

Problem 24 (Advanced)

Hydrogen in a star. A stellar atmosphere contains hydrogen at temperature $T = 10{,}000$ K.

(a) Using the Boltzmann factor $N_n/N_1 = (n^2/1^2)e^{-(E_n - E_1)/k_BT}$, compute the ratio $N_2/N_1$ of atoms in $n = 2$ to atoms in $n = 1$. (Include the degeneracy factor $g_n = n^2$.)

(b) What fraction of hydrogen atoms are in the $n = 2$ state? In $n = 3$?

(c) Explain why the Balmer series (requiring atoms in $n = 2$) is visible in this star's spectrum even though the vast majority of atoms are in the ground state.

Problem 25 (Advanced)

Reduced mass correction. The hydrogen energy levels should use the reduced mass $\mu = m_e m_p/(m_e + m_p)$ instead of $m_e$.

(a) Compute $\mu/m_e$ for hydrogen. By what fraction do the energy levels shift?

(b) Compute the corresponding quantity for deuterium ($m_D \approx 2m_p$). What is the isotope shift (difference in wavelength) for the Balmer-$\alpha$ line between hydrogen and deuterium?

(c) Harold Urey discovered deuterium in 1931 by observing this isotope shift. The measured shift was about 1.79 \AA. Compare with your calculation.

Section E: Orbital Identification and the Periodic Table (Problems 26--30)

Problem 26 (Basic)

For each of the following, state whether the quantum numbers $(n, l, m)$ are allowed, and if so, name the orbital:

(a) $(3, 2, 1)$ (b) $(2, 2, 0)$ (c) $(4, 3, -3)$ (d) $(1, 0, 1)$ (e) $(5, 0, 0)$ (f) $(3, 1, -2)$

Problem 27 (Basic)

Determine the electron configurations for the following atoms using the Aufbau principle:

(a) Carbon ($Z = 6$) (b) Silicon ($Z = 14$) (c) Iron ($Z = 26$) (d) Krypton ($Z = 36$)

For each, identify the outermost (valence) subshell.

Problem 28 (Intermediate)

(a) What is the maximum number of electrons that can have $n = 3$?

(b) What is the maximum number of electrons that can have $n = 3$ and $l = 2$?

(c) What is the maximum number of electrons that can have $n = 3$, $l = 2$, and $m_l = 1$?

(d) List all four quantum numbers $(n, l, m_l, m_s)$ for every electron in the ground state of nitrogen ($Z = 7$).

Problem 29 (Advanced)

Scaling for hydrogen-like ions. For a hydrogen-like ion with nuclear charge $Z$ (e.g., He$^+$, Li$^{2+}$), the energy levels are $E_n = -Z^2(13.6\;\text{eV})/n^2$ and the Bohr radius scales as $a_0/Z$.

(a) Compute the ground state energy and "Bohr radius" for He$^+$ ($Z = 2$) and Li$^{2+}$ ($Z = 3$).

(b) The Lyman-$\alpha$ transition for He$^+$ has the same energy as which hydrogen transition?

(c) Write the ground state wavefunction for a hydrogen-like ion with charge $Z$.

Problem 30 (Advanced)

Comprehensive hydrogen problem. A hydrogen atom is in the state:

$$\psi = \frac{1}{\sqrt{6}}\psi_{210} + \frac{1}{\sqrt{3}}\psi_{211} + \frac{1}{\sqrt{6}}\psi_{300} + \frac{1}{\sqrt{3}}\psi_{100}$$

(a) Verify that $\psi$ is normalized (use orthogonality of the $\psi_{nlm}$).

(b) If you measure the energy, what values might you get and with what probabilities?

(c) If you measure $\hat{L}^2$, what values might you get and with what probabilities?

(d) If you measure $\hat{L}_z$, what values might you get and with what probabilities?

(e) After measuring the energy and obtaining $E_2$, what is the state of the system? If you then measure $\hat{L}_z$, what values might you get?