Chapter 34 Quiz: Second Quantization

Multiple Choice

Q1. The fundamental commutation relation for bosonic creation and annihilation operators is:

(a) $\{\hat{a}_k, \hat{a}_{k'}^\dagger\} = \delta_{kk'}$ (b) $[\hat{a}_k, \hat{a}_{k'}^\dagger] = \delta_{kk'}$ (c) $[\hat{a}_k, \hat{a}_{k'}] = \delta_{kk'}$ (d) $[\hat{a}_k, \hat{a}_{k'}^\dagger] = i\hbar\delta_{kk'}$

Q2. The Pauli exclusion principle in second quantization follows directly from:

(a) The commutation relations of bosonic operators (b) The anticommutation relation $\{\hat{c}_k^\dagger, \hat{c}_k^\dagger\} = 0$ (c) The Schr\u00f6dinger equation (d) The normalization of the vacuum state

Q3. The vacuum state $|0\rangle$ in Fock space is defined by:

(a) $\hat{a}_k^\dagger|0\rangle = 0$ for all $k$ (b) $\hat{a}_k|0\rangle = |0\rangle$ for all $k$ (c) $\hat{a}_k|0\rangle = 0$ for all $k$ (d) $\langle 0|0\rangle = 0$

Q4. For fermionic modes, the occupation number $n_k = \hat{c}_k^\dagger\hat{c}_k$ can take values:

(a) $0, 1, 2, 3, \ldots$ (b) $0, 1$ (c) $-1, 0, 1$ (d) $\frac{1}{2}, -\frac{1}{2}$

Q5. The one-body operator $\hat{O}_1 = \sum_i \hat{o}(i)$ in second-quantized form is:

(a) $\sum_k \langle k|\hat{o}|k\rangle\, \hat{a}_k$ (b) $\sum_{k,k'} \langle k|\hat{o}|k'\rangle\, \hat{a}_k^\dagger \hat{a}_{k'}$ (c) $\sum_k \hat{o}\, \hat{a}_k^\dagger \hat{a}_k$ (d) $\sum_{k,k'} \langle k|\hat{o}|k'\rangle\, \hat{a}_k^\dagger \hat{a}_{k'}^\dagger$

Q6. The ground state of a free electron gas at zero temperature is called the Fermi sea. In second quantization, it is constructed by:

(a) Applying bosonic creation operators for all states up to the Fermi energy (b) Applying fermionic creation operators for all states up to the Fermi energy to the vacuum (c) Applying the Hamiltonian to the vacuum state (d) Taking the tensor product of all single-particle ground states

Q7. The Fermi energy of copper ($n \approx 8.5 \times 10^{28}$ m$^{-3}$) is approximately:

(a) $0.07$ eV (b) $0.7$ eV (c) $7$ eV (d) $70$ eV

Q8. A phonon is:

(a) A photon that propagates through a crystal (b) A quantized excitation of a lattice vibration mode (c) An electron bound to a lattice site (d) A classical sound wave in a solid

Q9. Phonons obey Bose-Einstein statistics because:

(a) They have integer spin (b) Their creation operators satisfy commutation (not anticommutation) relations (c) They are massless particles (d) They travel at the speed of sound

Q10. The Debye model predicts that the low-temperature specific heat of a solid varies as:

(a) $C_V \propto T$ (b) $C_V \propto T^2$ (c) $C_V \propto T^3$ (d) $C_V$ is constant (Dulong-Petit)

Q11. Fock space is defined as:

(a) The Hilbert space of a single particle (b) The tensor product of all single-particle Hilbert spaces (c) The direct sum of $n$-particle Hilbert spaces for $n = 0, 1, 2, \ldots$ (d) The space of all classical fields

Q12. The field operator $\hat{\psi}(\mathbf{r}) = \sum_k \phi_k(\mathbf{r})\hat{c}_k$ acting on the vacuum creates:

(a) Nothing (it annihilates the vacuum) (b) A particle at position $\mathbf{r}$ (c) A particle in mode $k$ (d) A superposition of all single-particle states

True or False

Q13. True or False: The name "second quantization" means that the system is quantized twice — first the particles, then the fields.

Q14. True or False: In second quantization, the total particle number is always conserved.

Q15. True or False: For bosons, the occupation number $n_k$ of any single-particle state can be arbitrarily large.

Q16. True or False: The vacuum state $|0\rangle$ has zero energy in a system of quantum harmonic oscillators.

Short Answer

Q17. Explain in 2-3 sentences why second quantization is "more natural" than first quantization for identical particles.

Q18. State the dispersion relation for phonons in a 1D monatomic chain and identify the maximum frequency.

Q19. What is the physical meaning of the operator $\hat{a}_k^\dagger\hat{a}_{k'}$ (with $k \neq k'$) in the context of second quantization?

Q20. Explain why the zero-point energy of the vacuum ($\sum_k \frac{1}{2}\hbar\omega_k$) is formally infinite and what physical consequence survives after renormalization (give one example).

Answer Key

Q1: (b) — The commutation relation $[\hat{a}_k, \hat{a}_{k'}^\dagger] = \delta_{kk'}$ defines bosonic operators. The anticommutator (a) is for fermions.

Q2: (b) — $\{\hat{c}_k^\dagger, \hat{c}_k^\dagger\} = 0$ implies $(\hat{c}_k^\dagger)^2 = 0$, so no state can hold two identical fermions.

Q3: (c) — The vacuum is defined by $\hat{a}_k|0\rangle = 0$ for all $k$ — all annihilation operators give zero.

Q4: (b) — Fermionic occupation numbers are restricted to 0 or 1 by the Pauli exclusion principle.

Q5: (b) — The second-quantized form involves matrix elements $\langle k|\hat{o}|k'\rangle$ multiplied by $\hat{a}_k^\dagger\hat{a}_{k'}$.

Q6: (b) — The Fermi sea is built by applying fermionic creation operators for all states below $E_F$ to the vacuum.

Q7: (c) — $E_F \approx 7.0$ eV for copper.

Q8: (b) — A phonon is a quantum of vibrational energy in a lattice mode.

Q9: (b) — Phonon operators satisfy bosonic commutation relations, which determine Bose-Einstein statistics.

Q10: (c) — The Debye $T^3$ law for specific heat at low temperatures.

Q11: (c) — Fock space is the direct sum $\mathcal{F} = \bigoplus_{n=0}^\infty \mathcal{H}_n$.

Q12: (a) — $\hat{\psi}(\mathbf{r})|0\rangle = 0$ because $\hat{c}_k|0\rangle = 0$ for all $k$. The creation field operator $\hat{\psi}^\dagger(\mathbf{r})$ creates a particle at $\mathbf{r}$.

Q13: False — "Second quantization" is a misnomer. It refers to the quantization of fields (or equivalently, the promotion of particle number to a dynamical variable), not a second round of quantization.

Q14: False — In general, particle number can change (e.g., photon emission/absorption). The total number operator $\hat{N}$ commutes with the Hamiltonian only in special cases (non-relativistic theories without pair creation).

Q15: True — Bosonic occupation numbers are unrestricted: $n_k = 0, 1, 2, 3, \ldots$

Q16: False — The vacuum energy is $E_0 = \sum_k \frac{1}{2}\hbar\omega_k$, which is nonzero (and formally infinite). The zero-point energy of each mode contributes.

Q17: In first quantization, identical particles are labeled ($1, 2, 3, \ldots$), and we must manually symmetrize or antisymmetrize the wavefunction. In second quantization, we ask "how many particles are in each state?" — no labels, no symmetrization needed. The indistinguishability is built into the algebra of creation/annihilation operators from the start.

Q18: $\omega(k) = 2\sqrt{\kappa/m}\,|\sin(ka/2)|$, where $\kappa$ is the spring constant, $m$ is the atomic mass, and $a$ is the lattice spacing. The maximum frequency is $\omega_{\max} = 2\sqrt{\kappa/m}$ at the Brillouin zone boundary $k = \pi/a$.

Q19: The operator $\hat{a}_k^\dagger\hat{a}_{k'}$ destroys a particle in mode $k'$ and creates one in mode $k$. It describes "hopping" or "scattering" — a transition from single-particle state $k'$ to state $k$.

Q20: The sum $\sum_k \frac{1}{2}\hbar\omega_k$ diverges because there are infinitely many modes with arbitrarily high frequency. In renormalized quantum field theory, only differences in vacuum energy are physical. One measurable consequence is the Casimir effect: two conducting plates in vacuum experience an attractive force due to the difference in vacuum energy between the constrained (between plates) and unconstrained (outside) field modes.