2.1.3 Second Quantization#

Prompts

What is the key difference between first-quantized and second-quantized descriptions of a many-body system? Why does occupation-number representation suit identical particles better?
Define the creation operator \(\hat{a}^\dagger\) and annihilation operator \(\hat{a}\). Describe their differing actions on Fock states for bosons versus fermions.
Why do boson operators satisfy commutation relations while fermion operators satisfy anticommutation relations? How does each algebra enforce correct quantum statistics?
If you apply \(\hat{b}^\dagger\) to a state with \(n\) bosons, why is the amplitude \(\sqrt{n+1}\), and what physical phenomenon does this describe?
Why does \((\hat{c}^\dagger_\alpha)^2 = 0\) for fermions? How does this operator relation encode the Pauli exclusion principle?

Lecture Notes#

Overview#

Second quantization reformulates quantum mechanics so that identical-particle statistics are automatic. Instead of labeling individual particles and then symmetrizing or antisymmetrizing by hand, we simply count how many particles occupy each single-particle mode. This shift from “particle labels” to “occupation numbers” handles variable particle number naturally and leads directly to the operator algebra of creation and annihilation.

First vs Second Quantization#

First quantization asks: “Which particle is in which state?” An \(N\)-particle state is a tensor product

\[ \vert\Psi\rangle = \vert\psi_1\rangle \otimes \vert\psi_2\rangle \otimes \cdots \otimes \vert\psi_N\rangle \]

that must be manually symmetrized (bosons) or antisymmetrized (fermions).

Second quantization asks: “How many particles occupy each state?” A state is specified by occupation numbers

\[ \vert n_1, n_2, \ldots, n_D\rangle \]

where \(n_\alpha\) counts particles in single-particle mode \(\alpha\). Symmetry is built into the operator algebra—no manual (anti)symmetrization needed.

Fock Space#

Fock Space

The Fock space is the direct sum of all \(N\)-particle Hilbert spaces:

\[ \mathcal{F} = \bigoplus_{N=0}^\infty \mathcal{H}_N \]

Its basis states are Fock states \(\vert n_1, n_2, \ldots, n_D\rangle\), labeled by occupation numbers \(\{n_\alpha\}\). These form a complete orthonormal basis:

\[ \langle n_1', n_2', \ldots \vert n_1, n_2, \ldots \rangle = \prod_\alpha \delta_{n_\alpha' n_\alpha} \]

The vacuum \(\vert 0\rangle \equiv \vert 0, 0, \ldots, 0\rangle\) has no particles.

Creation and Annihilation Operators#

The creation operator \(\hat{a}^\dagger_\alpha\) adds a particle to mode \(\alpha\); the annihilation operator \(\hat{a}_\alpha\) removes one. Their action on Fock states differs for bosons and fermions.

Bosonic Operators.

For bosons, the creation and annihilation operators act on occupation number states as:

\[ \hat{b}^\dagger_\alpha \vert \ldots, n_\alpha, \ldots\rangle = \sqrt{n_\alpha + 1}\;\vert \ldots, n_\alpha + 1, \ldots\rangle \]

\[ \hat{b}_\alpha \vert \ldots, n_\alpha, \ldots\rangle = \sqrt{n_\alpha}\;\vert \ldots, n_\alpha - 1, \ldots\rangle \]

The \(\sqrt{n_\alpha \pm 1}\) factors emerge from the symmetric insertion/deletion structure—two orderings of \(\hat{b}^\dagger_\alpha \hat{b}^\dagger_\beta\) produce the same state with the same coefficient, so they commute.

Why the square-root factors?

The square roots come from normalization when inserting or removing particles symmetrically. When adding a boson to mode \(\alpha\) in an \(N\)-particle state, there are \(N+1\) symmetric insertion slots (before the 1st particle, between adjacent particles, and after the last). The creation operator distributes the amplitude equally among them, giving a prefactor \(1/\sqrt{N+1}\). When paired with the rising ladder \(\vert \ldots,n_\alpha,\ldots\rangle\to\vert \ldots,n_\alpha+1,\ldots\rangle\), this yields \(\sqrt{n_\alpha+1}\).

Annihilation reverses the process with \(N\) deletion choices, giving \(\sqrt{n_\alpha}\).

Fermionic Operators.

For fermions (\(n_\alpha \in \{0,1\}\) due to Pauli exclusion), the operators are:

\[\begin{split} \hat{c}^\dagger_\alpha \vert \ldots, n_\alpha, \ldots\rangle = \begin{cases} (-1)^{P_\alpha} \vert \ldots, 1, \ldots\rangle & n_\alpha = 0 \\ 0 & n_\alpha = 1 \end{cases} \end{split}\]

\[\begin{split} \hat{c}_\alpha \vert \ldots, n_\alpha, \ldots\rangle = \begin{cases} (-1)^{P_\alpha} \vert \ldots, 0, \ldots\rangle & n_\alpha = 1 \\ 0 & n_\alpha = 0 \end{cases} \end{split}\]

where \(P_\alpha = \sum_{i<\alpha} n_i\) counts occupied modes before \(\alpha\) in canonical order. The sign \((-1)^{P_\alpha}\) arises because permuting the new fermion past occupied modes in antisymmetric insertion picks up a minus sign for each crossing.

Algebraic Relations.

These operator definitions are equivalent to canonical commutation/anticommutation algebras:

Commutation Relations (Bosons)

(34)#\[ [\hat{b}_\alpha, \hat{b}^\dagger_\beta] = \delta_{\alpha\beta}, \quad [\hat{b}_\alpha, \hat{b}_\beta] = 0, \quad [\hat{b}^\dagger_\alpha, \hat{b}^\dagger_\beta] = 0 \]

Anticommutation Relations (Fermions)

(35)#\[ \{\hat{c}_\alpha, \hat{c}^\dagger_\beta\} = \delta_{\alpha\beta}, \quad \{\hat{c}_\alpha, \hat{c}_\beta\} = 0, \quad \{\hat{c}^\dagger_\alpha, \hat{c}^\dagger_\beta\} = 0 \]

Number Operator.

The number operator \(\hat{n}_\alpha = \hat{a}^\dagger_\alpha \hat{a}_\alpha\) counts particles in mode \(\alpha\):

\[ \hat{n}_\alpha \vert \ldots, n_\alpha, \ldots\rangle = n_\alpha \vert \ldots, n_\alpha, \ldots\rangle \]

For bosons: \(\hat{n}_\alpha = \hat{b}^\dagger_\alpha \hat{b}_\alpha\) gives \(\hat{b}^\dagger_\alpha \hat{b}_\alpha \vert n_\alpha\rangle = n_\alpha \vert n_\alpha\rangle\).

For fermions: \(\hat{n}_\alpha = \hat{c}^\dagger_\alpha \hat{c}_\alpha\) also gives the occupation (0 or 1).

Fock states are simultaneous eigenstates of every \(\hat{n}_\alpha\).

Boson Enhancement and Stimulated Emission#

Apply \(\hat{b}^\dagger\) to a Fock state:

\[ \hat{b}^\dagger_\alpha \vert n_\alpha\rangle = \sqrt{n_\alpha + 1}\;\vert n_\alpha+1\rangle \]

The \(\sqrt{n_\alpha+1}\) prefactor means adding a boson to an occupied mode is more probable than to an empty mode. This “rich get richer” effect is the origin of stimulated emission—the transition rate into a mode with \(n\) photons is proportional to \(n+1\) (one factor spontaneous, \(n\) factors stimulated). It also drives Bose-Einstein condensation.

Example: Stimulated Emission

A cavity mode contains \(n\) photons. An excited atom can emit a photon into the mode via \(\hat{b}^\dagger \vert n\rangle = \sqrt{n+1}\;\vert n+1\rangle\). The emission amplitude is \(\sqrt{n+1}\), so the rate \(\propto n+1\). The “\(+1\)” is spontaneous emission; the “\(n\)” term is stimulated. This quadratic feedback loop is why lasers self-amplify.

Pauli Exclusion from Anticommutation#

For fermions, the anticommutation relation \(\{\hat{c}^\dagger_\alpha, \hat{c}^\dagger_\alpha\} = 0\) implies:

\[ (\hat{c}^\dagger_\alpha)^2 = 0 \]

Attempting to create a second fermion in the same mode gives zero. This is the Pauli exclusion principle encoded directly in the algebra.

Bosons vs Fermions: Key Contrasts#

Aspect	Bosons	Fermions
Occupation	\(n_\alpha \in \{0,1,2,\ldots\}\)	\(n_\alpha \in \{0,1\}\)
Creation action	\(\hat{b}^\dagger \vert n\rangle = \sqrt{n+1}\,\vert n{+}1\rangle\)	\(\hat{c}^\dagger \vert 0\rangle = \vert 1\rangle\), \(\hat{c}^\dagger \vert 1\rangle = 0\)
Algebra	\([\hat{b},\hat{b}^\dagger]=1\) (commute)	\(\{\hat{c},\hat{c}^\dagger\}=1\) (anticommute)
Phase sign	None	\((-1)^{P_\alpha}\) (parity)
Key physics	Stimulated emission, BEC, coherence	Pauli exclusion, degeneracy pressure
Examples	Photons, phonons, \(^4\text{He}\)	Electrons, quarks, neutrinos

Discussion: same formalism, different physics

Why does second quantization treat bosons and fermions on equal footing at the level of formalism—both use creation and annihilation operators acting on Fock space—yet produce radically different physics? The algebra changes only by a sign (commutator vs anticommutator), but this sign leads to Bose-Einstein condensation in one case and the Pauli exclusion principle in the other. Is there a deeper reason why nature picks exactly these two statistics, or could other algebraic relations (e.g., \(q\)-deformed commutation relations) describe physical particles?

Poll: Pauli exclusion from antisymmetry

Two identical fermions cannot occupy the same single-particle state. This follows from antisymmetry because the state \(\vert i\rangle\otimes\vert i\rangle\) satisfies which property under exchange?

(A) It is invariant: \(\hat{P}\vert i\rangle\otimes\vert i\rangle = \vert i\rangle\otimes\vert i\rangle\).

(B) It changes sign: \(\hat{P}\vert i\rangle\otimes\vert i\rangle = -\vert i\rangle\otimes\vert i\rangle\).

(D) Both (A) and (B) - a state cannot be both symmetric and antisymmetric.

Summary#

Second quantization uses occupation numbers \(\{n_\alpha\}\) instead of coordinates; Fock space is the direct sum of all particle-number sectors.
Creation and annihilation operators (\(\hat{b}^\dagger_\alpha\), \(\hat{b}_\alpha\)) add/remove particles from state \(\alpha\) with the symmetric or antisymmetric sign convention.
Canonical commutation/anticommutation relations: \([\hat{b},\hat{b}^\dagger]=1\) (bosons) or \(\{\hat{c},\hat{c}^\dagger\}=1\) (fermions), with boson enhancement (\(\sqrt{n+1}\) factor) driving stimulated emission and Pauli blocking for fermions.
The number operator \(\hat{n}_\alpha=\hat{a}^\dagger_\alpha\hat{a}_\alpha\) counts particles; total particle number \(\hat{N}=\sum_\alpha\hat{n}_\alpha\) commutes with \(\hat{H}\) when no creation/destruction processes occur..

Homework#

1. Fock states from the vacuum. Use the bosonic action rules \(\hat{b}^\dagger\vert n\rangle = \sqrt{n+1}\vert n+1\rangle\), \(\hat{b}\vert n\rangle = \sqrt n\vert n-1\rangle\), and the vacuum condition \(\hat{b}\vert 0\rangle = 0\).

(a) Apply \(\hat{b}^\dagger\) to \(\vert 0\rangle\) once, twice, and three times, tracking the \(\sqrt{n+1}\) prefactor at each step. Use induction to show that

\[ \vert n\rangle = \frac{(\hat{b}^\dagger)^n}{\sqrt{n!}}\,\vert 0\rangle. \]

(b) Compute the matrix elements \(\langle m\vert\hat{b}^\dagger\vert n\rangle\) and \(\langle m\vert\hat{b}\vert n\rangle\) in the Fock basis. Conclude that \(\hat{b}^\dagger\) is super-diagonal and \(\hat{b}\) is sub-diagonal — they connect only Fock states differing by one quantum.

(c) Verify directly from the action rules that \(\hat{b}^\dagger\hat{b}\vert n\rangle = n\vert n\rangle\) and \(\hat{b}\hat{b}^\dagger\vert n\rangle = (n+1)\vert n\rangle\). Subtract to recover \([\hat{b},\hat{b}^\dagger]\vert n\rangle = \vert n\rangle\) on every Fock state — the algebra emerges from the state action, not the other way around.

(d) The harmonic-oscillator position operator is \(\hat{x} = x_0(\hat{b} + \hat{b}^\dagger)\) with \(x_0 = \sqrt{\hbar/(2m\omega)}\). Compute \(\langle m\vert\hat{x}\vert n\rangle\) and identify the dipole selection rule: \(\hat{x}\) connects only Fock states with \(\vert m - n\vert = 1\).

2. Number-ladder commutators. Compute the commutators \([\hat{n}_\alpha, \hat{b}^\dagger_\beta]\) and \([\hat{n}_\alpha, \hat{b}_\beta]\) from \([\hat{b}_\alpha, \hat{b}^\dagger_\beta] = \delta_{\alpha\beta}\). Interpret: how does \(\hat{b}^\dagger_\beta\) change the eigenvalue of \(\hat{n}_\alpha\), and what is the role of \(\delta_{\alpha\beta}\)?

3. Coherent states. Define the coherent state with complex amplitude \(\alpha\) by

\[ \vert\alpha\rangle = \mathrm{e}^{-\vert\alpha\vert^2/2}\sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}}\,\vert n\rangle. \]

(a) Verify that \(\vert\alpha\rangle\) is normalised: \(\langle\alpha\vert\alpha\rangle = 1\).

(b) Show that \(\vert\alpha\rangle\) is an eigenstate of the annihilation operator \(\hat b\) with eigenvalue \(\alpha\): \(\hat b\vert\alpha\rangle = \alpha\vert\alpha\rangle\).

(c) Compute the mean and variance of the particle number, \(\langle\hat n\rangle\) and \((\Delta\hat n)^2\). Show that the relative fluctuation \(\Delta\hat n / \langle\hat n\rangle = 1/\vert\alpha\vert\) — Poisson statistics — and that fluctuations become negligible in the classical limit \(\vert\alpha\vert\to\infty\).

(d) Compute the overlap \(\vert\langle\alpha\vert\beta\rangle\vert^2\) between two coherent states. Show that it equals \(\mathrm{e}^{-\vert\alpha-\beta\vert^2}\). Are coherent states orthogonal?

4. Stimulated emission and the boson enhancement factor. A single bosonic mode contains \(n\) existing quanta. Consider a coupling that drives transitions in or out of the mode through the ladder operators.

(a) Compute the transition amplitude \(\langle n+1\vert\hat b^\dagger\vert n\rangle\) (emission) and the absorption amplitude \(\langle n-1\vert\hat b\vert n\rangle\). Square each to obtain the corresponding transition rate per unit coupling.

(b) Decompose the emission rate as \((n+1) = 1 + n\). Identify the “\(1\)” as the spontaneous rate and the “\(n\)” as the stimulated rate proportional to the existing field.

(c) Compute the net emission rate (emission minus absorption) and show it equals \(1\), independent of \(n\). Interpret physically: stimulated emission and absorption have the same matrix element, so their difference picks out only the spontaneous “vacuum” contribution.

(d) Identify the laser regime: when does stimulated dominate spontaneous, and what does this regime predict about the field amplitude inside an active laser cavity?

5. Slater determinant from creation operators. For two fermions in orthonormal single-particle modes \(\vert\alpha\rangle\) and \(\vert\beta\rangle\) (\(\alpha\neq\beta\)), define the second-quantized two-particle state

\[ \vert\alpha,\beta\rangle_F \equiv \hat c^\dagger_\alpha \hat c^\dagger_\beta \vert 0\rangle. \]

(a) Show \(\vert\beta,\alpha\rangle_F = -\vert\alpha,\beta\rangle_F\), i.e. exchange flips the sign. Identify which anticommutation relation is responsible.

(b) Show \(\hat c^\dagger_\alpha \hat c^\dagger_\alpha \vert 0\rangle = 0\) — Pauli exclusion in operator form. Identify the responsible relation.

(c) Use the anticommutator \(\{\hat c_\alpha, \hat c^\dagger_\beta\} = \delta_{\alpha\beta}\) together with \(\hat c_\alpha\vert 0\rangle = 0\) to compute the inner product \(\langle\alpha',\beta'\vert\alpha,\beta\rangle_F\). Show it equals the \(2\times 2\) Slater determinant \(\det\!\begin{pmatrix}\langle\alpha'\vert\alpha\rangle & \langle\alpha'\vert\beta\rangle \\ \langle\beta'\vert\alpha\rangle & \langle\beta'\vert\beta\rangle\end{pmatrix}\), recovering 2.1.2 Problem 6.

(d) Argue in one sentence why the second-quantized formalism makes the manual antisymmetrisation of 2.1.2 automatic: the algebra encodes Pauli exclusion structurally.

6. Particle-hole conjugation. In a fermionic Fock space, define the hole operator \(\hat d_\alpha \equiv \hat c^\dagger_\alpha\) (and its conjugate \(\hat d^\dagger_\alpha = \hat c_\alpha\)).

(a) Compute the four anticommutators \(\{\hat d_\alpha,\hat d^\dagger_\beta\}\), \(\{\hat d_\alpha,\hat d_\beta\}\), \(\{\hat d^\dagger_\alpha,\hat d^\dagger_\beta\}\) from the fermionic algebra of \(\hat c\). Show that the hole operators satisfy the same fermionic anticommutation algebra.

(b) For a finite set of modes \(\alpha = 1,\ldots,D\), define the filled state \(\vert F\rangle = \hat c^\dagger_1\hat c^\dagger_2\cdots\hat c^\dagger_D\vert 0\rangle\). Show that \(\vert F\rangle\) is the hole vacuum: \(\hat d_\alpha\vert F\rangle = 0\) for every \(\alpha\).

(c) The hole number operator is \(\hat n^h_\alpha = \hat d^\dagger_\alpha\hat d_\alpha\). Show that \(\hat n^h_\alpha = 1 - \hat n_\alpha\) (a hole exists in mode \(\alpha\) exactly when the mode is empty of fermions).

(d) Interpret physically: in solid-state physics, removing an electron from a filled valence band creates a “hole” that behaves as a positive-charge fermion. Argue that this physical picture is exactly the algebraic statement that the hole operators obey the same fermionic algebra as the particle operators, just relative to a different vacuum (\(\vert F\rangle\) instead of \(\vert 0\rangle\)).

7. Two-mode boson tunnelling. Consider two bosonic modes labelled L (left) and R (right), with Hamiltonian

\[ \hat H = -t\bigl(\hat b_L^\dagger\hat b_R + \hat b_R^\dagger\hat b_L\bigr), \qquad t > 0. \]

This is the simplest tunnelling model — particles “hop” between two sites with amplitude \(t\).

(a) Single particle (\(N = 1\)). The Hilbert space at fixed \(N=1\) is spanned by \(\vert 1,0\rangle = \hat b_L^\dagger\vert 0\rangle\) and \(\vert 0,1\rangle = \hat b_R^\dagger\vert 0\rangle\). Compute the action of \(\hat H\) on each basis state, and write \(\hat H\) as a \(2\times 2\) matrix in this basis.

(b) Diagonalise \(\hat H\). Show that the eigenstates are the bonding state \(\vert +\rangle = (\vert 1,0\rangle + \vert 0,1\rangle)/\sqrt 2\) with energy \(-t\) and the antibonding state \(\vert -\rangle = (\vert 1,0\rangle - \vert 0,1\rangle)/\sqrt 2\) with energy \(+t\).

(c) Starting from \(\vert\psi(0)\rangle = \vert 1,0\rangle\), compute the probability \(P_R(t)\) of finding the particle in the right well at time \(t\). Identify the tunnelling angular frequency.

(d) Two particles (\(N = 2\)). The Hilbert space at \(N=2\) has dimension \(3\) (states \(\vert 2,0\rangle\), \(\vert 1,1\rangle\), \(\vert 0,2\rangle\)). Compute the action of \(\hat H\) on each — paying attention to the boson enhancement factors \(\sqrt{n+1}\) — and write \(\hat H\) as a \(3\times 3\) matrix.

8. Fock space dimensions. From 2.1.2 Problems 7 and 8, the number of \(N\)-particle states in \(D\) modes is \(\binom{D}{N}\) for fermions and \(\binom{N+D-1}{N}\) for bosons. Recover these counts from the operator formalism, then assemble the full Fock space.

(a) The most general fermion Fock state in \(D\) modes is \(\prod_\alpha (\hat c^\dagger_\alpha)^{n_\alpha}\vert 0\rangle\) with each \(n_\alpha\in\{0,1\}\) (by \((\hat c^\dagger_\alpha)^2 = 0\)). Count the choices of \(\{n_\alpha\}\) with \(\sum_\alpha n_\alpha = N\) to recover \(\binom{D}{N}\).

(b) The general boson Fock state is \(\prod_\alpha\frac{(\hat b^\dagger_\alpha)^{n_\alpha}}{\sqrt{n_\alpha!}}\vert 0\rangle\) with each \(n_\alpha\in\{0,1,2,\ldots\}\). Count the choices with \(\sum_\alpha n_\alpha = N\) to recover \(\binom{N+D-1}{N}\).

(c) Sum over \(N\) to find the total dimension of the Fock space \(\mathcal F_D = \bigoplus_{N=0}^\infty \mathcal H_N\). Show that for fermions \(\dim\mathcal F_D = 2^D\) (finite!), while for bosons \(\dim\mathcal F_D = \infty\).

(d) Interpret physically: the same algebraic distinction (anticommutator vs commutator) that gives the Pauli exclusion principle also constrains the size of the fermionic Hilbert space. Why is a finite-dimensional Fock space the natural setting for “lattice fermion” problems (electrons on a crystal), and why does the infinite boson Fock space correspond to the more delicate analytical issues of bosonic theories (BEC, photon counting)?

★ 9. Equal partition theorem. Consider the harmonic oscillator \(\hat{H} = \hbar\omega(\hat{a}^\dagger\hat{a} + \tfrac{1}{2})\) with position \(\hat{x} = \sqrt{\frac{\hbar}{2m\omega}}(\hat{a} + \hat{a}^\dagger)\) and momentum \(\hat{p} = \mathrm{i}\sqrt{\frac{m\hbar\omega}{2}}(\hat{a}^\dagger - \hat{a})\).

(a) Show that in the energy eigenstate \(\vert n\rangle\), \(\langle n\vert\hat{x}^2\vert n\rangle = \frac{\hbar}{2m\omega}(2n+1)\) and \(\langle n\vert\hat{p}^2\vert n\rangle = \frac{m\hbar\omega}{2}(2n+1)\).

(b) Verify that \(\frac{1}{2m}\langle\hat{p}^2\rangle = \frac{1}{2}m\omega^2\langle\hat{x}^2\rangle = \frac{1}{2}E_n\) in each eigenstate \(\vert n\rangle\). This is the quantum equal partition of energy between kinetic and potential.

(c) For a superposition \(\vert\psi(0)\rangle = c_0\vert 0\rangle + c_1\vert 1\rangle\), compute \(\langle\hat{x}^2\rangle(t)\) and \(\langle\hat{p}^2/(m^2\omega^2)\rangle(t)\). Show that the time-averaged values over one period still satisfy equal partition.

★ 10. Schwinger boson. Define two independent bosonic modes with operators \((\hat{a}, \hat{a}^\dagger)\) and \((\hat{b}, \hat{b}^\dagger)\), and construct

\[ \hat{S}_+ = \hat{a}^\dagger\hat{b}, \quad \hat{S}_- = \hat{b}^\dagger\hat{a}, \quad \hat{S}_z = \tfrac{1}{2}(\hat{a}^\dagger\hat{a} - \hat{b}^\dagger\hat{b}). \]

(a) Verify that these satisfy the angular momentum commutation relations \([\hat{S}_z, \hat{S}_\pm] = \pm\hat{S}_\pm\) and \([\hat{S}_+, \hat{S}_-] = 2\hat{S}_z\).

(b) Show that \(\hat{\boldsymbol{S}}^2 = \hat{S}_z^2 + \frac{1}{2}(\hat{S}_+\hat{S}_- + \hat{S}_-\hat{S}_+) = \frac{\hat{N}}{2}\bigl(\frac{\hat{N}}{2} + 1\bigr)\), where \(\hat{N} = \hat{a}^\dagger\hat{a} + \hat{b}^\dagger\hat{b}\) is the total boson number.

(c) In the subspace with \(\hat{N} = 2\) (three Fock states \(\vert 2,0\rangle\), \(\vert 1,1\rangle\), \(\vert 0,2\rangle\)), write \(\hat{S}_z\) and \(\hat{S}_+\) as \(3\times 3\) matrices. Verify these are the spin-1 representation.

(d) Identify the general correspondence: each Fock state \(\vert n_a, n_b\rangle\) with \(n_a + n_b = 2s\) maps to \(\vert s, m\rangle\) with \(m = (n_a - n_b)/2\). What value of \(s\) does the \(N\)-boson subspace carry?