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. Fermionic ladder operators. Using the bosonic commutation relation \([\hat{b}_\alpha, \hat{b}^\dagger_\alpha] = 1\), show that \(\hat{n}_\alpha = \hat{b}^\dagger_\alpha \hat{b}_\alpha\) satisfies \(\hat{n}_\alpha \vert n_\alpha\rangle = n_\alpha \vert n_\alpha\rangle\), with \(n_\alpha = 0, 1, 2, \ldots\) Verify explicitly for \(n_\alpha = 0\) and \(n_\alpha = 1\).

2. Commutation relation. Compute the commutators \([\hat{n}_\alpha, \hat{b}^\dagger_\beta]\) and \([\hat{n}_\alpha, \hat{b}_\beta]\) using \([\hat{b}_\alpha, \hat{b}^\dagger_\beta] = \delta_{\alpha\beta}\). Interpret: how does \(\hat{b}^\dagger_\beta\) change the eigenvalue of \(\hat{n}_\alpha\)?

3. Boson ladder operators. Show directly from \(\{\hat{c}^\dagger_\alpha, \hat{c}^\dagger_\alpha\} = 0\) that \((\hat{c}^\dagger_\alpha)^2 = 0\). Explain why this is the Pauli exclusion principle in operator language.

4. Occupation number basis. Compute \(\langle 0 \vert \hat{b}_\alpha \hat{b}^\dagger_\alpha \vert 0\rangle\) and \(\langle 0 \vert \hat{b}^\dagger_\alpha \hat{b}_\alpha \vert 0\rangle\). What is the physical meaning of their difference?

5. Creation annihilation. For a two-mode bosonic system, list all Fock states with total particle number \(N = 2\). Do the same for a two-mode fermionic system. How many states exist in each case?

6. Number operator expectation. For non-interacting particles with single-particle energies \(\epsilon_\alpha\), the Hamiltonian is \(\hat{H} = \sum_\alpha \epsilon_\alpha \hat{n}_\alpha\). Compute \(\langle n_1, n_2, \ldots \vert \hat{H} \vert n_1, n_2, \ldots\rangle\).

7. Two-body interaction. Show that \([\hat{H}, \hat{N}] = 0\) for the Hamiltonian in problem 6, where \(\hat{N} = \sum_\alpha \hat{n}_\alpha\) is the total number operator. What conservation law does this express?

8. Second quantization Hamiltonian. The single-particle kinetic energy eigenvalue in a plane-wave basis is \(\epsilon_{\boldsymbol{k}} = \hbar^2 k^2 / 2m\). Write the second-quantized kinetic energy operator \(\hat{T} = \sum_{\boldsymbol{k}} \epsilon_{\boldsymbol{k}} \hat{b}^\dagger_{\boldsymbol{k}} \hat{b}_{\boldsymbol{k}}\) for bosons. Compare this to the first-quantized form \(\sum_{i=1}^N \hat{\boldsymbol{p}}_i^2 / 2m\): which is simpler and why?

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?