Chapter 15 Exercises: Identical Particles

Part A: Conceptual Questions (7 problems)

These questions test your understanding of the core ideas. No calculations required.

A.1 A friend says: "Two electrons are identical because we can't build instruments precise enough to tell them apart." Explain what is wrong with this statement. What does genuine quantum indistinguishability mean, and how does it differ from mere practical indistinguishability?

A.2 Consider two distinguishable spin-1/2 particles (say, an electron and a muon) both in a box. Do their wavefunctions need to be symmetrized or antisymmetrized? Explain why or why not. What changes if both particles are electrons?

A.3 Explain in your own words why the Pauli exclusion principle is a theorem in the formalism of identical particles, not an additional postulate. What is the actual postulate from which it follows?

A.4 The $\Delta^{++}$ baryon consists of three up quarks, all with spin up. This appears to put three identical fermions in the same quantum state, violating the Pauli exclusion principle. How is this resolved? What additional quantum number must exist?

A.5 Liquid $^4$He becomes a superfluid below 2.17 K, but liquid $^3$He does not become a superfluid until about 0.0025 K. Both isotopes have the same electronic structure and nearly the same mass. What accounts for the dramatically different behavior?

A.6 The exchange interaction makes the triplet state of two electrons lower in energy than the singlet state (Hund's first rule). Yet the ground state of helium has the electrons in a singlet. Is this a contradiction? Explain carefully.

A.7 A student claims: "The exchange interaction is a new fundamental force, like gravity or electromagnetism." Critique this statement. What is the actual origin of the exchange interaction?

A.8 Explain why the symmetrization postulate restricts the physically realizable states to a subspace of the full tensor product space $\mathcal{H}_1 \otimes \mathcal{H}_2$. Is this restriction analogous to any constraint in classical mechanics? Why or why not?

Part B: Applied Problems (11 problems)

These problems require direct application of the chapter's key equations and techniques.

B.1: Exchange Operator Properties

Let $\hat{P}_{12}$ be the exchange operator for two particles.

(a) Show that $\hat{P}_{12}$ is Hermitian: $\hat{P}_{12}^\dagger = \hat{P}_{12}$. Hint: Show that $\langle \Phi | \hat{P}_{12} | \Psi \rangle = \langle \hat{P}_{12}\Phi | \Psi \rangle$ for arbitrary product states.

(b) Show that $\hat{P}_{12}$ is unitary: $\hat{P}_{12}^\dagger \hat{P}_{12} = \hat{I}$.

(c) Verify that $\hat{P}_{12}^2 = \hat{I}$ and conclude that the eigenvalues of $\hat{P}_{12}$ are $\pm 1$.

(d) If $\hat{H}$ is a two-particle Hamiltonian of the form $\hat{H} = \hat{h}(\mathbf{r}_1) + \hat{h}(\mathbf{r}_2) + V(|\mathbf{r}_1 - \mathbf{r}_2|)$, show that $[\hat{H}, \hat{P}_{12}] = 0$.

B.2: Symmetrized States

Consider two identical bosons that can each be in one of three single-particle states: $|a\rangle$, $|b\rangle$, or $|c\rangle$.

(a) Write all possible two-particle states. How many are there?

(b) Now consider two identical fermions in the same three states. How many antisymmetric two-particle states are there?

(c) How many two-particle states would there be if the particles were distinguishable?

(d) Verify that the number of symmetric states is $\binom{n+N-1}{N}$ and the number of antisymmetric states is $\binom{n}{N}$ for $n=3$ single-particle states and $N=2$ particles.

B.3: Two Particles in a Box

Two identical particles occupy a one-dimensional infinite square well of width $L$. The single-particle eigenstates are $\phi_n(x) = \sqrt{2/L}\sin(n\pi x/L)$ with energies $E_n = n^2 \pi^2 \hbar^2/(2mL^2)$.

(a) Write the ground-state wavefunction if the particles are identical bosons (ignore spin). What is the ground-state energy?

(b) Write the ground-state wavefunction if the particles are identical fermions with the same spin projection (e.g., both spin-up). What is the ground-state energy?

(c) For the fermionic case in (b), show explicitly that the wavefunction vanishes when $x_1 = x_2$.

(d) Calculate $\langle (x_1 - x_2)^2 \rangle$ for both the bosonic and fermionic ground states. Which particles are, on average, farther apart?

B.4: Three-Electron Slater Determinant

Three electrons occupy the single-particle spin-orbitals $|1s\uparrow\rangle$, $|1s\downarrow\rangle$, and $|2s\uparrow\rangle$ (the ground-state configuration of lithium).

(a) Write the full $3 \times 3$ Slater determinant.

(b) Expand the determinant explicitly into six terms.

(c) Verify that exchanging particles 1 and 2 changes the sign of the wavefunction.

(d) What would happen if you tried to put all three electrons in the $1s$ orbital? Why is this impossible?

B.5: Normalization of Symmetrized States

Consider a symmetric two-particle state built from possibly identical single-particle states:

$$|\Psi_S\rangle = \mathcal{N}\left(|\alpha\rangle_1 |\beta\rangle_2 + |\beta\rangle_1 |\alpha\rangle_2\right)$$

(a) If $\langle \alpha | \beta \rangle = 0$, find $\mathcal{N}$.

(b) If $|\alpha\rangle = |\beta\rangle$, find $\mathcal{N}$.

(c) For the general case where $\langle \alpha | \beta \rangle = s$ (a complex number with $|s| < 1$), find $\mathcal{N}$ in terms of $s$.

B.6: Projection Operators

Define the symmetrization and antisymmetrization projectors:

$$\hat{\Pi}_S = \frac{1}{2}(\hat{I} + \hat{P}_{12}), \qquad \hat{\Pi}_A = \frac{1}{2}(\hat{I} - \hat{P}_{12})$$

(a) Show that $\hat{\Pi}_S^2 = \hat{\Pi}_S$ and $\hat{\Pi}_A^2 = \hat{\Pi}_A$ (idempotency).

(b) Show that $\hat{\Pi}_S \hat{\Pi}_A = 0$ (orthogonality).

(c) Show that $\hat{\Pi}_S + \hat{\Pi}_A = \hat{I}$ (completeness).

(d) If $|\Phi\rangle$ is an arbitrary two-particle state, what are $\hat{\Pi}_S|\Phi\rangle$ and $\hat{\Pi}_A|\Phi\rangle$? Verify that $\hat{P}_{12}(\hat{\Pi}_S|\Phi\rangle) = +\hat{\Pi}_S|\Phi\rangle$ and $\hat{P}_{12}(\hat{\Pi}_A|\Phi\rangle) = -\hat{\Pi}_A|\Phi\rangle$.

B.7: Helium Singlet-Triplet Splitting

For helium with one electron in the $1s$ state and one in the $2s$ state, the direct integral is $J = 1.19$ eV and the exchange integral is $K = 0.42$ eV.

(a) What are the energies of the singlet ($^1S_0$) and triplet ($^3S_1$) states, relative to $E_{1s} + E_{2s}$?

(b) Which state has lower energy? Explain physically why (consider the spatial probability distribution).

(c) The triplet state has degeneracy 3 (one for each $m_S = -1, 0, +1$). All three have the same energy. Why?

(d) If a magnetic field is applied, the triplet splits into three levels. What is this splitting called, and how does it depend on $m_S$?

B.8: Fermion Antibunching

Two identical fermions (same spin state) are in the antisymmetric spatial state:

$$\Psi_A(x_1, x_2) = \frac{1}{\sqrt{2}}[\phi_1(x_1)\phi_2(x_2) - \phi_2(x_1)\phi_1(x_2)]$$

where $\phi_1$ and $\phi_2$ are real, orthonormal functions.

(a) Compute the probability density $|\Psi_A(x_1, x_2)|^2$ and expand it.

(b) For distinguishable particles in the unsymmetrized state $\phi_1(x_1)\phi_2(x_2)$, compute $|\Psi|^2$.

(c) Show that the cross term in (a) is negative when $x_1 = x_2$, reducing the probability of finding both particles at the same point. Interpret this as "fermion antibunching."

(d) Compute $\langle x_1 x_2 \rangle$ for the antisymmetric state and compare it to the distinguishable case. Show that the difference is the exchange term.

B.9: Composite Particle Statistics

Determine whether each of the following composite particles is a boson or a fermion, and state why:

(a) A hydrogen atom ($^1$H: 1 proton + 1 electron)

(b) A deuterium atom ($^2$H: 1 proton + 1 neutron + 1 electron)

(c) An alpha particle ($^4$He nucleus: 2 protons + 2 neutrons)

(d) A $^6$Li atom (3 protons + 3 neutrons + 3 electrons)

(e) A Cooper pair (two electrons bound together in a superconductor)

(f) A positronium atom (1 electron + 1 positron)

B.10: Fermi-Dirac vs. Bose-Einstein

The average occupation number for a state of energy $\epsilon$ is:

$$\bar{n}_{\text{FD}} = \frac{1}{e^{(\epsilon-\mu)/k_BT}+1}, \qquad \bar{n}_{\text{BE}} = \frac{1}{e^{(\epsilon-\mu)/k_BT}-1}$$

(a) Show that $0 \leq \bar{n}_{\text{FD}} \leq 1$ for all $\epsilon$, but $\bar{n}_{\text{BE}}$ can be arbitrarily large.

(b) At $T = 0$, show that $\bar{n}_{\text{FD}}$ is a step function: $\bar{n}_{\text{FD}} = 1$ for $\epsilon < \mu$ and $\bar{n}_{\text{FD}} = 0$ for $\epsilon > \mu$.

(c) Show that both distributions reduce to the Maxwell-Boltzmann distribution $\bar{n}_{\text{MB}} = e^{-(\epsilon-\mu)/k_BT}$ when $e^{(\epsilon-\mu)/k_BT} \gg 1$.

(d) For the Bose-Einstein distribution, what is the maximum allowed value of $\mu$ if all states have $\epsilon \geq 0$? What happens as $\mu \to 0^-$?

B.11: Thermal de Broglie Wavelength

The thermal de Broglie wavelength is $\lambda_{\text{th}} = h/\sqrt{2\pi m k_B T}$.

(a) Calculate $\lambda_{\text{th}}$ for $^{87}$Rb atoms ($m = 1.44 \times 10^{-25}$ kg) at $T = 300$ K and at $T = 170$ nK.

(b) For $^{87}$Rb at density $n = 10^{19}$ m$^{-3}$, what is the average inter-particle spacing $d = n^{-1/3}$?

(c) At what temperature does $\lambda_{\text{th}} = d$? Compare this to the critical temperature for BEC: $T_c = \frac{2\pi\hbar^2}{mk_B}\left(\frac{n}{\zeta(3/2)}\right)^{2/3}$ where $\zeta(3/2) \approx 2.612$.

(d) Repeat (a) for electrons in a metal ($m = 9.11 \times 10^{-31}$ kg) at $T = 300$ K. Compare $\lambda_{\text{th}}$ to the typical inter-electron spacing in copper ($d \approx 0.23$ nm). Are electrons in metals in the quantum or classical regime?

Part C: Advanced Problems (9 problems)

These problems require deeper analysis, multi-step reasoning, or synthesis of ideas from multiple sections.

C.1: Exchange Force and Probability Distributions ⭐⭐⭐

Two identical particles are in single-particle states described by Gaussian wavefunctions centered at $\pm d/2$:

$$\phi_L(x) = \left(\frac{1}{\pi\sigma^2}\right)^{1/4} e^{-(x+d/2)^2/(2\sigma^2)}, \qquad \phi_R(x) = \left(\frac{1}{\pi\sigma^2}\right)^{1/4} e^{-(x-d/2)^2/(2\sigma^2)}$$

(a) Compute $\langle (x_1 - x_2)^2 \rangle$ for the symmetric and antisymmetric spatial wavefunctions. Express your answer in terms of $\sigma$ and $d$ and the overlap integral $s = \int \phi_L(x)\phi_R(x) \, dx$.

(b) Show that the difference $\langle (x_1 - x_2)^2 \rangle_A - \langle (x_1 - x_2)^2 \rangle_S$ is always positive, confirming that fermions (in the antisymmetric spatial state) are farther apart on average.

(c) Evaluate $s$ analytically (Gaussian integral). Plot $\langle (x_1-x_2)^2\rangle_S$ and $\langle (x_1-x_2)^2\rangle_A$ as functions of $d/\sigma$.

(d) In the limit $d \gg \sigma$ (well-separated particles), show that both $\langle (x_1 - x_2)^2 \rangle_{S,A} \to d^2 + 2\sigma^2$. Interpret: when the particles are far apart, exchange symmetry doesn't matter.

C.2: N-Fermion Slater Determinant and Vandermonde ⭐⭐⭐

Consider $N$ fermions in a one-dimensional harmonic oscillator. The single-particle states are $\phi_n(x) = H_n(\xi)e^{-\xi^2/2}/(\pi^{1/4}\sqrt{2^n n!})$ where $\xi = x\sqrt{m\omega/\hbar}$ and $H_n$ is the Hermite polynomial.

(a) Write the Slater determinant for $N$ fermions occupying the lowest $N$ states ($n = 0, 1, \ldots, N-1$).

(b) Factor out the common Gaussian $\exp(-\sum_i \xi_i^2/2)$ and the normalization. Show that the remaining determinant is a Vandermonde determinant:

$$\det\begin{pmatrix} 1 & \xi_1 & \xi_1^2 & \cdots & \xi_1^{N-1} \\ 1 & \xi_2 & \xi_2^2 & \cdots & \xi_2^{N-1} \\ \vdots & & & & \vdots \\ 1 & \xi_N & \xi_N^2 & \cdots & \xi_N^{N-1} \end{pmatrix} = \prod_{i

(c) The Vandermonde determinant vanishes whenever $\xi_i = \xi_j$ for any $i \neq j$. Explain why this is the Pauli exclusion principle in disguise.

(d) This Slater determinant is, up to normalization, the ground state of the Calogero-Sutherland model and appears in the fractional quantum Hall effect. Comment on how a wavefunction that vanishes at coincidence points creates an effective repulsion between particles.

C.3: Exchange Integral for Hydrogen 1s-2s ⭐⭐⭐

Using the hydrogen wavefunctions $\phi_{1s}(r) = (1/\sqrt{\pi a_0^3})e^{-r/a_0}$ and $\phi_{2s}(r) = (1/(4\sqrt{2\pi a_0^3}))(2-r/a_0)e^{-r/(2a_0)}$:

(a) Set up the exchange integral: $$K = \int\!\!\int \phi_{1s}^*(\mathbf{r}_1)\phi_{2s}^*(\mathbf{r}_2)\frac{e^2}{4\pi\epsilon_0|\mathbf{r}_1-\mathbf{r}_2|}\phi_{2s}(\mathbf{r}_1)\phi_{1s}(\mathbf{r}_2) \, d^3r_1 \, d^3r_2$$

(b) Expand $1/|\mathbf{r}_1 - \mathbf{r}_2|$ in terms of Legendre polynomials and show that only the $\ell = 0$ term survives (since both states are $s$-states).

(c) The resulting radial integrals can be evaluated analytically. Show that $K = (16/729)(e^2/(4\pi\epsilon_0 a_0)) = (16/729) \times 27.2$ eV $\approx 0.60$ eV for hydrogen. (For helium with $Z=2$, this scales as $Z$, giving $K \approx 1.2$ eV.)

(d) Compute the singlet-triplet energy splitting $2K$ and compare to the experimental value for helium ($\approx 0.80$ eV for the $(1s)(2s)$ configuration).

C.4: Three-Particle Permutation Group ⭐⭐⭐

For three identical particles, the permutation group $S_3$ has 6 elements.

(a) List all 6 permutations and classify each as even or odd.

(b) Show that the antisymmetrization operator $\hat{\mathcal{A}} = \frac{1}{6}\sum_{P \in S_3}(-1)^P \hat{P}$ is idempotent: $\hat{\mathcal{A}}^2 = \hat{\mathcal{A}}$.

(c) Apply $\hat{\mathcal{A}}$ to the state $|a\rangle_1|b\rangle_2|c\rangle_3$ and show that the result is the Slater determinant.

(d) For three identical bosons, the symmetrization operator is $\hat{\mathcal{S}} = \frac{1}{6}\sum_{P \in S_3}\hat{P}$. Apply $\hat{\mathcal{S}}$ to $|a\rangle_1|b\rangle_2|c\rangle_3$ where all three states are distinct. How many terms are in the normalized result?

C.5: Density Matrix of Indistinguishable Particles ⭐⭐⭐

Consider two identical particles in the antisymmetric state:

$$|\Psi_A\rangle = \frac{1}{\sqrt{2}}(|\alpha\rangle_1|\beta\rangle_2 - |\beta\rangle_1|\alpha\rangle_2)$$

(a) Compute the full two-particle density matrix $\hat{\rho}_{12} = |\Psi_A\rangle\langle\Psi_A|$.

(b) Trace over particle 2 to obtain the reduced density matrix for particle 1: $\hat{\rho}_1 = \text{Tr}_2(\hat{\rho}_{12})$.

(c) Show that $\hat{\rho}_1 = \frac{1}{2}(|\alpha\rangle\langle\alpha| + |\beta\rangle\langle\beta|)$ — a maximally mixed state in the $\{|\alpha\rangle, |\beta\rangle\}$ subspace.

(d) Compute the von Neumann entropy $S(\hat{\rho}_1) = -\text{Tr}(\hat{\rho}_1 \ln \hat{\rho}_1)$. Interpret: antisymmetrization forces entanglement, and the reduced state of a single fermion is mixed.

C.6: Pauli Paramagnetism ⭐⭐⭐

A gas of $N$ non-interacting spin-1/2 fermions at $T=0$ fills states up to the Fermi energy $\epsilon_F$. A weak magnetic field $B$ is applied, shifting spin-up and spin-down energies by $\mp \mu_B B$.

(a) Show that the number of spin-up and spin-down electrons that "flip" in response to the field is approximately $\frac{1}{2}g(\epsilon_F)\mu_B B$, where $g(\epsilon_F)$ is the density of states at the Fermi energy.

(b) Show that the resulting magnetization is $M = \mu_B^2 g(\epsilon_F) B$.

(c) For a free electron gas in 3D, $g(\epsilon) = \frac{V}{2\pi^2}\left(\frac{2m}{\hbar^2}\right)^{3/2}\epsilon^{1/2}$. Show that the Pauli spin susceptibility is:

$$\chi_{\text{Pauli}} = \mu_0 \mu_B^2 g(\epsilon_F) = \frac{3\mu_0 n \mu_B^2}{2\epsilon_F}$$

(d) Explain why Pauli paramagnetism is much weaker than classical Curie paramagnetism ($\chi \propto 1/T$) at room temperature. Why does only a thin shell of electrons near the Fermi surface contribute?

C.7: Hong-Ou-Mandel Effect ⭐⭐⭐

Two identical photons enter a 50/50 beam splitter from different input ports. The beam splitter transformation is:

$$\hat{a}_1^\dagger \to \frac{1}{\sqrt{2}}(\hat{a}_3^\dagger + i\hat{a}_4^\dagger), \qquad \hat{a}_2^\dagger \to \frac{1}{\sqrt{2}}(i\hat{a}_3^\dagger + \hat{a}_4^\dagger)$$

where 1,2 are input ports and 3,4 are output ports.

(a) The input state is $|1,1\rangle = \hat{a}_1^\dagger \hat{a}_2^\dagger |0\rangle$ (one photon in each input port). Apply the beam splitter transformation to find the output state.

(b) Show that the output state is $\frac{i}{\sqrt{2}}(|2,0\rangle + |0,2\rangle)$. Both photons always exit the same port!

(c) Explain why the $|1,1\rangle$ output (one photon in each output port) has zero amplitude. This is the Hong-Ou-Mandel effect — destructive interference of the two paths that would produce one photon in each output.

(d) What would happen if the two input photons were distinguishable (e.g., different polarizations)? Show that the $|1,1\rangle$ output probability becomes 1/2.

C.8: Degeneracy Pressure and White Dwarfs ⭐⭐⭐

A white dwarf star is supported against gravitational collapse by the degeneracy pressure of electrons. Model the electrons as a non-interacting Fermi gas at $T = 0$.

(a) The Fermi energy of a free electron gas at density $n$ is $\epsilon_F = \frac{\hbar^2}{2m_e}(3\pi^2 n)^{2/3}$. For a white dwarf with density $\rho \approx 10^9$ kg/m$^3$ and assuming approximately one electron per two nucleons ($n_e \approx \rho/(2m_p)$), compute $\epsilon_F$ in eV.

(b) The degeneracy pressure at $T = 0$ is $P = \frac{2}{5}n\epsilon_F = \frac{\hbar^2}{5m_e}(3\pi^2)^{2/3} n^{5/3}$. Compute $P$ for the density in part (a).

(c) For the white dwarf to be in equilibrium, the degeneracy pressure must balance the gravitational pressure $P_{\text{grav}} \sim GM^2/(R^4)$ (dimensional estimate). Show that this gives a mass-radius relation $R \propto M^{-1/3}$: more massive white dwarfs are smaller.

(d) The Chandrasekhar limit arises when the electrons become relativistic ($\epsilon_F \sim m_e c^2$). Estimate the maximum mass of a white dwarf by setting $\epsilon_F = m_e c^2$ and using the result from (c). Compare to the accepted value of $\sim 1.4 M_\odot$.

C.9: Entanglement Entropy of Identical Particles ⭐⭐⭐

Consider $N$ identical fermions occupying $N$ out of $2N$ available single-particle states (half-filling).

(a) For $N = 1$ (one fermion in 2 states, in a Slater determinant with one state): the reduced density matrix of a subregion containing one state is trivially $\rho_{\text{sub}} = |1\rangle\langle 1|$ (pure state). What is the entanglement entropy?

(b) For $N = 2$ fermions in 4 states, consider the Slater determinant where the fermions occupy states 1 and 2. Partition the states into two groups: $A = \{1, 2\}$ and $B = \{3, 4\}$. Show that the reduced density matrix $\rho_A$ is a pure state (since both fermions are entirely in region $A$). What is the entanglement entropy?

(c) Now consider $N = 2$ fermions in states 1 and 3, with the same partition $A = \{1, 2\}$, $B = \{3, 4\}$. One fermion is in region $A$ and one in region $B$. Compute the reduced density matrix $\rho_A$ and show it has entropy $S = \ln 2$.

(d) Generalize: for a Slater determinant of $N$ fermions at half-filling, how does the entanglement entropy between two halves of the system scale with the number of states? Compare to the "volume law" and "area law" expectations.

Solutions Notes

Detailed solutions to selected problems (B.1, B.3, B.7, C.1, C.3, C.7) are available in the Instructor Guide.