Chapter 15 Quiz: Identical Particles

Instructions: This quiz covers the core concepts from Chapter 15. For multiple choice, select the single best answer. For true/false, provide a brief justification (1-2 sentences). For short answer, aim for 3-5 sentences. For applied scenarios, show your work.

Multiple Choice (10 questions)

Q1. The eigenvalues of the exchange operator $\hat{P}_{12}$ are:

(a) $0$ and $1$ (b) $+1$ and $-1$ (c) $+1$ and $+i$ (d) Any complex number of modulus 1

Q2. Two identical bosons can each be in one of $n$ single-particle states. The number of distinct two-boson states is:

(a) $n^2$ (b) $n(n-1)/2$ (c) $n(n+1)/2$ (d) $2n$

Q3. A Slater determinant for $N$ fermions automatically vanishes when:

(a) Any two rows are identical (b) Any two columns are identical (c) $N$ is odd (d) The single-particle states are orthogonal

Q4. In the helium atom with configuration $(1s)(2s)$, the triplet state has lower energy than the singlet state. This is because:

(a) The exchange integral $K$ is negative (b) The antisymmetric spatial wavefunction keeps the electrons farther apart, reducing Coulomb repulsion (c) Parallel spins attract each other magnetically (d) The Pauli exclusion principle forces the triplet to have less kinetic energy

Q5. The spin-statistics theorem states that particles with half-integer spin are:

(a) Bosons with symmetric wavefunctions (b) Fermions with antisymmetric wavefunctions (c) Either bosons or fermions depending on the interaction (d) Described by Maxwell-Boltzmann statistics

Q6. Which of the following is a boson?

(a) Electron (b) Proton (c) Neutron (d) $^4$He atom

Q7. The Pauli exclusion principle in the formalism of identical particles is:

(a) An independent postulate of quantum mechanics (b) A consequence of the antisymmetry of the fermionic wavefunction (c) A consequence of the Heisenberg uncertainty principle (d) Only valid for electrons, not other fermions

Q8. For two identical particles in single-particle states $|\alpha\rangle$ and $|\beta\rangle$, the antisymmetric state is:

(a) $|\alpha\rangle_1 |\beta\rangle_2 - |\beta\rangle_1 |\alpha\rangle_2$ (b) $\frac{1}{\sqrt{2}}(|\alpha\rangle_1 |\beta\rangle_2 - |\beta\rangle_1 |\alpha\rangle_2)$ (c) $\frac{1}{\sqrt{2}}(|\alpha\rangle_1 |\beta\rangle_2 + |\beta\rangle_1 |\alpha\rangle_2)$ (d) $|\alpha\rangle_1 |\beta\rangle_2$

Q9. The exchange integral $K$ in the two-electron problem:

(a) Has a simple classical interpretation as the electrostatic energy between charge distributions (b) Is always negative (c) Has no classical analogue and arises purely from the symmetrization requirement (d) Depends on the magnetic interaction between electron spins

Q10. At $T = 0$, the Fermi-Dirac distribution $\bar{n}_{\text{FD}}(\epsilon)$ is:

(a) A smooth exponential decay (b) A step function: 1 below $\epsilon_F$ and 0 above (c) Zero for all energies (d) Equal to the Bose-Einstein distribution

True/False (4 questions)

For each statement, indicate whether it is TRUE or FALSE, and provide a brief justification (1-2 sentences).

Q11. TRUE or FALSE: Two identical bosons can both occupy the same single-particle quantum state.

Q12. TRUE or FALSE: The ground state of helium has a symmetric spatial wavefunction and an antisymmetric spin state (singlet).

Q13. TRUE or FALSE: The spin-statistics theorem can be proved using only non-relativistic quantum mechanics.

Q14. TRUE or FALSE: A composite particle made of an even number of fermions is always a boson.

Short Answer (4 questions)

Q15. Explain the physical meaning of the statement "$[\hat{H}, \hat{P}_{12}] = 0$ for identical particles." Why must this commutator vanish, and what does it imply about the time evolution of the exchange symmetry?

Q16. Without the Pauli exclusion principle, what would happen to the structure of atoms? Describe at least two specific consequences for the periodic table and for chemistry.

Q17. Explain the difference between the direct integral $J$ and the exchange integral $K$ in a two-electron system. Which has a classical analogue, and which does not?

Q18. Why is the thermal de Broglie wavelength $\lambda_{\text{th}} = h/\sqrt{2\pi m k_B T}$ the relevant quantity for determining when quantum statistics becomes important? What happens when $\lambda_{\text{th}}$ is comparable to the inter-particle spacing?

Applied Scenarios (2 questions)

Q19. Three electrons are placed in a one-dimensional infinite square well. The single-particle energy levels are $E_n = n^2 E_1$, where $E_1 = \pi^2\hbar^2/(2mL^2)$.

(a) What is the ground-state energy of the three-electron system? Explain your reasoning using the Pauli exclusion principle.

(b) List the quantum numbers $(n, m_s)$ for each electron in the ground state.

(c) What is the energy of the first excited state? How many degenerate configurations does it have?

Q20. Consider two identical spin-1/2 fermions in an attractive harmonic oscillator potential with single-particle energies $E_n = (n + 1/2)\hbar\omega$.

(a) What is the ground-state energy? Write the ground-state wavefunction as a product of spatial and spin parts.

(b) What is the energy of the first excited state? Is the first excited state a singlet or a triplet (or both)?

(c) For the first excited state, which has lower energy — the singlet or the triplet? Explain qualitatively in terms of the spatial probability distribution.

Answer Key

Q1: (b) — $\hat{P}_{12}^2 = \hat{I}$ implies $\lambda^2 = 1$, so $\lambda = \pm 1$.

Q2: (c) — $\binom{n+1}{2} = n(n+1)/2$ (stars and bars with repetition allowed).

Q3: (b) — Two identical columns means two fermions in the same state; the determinant vanishes.

Q4: (b) — The antisymmetric spatial wavefunction has a node at $\mathbf{r}_1 = \mathbf{r}_2$, keeping electrons apart and reducing Coulomb repulsion.

Q5: (b)

Q6: (d) — $^4$He has 2 protons + 2 neutrons + 2 electrons = 6 fermions (even), so it is a boson.

Q7: (b)

Q8: (b)

Q9: (c)

Q10: (b)

Q11: TRUE — Bosons have symmetric wavefunctions. The state $|\alpha\rangle_1|\alpha\rangle_2$ is already symmetric and is an allowed state.

Q12: TRUE — Both electrons in the $1s$ orbital (symmetric spatial part) requires the spin part to be antisymmetric (singlet) to make the total wavefunction antisymmetric.

Q13: FALSE — The spin-statistics theorem requires Lorentz invariance, locality/causality, and positive energy — features of relativistic quantum field theory, not non-relativistic QM.

Q14: TRUE — Exchanging two composites exchanges all constituents. An even number of fermion exchanges gives $(-1)^{\text{even}} = +1$, corresponding to bosonic behavior.

Q15: The commutator $[\hat{H}, \hat{P}_{12}] = 0$ means the Hamiltonian is invariant under particle exchange — it does not depend on which particle is labeled "1" and which is "2." This must hold because the labels are unphysical for identical particles. The consequence is that exchange symmetry is conserved: if a state starts symmetric (antisymmetric), it remains symmetric (antisymmetric) under time evolution, because $\hat{P}_{12}$ and $\hat{H}$ share simultaneous eigenstates.

Q16: Without the exclusion principle, all electrons would collapse into the $1s$ orbital. All atoms would have roughly the same size ($\sim a_0/Z^{1/3}$) and the same chemical properties. There would be no shell structure, no periodic table, no distinct elements with different reactivities, no chemistry, and no complex molecules.

Q17: The direct integral $J$ represents the classical electrostatic repulsion between two charge distributions $|\phi_a|^2$ and $|\phi_b|^2$. It is always positive. The exchange integral $K$ has no classical analogue — its integrand involves the exchange of orbital labels between the two coordinates. It arises entirely from the requirement to symmetrize or antisymmetrize the wavefunction. For the Coulomb interaction, $K > 0$, and it creates the singlet-triplet energy splitting $2K$.

Q18: The thermal de Broglie wavelength $\lambda_{\text{th}}$ measures the quantum "size" of a particle at temperature $T$ — the spatial extent of its wave packet. When $\lambda_{\text{th}}$ is much smaller than the average inter-particle spacing $d = n^{-1/3}$, the wavefunctions of different particles don't overlap, and the particles behave as distinguishable (classical statistics applies). When $\lambda_{\text{th}} \gtrsim d$, wavefunctions overlap significantly, and the particles "know" they are identical — quantum statistics (Bose-Einstein or Fermi-Dirac) becomes essential.

Q19: (a) The first two electrons fill $n=1$ (one spin-up, one spin-down), and the third must go to $n=2$ (since the $n=1$ level is full). Ground-state energy: $E_1 + E_1 + 4E_1 = 6E_1$. (b) $(1, +1/2)$, $(1, -1/2)$, $(2, +1/2)$ or $(2, -1/2)$. (c) First excited state has one electron promoted to $n=2$ with the other spin, or the third electron in $n=2$ is promoted to $n=3$. If we interpret "first excited" as the third electron going from $n=2$ to $n=3$: energy $= 2E_1 + 9E_1 = 11E_1$. This state has degeneracy 2 (the $n=2$ and $n=3$ electrons can each be spin-up or spin-down, with constraints from the filled $n=1$ shell).

Q20: (a) $E_0 = \frac{1}{2}\hbar\omega + \frac{1}{2}\hbar\omega = \hbar\omega$. Both in $n=0$ spatial state (symmetric spatial), paired with the spin singlet: $|\Psi\rangle = \phi_0(x_1)\phi_0(x_2) \cdot \frac{1}{\sqrt{2}}(|\!\uparrow\downarrow\rangle - |\!\downarrow\uparrow\rangle)$. (b) $E_1 = \frac{1}{2}\hbar\omega + \frac{3}{2}\hbar\omega = 2\hbar\omega$. One particle in $n=0$, one in $n=1$. Both singlet and triplet configurations exist. (c) For an attractive potential (no electron-electron repulsion in this problem), singlet and triplet have the same energy $2\hbar\omega$. If we add a repulsive interaction between the particles, the triplet (antisymmetric spatial, particles farther apart) would have lower energy.