Chapter 5 Quiz: The Hydrogen Atom
Multiple Choice (Questions 1--15)
1. The Laplacian operator $\nabla^2$ in spherical coordinates naturally separates into a radial part and an angular part because:
(a) Spherical coordinates are more mathematically convenient than Cartesian (b) The Coulomb potential is proportional to $1/r$ (c) The angular part of $\nabla^2$ does not involve $r$, allowing separation of variables for any central potential (d) The Schrodinger equation is linear
2. The spherical harmonics $Y_l^m(\theta,\phi)$ are eigenfunctions of which operator(s)?
(a) $\hat{L}_z$ only (b) $\hat{L}^2$ only (c) Both $\hat{L}^2$ and $\hat{L}_z$ (d) $\hat{L}_x$, $\hat{L}_y$, and $\hat{L}_z$ simultaneously
3. For the quantum number $l = 3$, the allowed values of the magnetic quantum number $m$ are:
(a) $0, 1, 2, 3$ (b) $-3, -2, -1, 0, 1, 2, 3$ (c) $-2, -1, 0, 1, 2$ (d) $1, 2, 3$
4. The hydrogen atom energy levels $E_n = -13.6\;\text{eV}/n^2$ depend on:
(a) $n$, $l$, and $m$ (b) $n$ and $l$ only (c) $n$ only (d) $n$ and $m$ only
5. The centrifugal barrier $\hbar^2 l(l+1)/(2m_e r^2)$ in the effective potential:
(a) Attracts the electron toward the nucleus for all $l$ (b) Pushes the electron away from the nucleus for $l \geq 1$ but vanishes for $l = 0$ (c) Is always negligible compared to the Coulomb term (d) Is a classical force, not a quantum effect
6. The total degeneracy of the hydrogen $n = 3$ energy level (including spin) is:
(a) 3 (b) 9 (c) 18 (d) 27
7. Which of the following transitions is FORBIDDEN by electric dipole selection rules?
(a) $3p \to 2s$ (b) $4d \to 3p$ (c) $3d \to 2s$ ($\Delta l = -2$) (d) $2p \to 1s$
8. The radial probability density $P(r)$ for the hydrogen ground state has its maximum at:
(a) $r = 0$ (at the nucleus) (b) $r = a_0/2$ (c) $r = a_0$ (the Bohr radius) (d) $r = 2a_0$
9. The $3d$ orbital has how many radial nodes?
(a) 0 (b) 1 (c) 2 (d) 3
10. Which subshell fills before $3d$ in multi-electron atoms?
(a) $3p$ (b) $4s$ (c) $4p$ (d) $3s$
11. The Balmer series of hydrogen corresponds to transitions ending at:
(a) $n_f = 1$ (b) $n_f = 2$ (c) $n_f = 3$ (d) $n_f = \infty$
12. The fact that states with different $l$ but the same $n$ have the same energy in hydrogen is called:
(a) Essential degeneracy (b) Exchange degeneracy (c) Accidental degeneracy (d) Kramers degeneracy
13. The Bohr model gives the correct ground state angular momentum for hydrogen:
(a) True --- both give $L = \hbar$ (b) False --- Bohr gives $L = \hbar$ but quantum mechanics gives $L = 0$ (c) True --- both give $L = 0$ (d) False --- Bohr gives $L = 0$ but quantum mechanics gives $L = \hbar$
14. For the $4f$ subshell ($n = 4$, $l = 3$), the maximum number of electrons is:
(a) 6 (b) 10 (c) 14 (d) 18
15. The expectation value $\langle r \rangle$ for the hydrogen ground state is:
(a) $a_0$ (b) $\frac{3}{2}a_0$ (c) $2a_0$ (d) $\frac{1}{2}a_0$
Short Answer (Questions 16--22)
16. State the three quantum numbers $(n, l, m)$ for the hydrogen atom and give the allowed range for each.
17. How many total nodes (radial + angular) does the wavefunction $\psi_{nlm}$ have? Express your answer in terms of $n$ and $l$.
18. Write the Rydberg formula for the wavelength of a photon emitted when a hydrogen atom transitions from level $n_i$ to level $n_f$. Use this formula to compute the wavelength of the Lyman-$\alpha$ line ($n_i = 2 \to n_f = 1$).
19. Explain in 2--3 sentences why the $1/r$ Coulomb potential produces energy levels that depend only on $n$ (and not on $l$), whereas a generic central potential would have energy levels depending on both $n$ and $l$.
20. A hydrogen atom is in the $2p$ state ($n = 2$, $l = 1$). What are the possible results of measuring $L_z$? What is the magnitude of the total orbital angular momentum $|\mathbf{L}|$?
21. A hydrogen-like ion has nuclear charge $Z$. Write the formula for its energy levels and Bohr radius in terms of $Z$ and the corresponding hydrogen quantities. Compute the ionization energy of He$^+$.
22. A student claims: "The electron in a $2p$ orbital travels back and forth along the $z$-axis, spending most of its time near the top and bottom lobes." Identify and correct the misconception(s) in this statement.
Answer Key
1. (c) --- The angular Laplacian $\hat{\Lambda}^2$ contains only $\theta$ and $\phi$ derivatives, allowing separation for any central potential $V(r)$.
2. (c) --- $Y_l^m$ are simultaneous eigenfunctions of $\hat{L}^2$ (eigenvalue $\hbar^2 l(l+1)$) and $\hat{L}_z$ (eigenvalue $m\hbar$). They cannot be eigenstates of $\hat{L}_x$ or $\hat{L}_y$ simultaneously because $[\hat{L}_x, \hat{L}_y] \neq 0$.
3. (b) --- $m$ ranges from $-l$ to $+l$ in integer steps, giving $2l + 1 = 7$ values.
4. (c) --- The hydrogen energy levels depend only on $n$. This is the "accidental" degeneracy of the Coulomb potential.
5. (b) --- The centrifugal term is proportional to $l(l+1)$, which is zero for $l = 0$. For $l \geq 1$, it creates a repulsive barrier that prevents the electron from reaching $r = 0$.
6. (c) --- Without spin: $n^2 = 9$. With spin: $2n^2 = 18$.
7. (c) --- The selection rule is $\Delta l = \pm 1$. The $3d \to 2s$ transition has $\Delta l = -2$, violating this rule.
8. (c) --- The radial probability density $P(r) = r^2|R_{10}|^2$ has its maximum at $r = a_0$, even though $|R_{10}|^2$ is maximal at $r = 0$. The $r^2$ factor from the volume element shifts the peak outward.
9. (a) --- The number of radial nodes is $n - l - 1 = 3 - 2 - 1 = 0$.
10. (b) --- $4s$ fills before $3d$ due to penetration and shielding effects in multi-electron atoms.
11. (b) --- The Balmer series corresponds to transitions with final state $n_f = 2$, producing visible light.
12. (c) --- The $l$-degeneracy in hydrogen is called "accidental" because it is not required by the obvious rotational symmetry. It arises from a hidden $SO(4)$ symmetry unique to the $1/r$ potential.
13. (b) --- Bohr's model assigns $L = n\hbar$, giving $L = \hbar$ for the ground state ($n = 1$). Quantum mechanics gives $L = \hbar\sqrt{l(l+1)} = 0$ for the $1s$ state ($l = 0$). The ground state has zero orbital angular momentum.
14. (c) --- $2(2l + 1) = 2(7) = 14$.
15. (b) --- $\langle r \rangle_{10} = \frac{3}{2}a_0$. The most probable radius is $a_0$, but the long tail of the distribution pulls the mean outward.
16. $n = 1, 2, 3, \ldots$ (principal); $l = 0, 1, \ldots, n-1$ (azimuthal); $m = -l, -l+1, \ldots, l$ (magnetic). Including spin: $m_s = \pm 1/2$.
17. Total nodes $= (n - l - 1) + l = n - 1$. Radial nodes: $n - l - 1$. Angular nodes: $l$.
18. $1/\lambda = R_\infty(1/n_f^2 - 1/n_i^2)$. For Lyman-$\alpha$: $1/\lambda = R_\infty(1 - 1/4) = 3R_\infty/4$, giving $\lambda = 4/(3 \times 1.097 \times 10^7) = 121.5\;\text{nm}$ (ultraviolet).
19. The $1/r$ Coulomb potential has a hidden dynamical symmetry (the Laplace-Runge-Lenz vector is conserved) that goes beyond the obvious $SO(3)$ rotational symmetry. This extra symmetry enlarges the symmetry group to $SO(4)$, which forces states with different $l$ (but the same $n$) to be degenerate. A generic central potential lacks this additional conserved quantity, so the energy depends on both $n$ and $l$.
20. For $l = 1$: $m = -1, 0, +1$, so $L_z$ can be $-\hbar$, $0$, or $+\hbar$. The magnitude is $|\mathbf{L}| = \hbar\sqrt{l(l+1)} = \hbar\sqrt{2} \approx 1.49 \times 10^{-34}\;\text{J}\cdot\text{s}$.
21. $E_n(Z) = -Z^2 \times 13.6\;\text{eV}/n^2$; $a_0(Z) = a_0/Z$. For He$^+$ ($Z = 2$): $E_1 = -4 \times 13.6 = -54.4\;\text{eV}$. Ionization energy $= 54.4\;\text{eV}$.
22. Multiple misconceptions: (1) The electron does not "travel" along any path --- there is no trajectory in quantum mechanics. (2) The $2p$ orbital is a probability distribution, not an orbit. (3) The lobes represent regions of high probability density; the electron has some probability of being found anywhere (except at nodes). (4) For the complex spherical harmonics $Y_1^{\pm 1}$, the probability density is actually symmetric about the $z$-axis, not concentrated "at the top and bottom." The dumbbell shape along $z$ corresponds to the real combination $p_z$, but even then, the electron does not move back and forth between lobes.