Quiz — Chapter 5: Quantum Mechanics Review

Test your understanding of the quantum mechanical tools developed in this chapter. Answers appear at the end.

Question 1

A nucleon occupies the $1d_{3/2}$ single-particle state. What are the values of $l$, $j$, and the parity $\pi$?

(a) $l = 2$, $j = 3/2$, $\pi = +1$ \ (b) $l = 3$, $j = 3/2$, $\pi = -1$ \ (c) $l = 1$, $j = 3/2$, $\pi = -1$ \ (d) $l = 2$, $j = 3/2$, $\pi = -1$

Question 2

Two angular momenta $j_1 = 5/2$ and $j_2 = 3/2$ are coupled. Which of the following is NOT an allowed value of the total angular momentum $J$?

(a) $J = 1$ \ (b) $J = 2$ \ (c) $J = 5$ \ (d) $J = 4$

Question 3

Two identical neutrons occupy the $1g_{9/2}$ shell ($j = 9/2$). Which values of the total angular momentum $J$ are allowed by the Pauli principle?

(a) $J = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ \ (b) $J = 1, 3, 5, 7, 9$ \ (c) $J = 0, 2, 4, 6, 8$ \ (d) $J = 0, 1, 2, 3, 4$

Question 4

In the nuclear shell model, j-j coupling is preferred over L-S coupling because:

(a) The nuclear spin-orbit interaction is weak \ (b) The nuclear spin-orbit interaction is strong, comparable to level spacings \ (c) Nucleons have zero orbital angular momentum \ (d) L-S coupling violates parity conservation

Question 5

What is the Clebsch-Gordan coefficient $\langle j\, m;\, j\, {-m} | 0\, 0\rangle$ for $j = 1/2$, $m = 1/2$?

(a) $1/\sqrt{2}$ \ (b) $-1/\sqrt{2}$ \ (c) $1$ \ (d) $0$

Question 6

A nuclear state undergoes an E2 (electric quadrupole) gamma transition. What parity change does this transition require?

(a) Parity must change ($\pi_i \neq \pi_f$) \ (b) Parity must not change ($\pi_i = \pi_f$) \ (c) Parity change depends on the nuclear mass number \ (d) Parity is not conserved in electromagnetic transitions

Question 7

Which of the following transitions CANNOT proceed by single-photon emission?

(a) $2^+ \to 0^+$ \ (b) $3^- \to 2^+$ \ (c) $0^+ \to 0^+$ \ (d) $1^- \to 0^+$

Question 8

In Fermi's golden rule $\Gamma = (2\pi/\hbar)|V_{fi}|^2 \rho(E_f)$, the density of states $\rho(E_f)$ represents:

(a) The number of nucleons in the initial state \ (b) The probability of finding the system in the initial state \ (c) The number of available final states per unit energy \ (d) The energy of the emitted particle

Question 9

A beta decay is classified as "first forbidden" rather than "allowed." This means the half-life is expected to be:

(a) Shorter than an allowed transition by a factor of ~$10^3$ \ (b) Longer than an allowed transition by a factor of ~$10^3$--$10^4$ \ (c) The same as an allowed transition \ (d) Exactly twice as long as an allowed transition

Question 10

In the WKB approximation, the tunneling probability through a barrier is $T \approx e^{-2\gamma}$ where $\gamma$ is:

(a) The barrier height divided by the particle energy \ (b) An integral of $\sqrt{2m(V-E)}/\hbar$ across the classically forbidden region \ (c) The ratio of the nuclear radius to the Bohr radius \ (d) The Sommerfeld parameter $\eta$

Question 11

Alpha decay half-lives of heavy nuclei span approximately 25 orders of magnitude, while alpha particle energies vary by only a factor of about 2. This enormous range in half-lives is explained by:

(a) Variations in nuclear spin \ (b) The exponential sensitivity of quantum tunneling to barrier parameters \ (c) Differences in nuclear binding energy \ (d) Relativistic effects on the alpha particle

Question 12

The Gamow peak in stellar fusion arises from the competition between:

(a) Nuclear attraction and electromagnetic repulsion \ (b) The tunneling probability (increases with energy) and the Maxwell-Boltzmann distribution (decreases with energy) \ (c) Proton and neutron capture rates \ (d) Fission and fusion cross sections

Question 13

For the ground state of $^{17}$O, the shell model predicts $J^\pi = 5/2^+$. Which single nucleon determines this?

(a) The last proton, in the $1p_{1/2}$ orbit \ (b) The last neutron, in the $1d_{5/2}$ orbit \ (c) A proton-neutron pair in the $1p$ shell \ (d) The last neutron, in the $2s_{1/2}$ orbit

Question 14

A Slater determinant for an $A$-nucleon system is zero when:

(a) $A$ is odd \ (b) Any two single-particle states are orthogonal \ (c) Any two nucleons occupy the same quantum state \ (d) The total angular momentum is zero

Question 15

The nuclear level density $\rho(E^*)$ increases approximately as:

(a) Linearly with excitation energy: $\rho \propto E^*$ \ (b) Quadratically: $\rho \propto (E^*)^2$ \ (c) Exponentially: $\rho \propto \exp(2\sqrt{aE^*})$ \ (d) As a power law: $\rho \propto (E^*)^{A/4}$

Question 16

For two nucleons in the isospin formalism, the antisymmetry requirement is $(-1)^{l+S+T} = -1$. The deuteron has $T = 0$ and $S = 1$. What does this imply for the orbital angular momentum $l$?

(a) $l$ must be odd \ (b) $l$ must be even \ (c) $l$ can be any value \ (d) $l$ must be zero

Question 17

Which quantity does Fermi's golden rule NOT directly provide?

(a) Decay rate of an excited nuclear state \ (b) Cross section for a nuclear reaction (when combined with the incident flux) \ (c) Binding energy of a nucleus \ (d) Transition rate for gamma emission

Question 18

The Sommerfeld parameter $\eta = z_1 z_2 e^2/(4\pi\epsilon_0\hbar v)$ for alpha-nucleus scattering is large ($\eta \gg 1$) when:

(a) The relative velocity is large (high energy) \ (b) The charges are small and the velocity is large \ (c) The charges are large and/or the velocity is small (low energy) \ (d) The nuclear radius is large

Answers

  1. (a) — $d$-wave means $l = 2$, so $\pi = (-1)^2 = +1$. The subscript gives $j = 3/2 = l - 1/2$.

  2. (c) — $J = 5$ exceeds $j_1 + j_2 = 5/2 + 3/2 = 4$. The allowed values are $J = 1, 2, 3, 4$.

  3. (c) — Two identical fermions in the same $j$-shell: only even $J$ allowed. For $j = 9/2$: $J = 0, 2, 4, 6, 8$.

  4. (b) — The strong spin-orbit force in nuclei makes j-j coupling the natural scheme.

  5. (a) — $\langle 1/2\, 1/2;\, 1/2\, -1/2 | 0\, 0\rangle = (-1)^{1/2-1/2}/\sqrt{2} = 1/\sqrt{2}$.

  6. (b) — For E$\lambda$ transitions, $\pi_i \cdot \pi_f = (-1)^\lambda$. For E2: $(-1)^2 = +1$, so no parity change.

  7. (c) — A photon carries at least one unit of angular momentum ($\lambda \geq 1$). A $0 \to 0$ transition cannot conserve angular momentum with a single photon.

  8. (c) — The density of states counts the number of available final states per unit energy interval.

  9. (b) — Each degree of forbiddenness typically suppresses the decay rate by $\sim 10^3$--$10^4$, lengthening the half-life by the same factor.

  10. (b) — $\gamma = (1/\hbar)\int_a^b \sqrt{2m[V(x)-E]}\, dx$ is the integral across the forbidden region.

  11. (b) — The WKB tunneling probability $T \sim e^{-2\gamma}$ is exponentially sensitive to the Gamow factor, which depends on $E_\alpha$ and the Coulomb barrier.

  12. (b) — The Gamow peak is the product $e^{-b/\sqrt{E}} \times e^{-E/k_BT}$, peaked at $E_G$.

  13. (b) — $Z = 8$ is magic (closed proton shell). The 9th neutron ($N = 9$) fills the $1d_{5/2}$ orbit: $l = 2$, $j = 5/2$, $\pi = +1$.

  14. (c) — Two identical rows in a determinant make it zero. This is the Pauli exclusion principle.

  15. (c) — The Bethe formula gives $\rho \propto \exp(2\sqrt{aE^*})$, an exponential increase.

  16. (b) — $(-1)^{l+1+0} = -1$ requires $l+1$ odd, so $l$ must be even. The deuteron is primarily $l = 0$ ($S$-wave).

  17. (c) — Fermi's golden rule gives transition rates. Binding energy is a static property determined by the nuclear Hamiltonian, not a transition rate.

  18. (c) — $\eta \propto z_1 z_2 / v$. Large charges and/or low velocity give large $\eta$, indicating a strong Coulomb effect.