Chapter 34 Exercises: Second Quantization

Problems are organized by section and graded by difficulty: - [B] Basic — direct application of formulas and definitions - [I] Intermediate — requires multi-step reasoning or combining concepts - [A] Advanced — requires deeper analysis, proof, or synthesis

Section 34.1: Bosonic Creation and Annihilation Operators

Problem 34.1 [B]

Using the bosonic commutation relation $[\hat{a}, \hat{a}^\dagger] = 1$, prove that $\hat{a}^\dagger|n\rangle = \sqrt{n+1}\,|n+1\rangle$ and $\hat{a}|n\rangle = \sqrt{n}\,|n-1\rangle$.

(Hint: Use $\hat{n} = \hat{a}^\dagger\hat{a}$ and the commutation relation $[\hat{n}, \hat{a}^\dagger] = \hat{a}^\dagger$ to show that $\hat{a}^\dagger|n\rangle$ is an eigenstate of $\hat{n}$ with eigenvalue $n+1$. Then fix the normalization constant.)

Problem 34.2 [B]

Verify that the state $|n\rangle = \frac{(\hat{a}^\dagger)^n}{\sqrt{n!}}|0\rangle$ is normalized: $\langle n|n\rangle = 1$.

Problem 34.3 [B]

For two bosonic modes with operators $\hat{a}_1$, $\hat{a}_1^\dagger$, $\hat{a}_2$, $\hat{a}_2^\dagger$, compute the following commutators:

(a) $[\hat{a}_1, \hat{a}_2^\dagger]$

(b) $[\hat{n}_1, \hat{a}_2^\dagger]$

(c) $[\hat{a}_1^\dagger \hat{a}_2, \hat{a}_2^\dagger \hat{a}_1]$

Problem 34.4 [I]

Show that the bosonic commutation relations imply that the multi-particle state $|n_1, n_2\rangle = \frac{(\hat{a}_1^\dagger)^{n_1}}{\sqrt{n_1!}}\frac{(\hat{a}_2^\dagger)^{n_2}}{\sqrt{n_2!}}|0\rangle$ is symmetric under exchange of the two modes. That is, verify that creating particles in mode 1 first and then mode 2 gives the same state as creating in mode 2 first and then mode 1.

Problem 34.5 [I]

Compute the expectation value $\langle n|\hat{a}\hat{a}^\dagger|n\rangle$ using: (a) the commutation relation, and (b) by inserting a complete set of states. Verify that both methods agree.

Problem 34.6 [I]

A bosonic mode is in the coherent state $|\alpha\rangle = e^{-|\alpha|^2/2}\sum_n \frac{\alpha^n}{\sqrt{n!}}|n\rangle$. Compute:

(a) $\langle\hat{n}\rangle = \langle\alpha|\hat{a}^\dagger\hat{a}|\alpha\rangle$

(b) $\langle\hat{n}^2\rangle = \langle\alpha|(\hat{a}^\dagger\hat{a})^2|\alpha\rangle$

(c) $(\Delta n)^2 = \langle\hat{n}^2\rangle - \langle\hat{n}\rangle^2$

(d) Show that $\Delta n / \langle\hat{n}\rangle \to 0$ as $|\alpha| \to \infty$ (the relative fluctuation vanishes in the classical limit).

Section 34.2: Fermionic Creation and Annihilation Operators

Problem 34.7 [B]

Using the anticommutation relation $\{\hat{c}, \hat{c}^\dagger\} = 1$, show that $\hat{n}^2 = \hat{n}$ where $\hat{n} = \hat{c}^\dagger\hat{c}$. Conclude that the eigenvalues of $\hat{n}$ are 0 and 1.

Problem 34.8 [B]

For two fermionic modes, compute:

(a) $\hat{c}_1^\dagger\hat{c}_2^\dagger|0\rangle$ and $\hat{c}_2^\dagger\hat{c}_1^\dagger|0\rangle$. Verify that they differ by a sign.

(b) $(\hat{c}_1^\dagger)^2|0\rangle$. Verify that this is the zero vector.

Problem 34.9 [I]

Consider three fermionic modes. Write out the state $\hat{c}_3^\dagger\hat{c}_1^\dagger|0\rangle$ in the standard ordering convention where modes are created in increasing order. What sign is needed?

Problem 34.10 [I]

For a two-mode fermionic system, show that the operator $\hat{c}_1^\dagger\hat{c}_2 + \hat{c}_2^\dagger\hat{c}_1$ (a "hopping" operator) has the following effect on the Fock states:

(a) $(\hat{c}_1^\dagger\hat{c}_2 + \hat{c}_2^\dagger\hat{c}_1)|1, 0\rangle = ?$

(b) $(\hat{c}_1^\dagger\hat{c}_2 + \hat{c}_2^\dagger\hat{c}_1)|0, 1\rangle = ?$

(c) $(\hat{c}_1^\dagger\hat{c}_2 + \hat{c}_2^\dagger\hat{c}_1)|1, 1\rangle = ?$

(d) Find the eigenvalues and eigenstates of this operator.

Problem 34.11 [A]

Prove the fermionic "Wick's theorem" for the simplest nontrivial case: show that

$$\langle 0|\hat{c}_k\hat{c}_{k'}\hat{c}_{q'}^\dagger\hat{c}_q^\dagger|0\rangle = \delta_{kq}\delta_{k'q'} - \delta_{kq'}\delta_{k'q}$$

by anticommuting the operators to normal order. Interpret the two terms as "direct" and "exchange" contributions.

Problem 34.12 [A]

The Jordan-Wigner transformation maps spin-1/2 operators to fermionic operators. For a chain of $N$ sites, define:

$$\hat{c}_j = \left(\prod_{i=1}^{j-1}\hat{\sigma}_i^z\right)\hat{\sigma}_j^-, \qquad \hat{c}_j^\dagger = \left(\prod_{i=1}^{j-1}\hat{\sigma}_i^z\right)\hat{\sigma}_j^+$$

where $\hat{\sigma}^{\pm} = (\hat{\sigma}^x \pm i\hat{\sigma}^y)/2$. Verify that $\{\hat{c}_j, \hat{c}_k^\dagger\} = \delta_{jk}$ for the case $N = 2$.

Section 34.3: Fock Space

Problem 34.13 [B]

For a system with 3 available single-particle states and 2 bosonic particles, list all possible occupation number states $|n_1, n_2, n_3\rangle$ with $n_1 + n_2 + n_3 = 2$. How many are there? Verify using the formula $\binom{N+M-1}{N}$.

Problem 34.14 [B]

Repeat Problem 34.13 for 2 fermionic particles in 3 states. How many states are there? Verify using $\binom{M}{N}$.

Problem 34.15 [I]

The total Fock space for $M$ fermionic modes has dimension $2^M$. Interpret this as follows: each mode is either occupied or unoccupied (2 choices), and the modes are independent. For $M = 4$ fermionic modes, write out all 16 basis states, organized by particle number $N = 0, 1, 2, 3, 4$.

Section 34.4: Many-Particle Operators

Problem 34.16 [I]

The one-body potential energy operator in first quantization is $\hat{V} = \sum_{i=1}^N V(\hat{x}_i)$. In second quantization with a discrete position basis $\{|j\rangle\}$ (tight-binding model), this becomes $\hat{V} = \sum_j V_j\, \hat{c}_j^\dagger\hat{c}_j$.

(a) Show that $\langle 1_a, 1_b|\hat{V}|1_a, 1_b\rangle = V_a + V_b$ (the potential energy of two particles at sites $a$ and $b$).

(b) Show that $\hat{V}$ gives zero when applied to a state with no particle at site $j$.

Problem 34.17 [I]

The hopping Hamiltonian of a 1D tight-binding model is:

$$\hat{H} = -t\sum_{j,\sigma}(\hat{c}_{j+1,\sigma}^\dagger\hat{c}_{j,\sigma} + \hat{c}_{j,\sigma}^\dagger\hat{c}_{j+1,\sigma})$$

Diagonalize this by Fourier transform: define $\hat{c}_{k,\sigma} = \frac{1}{\sqrt{N}}\sum_j e^{ikja}\hat{c}_{j,\sigma}$. Show that the Hamiltonian becomes $\hat{H} = \sum_{k,\sigma}\epsilon_k\,\hat{c}_{k,\sigma}^\dagger\hat{c}_{k,\sigma}$ with $\epsilon_k = -2t\cos(ka)$.

Problem 34.18 [A]

For the Coulomb interaction in second quantization:

$$\hat{V} = \frac{1}{2}\sum_{\mathbf{k},\mathbf{k}',\mathbf{q}\neq 0}\frac{4\pi e^2}{Vq^2}\hat{c}_{\mathbf{k}+\mathbf{q}}^\dagger\hat{c}_{\mathbf{k}'-\mathbf{q}}^\dagger\hat{c}_{\mathbf{k}'}\hat{c}_{\mathbf{k}}$$

(a) Explain why the $\mathbf{q} = 0$ term is excluded.

(b) Show that the expectation value of $\hat{V}$ in the Fermi sea ground state (ignoring spin) can be written as a sum of direct and exchange terms.

Section 34.5: Free Electron Gas

Problem 34.19 [B]

For a free electron gas with density $n = 8.5 \times 10^{28}$ m$^{-3}$ (copper):

(a) Calculate the Fermi wavevector $k_F$.

(b) Calculate the Fermi energy $E_F$ in eV.

(c) Calculate the Fermi temperature $T_F = E_F/k_B$.

(d) How does $T_F$ compare to room temperature? What does this imply about the quantum nature of conduction electrons?

Problem 34.20 [I]

Derive the density of states for the 3D free electron gas:

$$g(\epsilon) = \frac{V}{2\pi^2}\left(\frac{2m}{\hbar^2}\right)^{3/2}\epsilon^{1/2}$$

by counting states in $k$-space and converting from $k$ to $\epsilon$.

Problem 34.21 [I]

Using the density of states from Problem 34.20, rederive the ground state energy per particle $E_0/N = \frac{3}{5}E_F$ by computing $E_0 = \int_0^{E_F}\epsilon\, g(\epsilon)\,d\epsilon$ and $N = \int_0^{E_F} g(\epsilon)\,d\epsilon$.

Problem 34.22 [A]

The ground state pressure of the free electron gas (degeneracy pressure) is:

$$P = -\frac{\partial E_0}{\partial V}$$

(a) Show that $P = \frac{2}{3}\frac{E_0}{V} = \frac{2}{5}nE_F$.

(b) Calculate the degeneracy pressure for copper. Compare it to atmospheric pressure.

(c) This pressure is what prevents white dwarf stars from gravitational collapse. For a white dwarf with electron density $n \sim 10^{36}$ m$^{-3}$, estimate $E_F$ and $P$.

Section 34.6: Phonons

Problem 34.23 [B]

For a linear chain of atoms with mass $m = 28\,m_u$ (silicon) and lattice spacing $a = 5.4$ \AA, the maximum phonon frequency is $\omega_{\max} = 2\sqrt{\kappa/m}$ where $\kappa \approx 50$ N/m.

(a) Calculate $\omega_{\max}$ and the corresponding temperature $\Theta = \hbar\omega_{\max}/k_B$.

(b) At $T = 300$ K, estimate the average phonon occupation number for the maximum-frequency mode using the Bose-Einstein distribution.

Problem 34.24 [B]

Show that the displacement operator $\hat{u}_n = \sum_k \sqrt{\frac{\hbar}{2Nm\omega_k}}\left(\hat{a}_k e^{ikna} + \hat{a}_k^\dagger e^{-ikna}\right)$ is Hermitian. Why is this important?

Problem 34.25 [I]

The Debye model approximates the phonon dispersion as $\omega_k = v|k|$ up to a cutoff $\omega_D$. Show that the density of states is:

$$g(\omega) = \frac{3V\omega^2}{2\pi^2 v^3}$$

for $\omega < \omega_D$ and zero otherwise. Determine $\omega_D$ by requiring that the total number of modes equals $3N$ (for $N$ atoms in 3D).

Problem 34.26 [I]

In the Debye model, show that the low-temperature specific heat is:

$$C_V = \frac{12\pi^4}{5}Nk_B\left(\frac{T}{\Theta_D}\right)^3$$

where $\Theta_D = \hbar\omega_D/k_B$ is the Debye temperature. (You will need the integral $\int_0^\infty \frac{x^3}{e^x - 1}dx = \pi^4/15$.)

Problem 34.27 [A]

Phonon-phonon interactions. When the Taylor expansion of the interatomic potential is extended beyond second order, cubic and quartic terms produce phonon-phonon interactions. The cubic term, in second-quantized form, has the structure:

$$\hat{H}_3 \propto \sum_{k_1, k_2, k_3} V_{k_1 k_2 k_3}\,(\hat{a}_{k_1} + \hat{a}_{-k_1}^\dagger)(\hat{a}_{k_2} + \hat{a}_{-k_2}^\dagger)(\hat{a}_{k_3} + \hat{a}_{-k_3}^\dagger)\,\delta_{k_1+k_2+k_3, G}$$

where $G$ is a reciprocal lattice vector. Identify the different types of three-phonon processes (absorption, emission, splitting) that arise from expanding the products.

Section 34.7: QFT Preview

Problem 34.28 [I]

The field operator for a free scalar field is $\hat{\phi}(\mathbf{r}) = \sum_k\sqrt{\frac{\hbar}{2V\omega_k}}(\hat{a}_k e^{i\mathbf{k}\cdot\mathbf{r}} + \hat{a}_k^\dagger e^{-i\mathbf{k}\cdot\mathbf{r}})$.

(a) Verify that $\hat{\phi}$ is Hermitian.

(b) Calculate $\langle 0|\hat{\phi}(\mathbf{r})|0\rangle$.

(c) Calculate $\langle 0|\hat{\phi}(\mathbf{r})^2|0\rangle$. Interpret the result (this is the vacuum fluctuation of the field).

Problem 34.29 [A]

Show that the equal-time commutation relation $[\hat{\psi}(\mathbf{r}), \hat{\psi}^\dagger(\mathbf{r}')] = \delta^{(3)}(\mathbf{r} - \mathbf{r}')$ for the fermionic field operator follows from the anticommutation relations $\{\hat{c}_k, \hat{c}_{k'}^\dagger\} = \delta_{kk'}$ and the completeness of the single-particle wavefunctions $\sum_k \phi_k(\mathbf{r})\phi_k^*(\mathbf{r}') = \delta^{(3)}(\mathbf{r} - \mathbf{r}')$.

Problem 34.30 [A]

The electron self-energy (Hartree-Fock). Consider the interacting electron gas Hamiltonian in second quantization. Using the variational principle with a Slater determinant trial state (the Fermi sea), show that the Hartree-Fock correction to the energy per electron in the free electron gas (the exchange energy) is:

$$\frac{E_{\text{ex}}}{N} = -\frac{3e^2 k_F}{4\pi}$$

(This is a classic result in many-body theory. Hint: compute the expectation value of the Coulomb interaction in the Fermi sea ground state.)