2.1.2 Symmetrization#

Prompts

What does it mean for particles to be identical in quantum mechanics? Why does indistinguishability constrain the wavefunction to be either symmetric or antisymmetric?
How do you construct a properly symmetrized (bosonic) or antisymmetrized (fermionic) multi-particle state from single-particle wavefunctions? What happens when two fermions are placed in the same state?
What are the insertion and deletion rules \(\rhd_\pm\) and \(\lhd_\pm\)? How do they provide a systematic procedure for adding or removing a particle while preserving symmetry?
Why does the Pauli exclusion principle follow automatically from antisymmetry? What physical consequences does it have for atomic structure and the stability of matter?

Lecture Notes#

Overview#

Particles of the same type (electrons, photons, etc.) are absolutely identical in quantum mechanics—they carry no identity tags and are completely interchangeable. This indistinguishability is not a practical limitation but a fundamental principle that restricts which quantum states are physical: multi-particle states must be either symmetric (bosons) or antisymmetric (fermions) under particle exchange.

Analogy. Money in a bank account is identical. You cannot ask “Is this the dollar I deposited today or yesterday?”—dollars are indistinguishable. Identical particles are the same: two electrons have no labels. This simple observation has profound consequences for the structure of matter.

Identical Particles#

Identical Particles

Particles of the same type have identical mass, charge, spin, and no inherent labels. This indistinguishability is a fundamental principle that constrains which quantum states are physical.

Physical requirement. Let \(\hat{\mathcal{P}}_{ij}\) be the permutation operator that swaps particles \(i\) and \(j\). For identical particles, all observables must be invariant under permutation: \(\langle\Psi\vert \hat{O}\vert\Psi\rangle = \langle\Psi\vert \hat{\mathcal{P}}_{ij}^\dagger \hat{O}\,\hat{\mathcal{P}}_{ij}\vert\Psi\rangle\) for any observable \(\hat{O}\). This requires the state to be an eigenstate of every transposition:

\[ \hat{\mathcal{P}}_{ij}\vert\Psi\rangle = \lambda\,\vert\Psi\rangle \]

Since \(\hat{\mathcal{P}}_{ij}^2 = \hat{I}\), the eigenvalue satisfies \(\lambda^2 = 1\), so \(\lambda = \pm 1\).

Symmetry Constraint

Under particle exchange, the many-body state must satisfy:

Symmetric (\(\lambda = +1\), bosons): \(\hat{\mathcal{P}}_{ij}\vert\Psi\rangle = +\vert\Psi\rangle\)
Antisymmetric (\(\lambda = -1\), fermions): \(\hat{\mathcal{P}}_{ij}\vert\Psi\rangle = -\vert\Psi\rangle\)

The choice is determined by particle spin (integer → boson, half-integer → fermion) via the spin-statistics theorem.

Discussion: Indistinguishability and Entanglement

A symmetrized or antisymmetrized state is always an entangled superposition—you cannot assign a definite single-particle state to a specific particle. Is this “entanglement” the same as in a Bell pair, or is it fundamentally different? Can you design an experiment that distinguishes indistinguishability-entanglement from preparation-entanglement?

Symmetric and Antisymmetric States#

Let \(\{\vert\alpha\rangle\}\) be an orthonormal basis for the single-particle Hilbert space \(\mathcal{H}\). A generic \(N\)-particle state in \(\mathcal{H}^{\otimes N}\) is

(31)#\[ \vert\Psi\rangle = \sum_{\alpha_1, \ldots, \alpha_N} \Psi_{\alpha_1 \cdots \alpha_N}\;\vert\alpha_1\rangle \otimes \vert\alpha_2\rangle \otimes \cdots \otimes \vert\alpha_N\rangle \]

where \(\Psi_{\alpha_1 \cdots \alpha_N}\) is the many-body amplitude. For identical particles, only the symmetric or antisymmetric subspace of \(\mathcal{H}^{\otimes N}\) is physical.

Bosonic (Symmetric) States.

For \(N\) bosons placed in distinct single-particle states \(\vert\alpha_1\rangle, \ldots, \vert\alpha_N\rangle\), the symmetrized state is:

(32)#\[ \vert\alpha_1, \ldots, \alpha_N\rangle_+ = \frac{1}{\sqrt{N!}} \sum_{\pi \in S_N} \vert\alpha_{\pi(1)}\rangle \otimes \vert\alpha_{\pi(2)}\rangle \otimes \cdots \otimes \vert\alpha_{\pi(N)}\rangle \]

The sum (all signs positive) is the ket-space analog of the permanent. Multiple bosons can occupy the same state.

Example: Two Bosons

Problem. Construct the symmetric state for two bosons in orthogonal states \(\vert\alpha_1\rangle, \vert\alpha_2\rangle\).

Solution.

\[ \vert\alpha_1, \alpha_2\rangle_+ = \frac{1}{\sqrt{2}} \bigl(\vert\alpha_1\rangle\vert\alpha_2\rangle + \vert\alpha_2\rangle\vert\alpha_1\rangle\bigr) \]

If both bosons occupy the same state \(\vert\alpha\rangle\): \(\vert\alpha, \alpha\rangle_+ = \vert\alpha\rangle\vert\alpha\rangle\) (already symmetric).

Normalization with Repeated States

More generally, when \(n_\alpha\) bosons share the same single-particle state, the permanent acquires \(n_\alpha !\) identical terms, which should be further normalized by \(\sqrt{n_\alpha !}\). So for \(N\) bosons with occupation numbers \(\{n_\alpha\}\), the normalization factor is:

\[ \mathcal{N}_+ = \frac{1}{\sqrt{N! \prod_\alpha n_\alpha!}} \]

Fermionic (Antisymmetric) States.

For \(N\) fermions placed in distinct single-particle states \(\vert\alpha_1\rangle, \ldots, \vert\alpha_N\rangle\), the antisymmetrized state is:

(33)#\[ \vert\alpha_1, \ldots, \alpha_N\rangle_- = \frac{1}{\sqrt{N!}} \sum_{\pi \in S_N} \mathrm{sgn}(\pi)\;\vert\alpha_{\pi(1)}\rangle \otimes \vert\alpha_{\pi(2)}\rangle \otimes \cdots \otimes \vert\alpha_{\pi(N)}\rangle \]

This is the ket-space analog of the Slater determinant. Antisymmetry is automatic: swapping any two particle labels flips the sign. If \(\vert\alpha_i\rangle = \vert\alpha_j\rangle\) for \(i \neq j\), the state vanishes—this is the Pauli exclusion principle.

Example: Two Fermions

Problem. Construct the antisymmetric state for two fermions in states \(\vert\alpha_1\rangle, \vert\alpha_2\rangle\).

Solution.

\[ \vert\alpha_1, \alpha_2\rangle_- = \frac{1}{\sqrt{2}} \bigl(\vert\alpha_1\rangle\vert\alpha_2\rangle - \vert\alpha_2\rangle\vert\alpha_1\rangle\bigr) \]

Check: Swap particle labels gives \(-\vert\alpha_1, \alpha_2\rangle_-\). Set \(\vert\alpha_1\rangle = \vert\alpha_2\rangle\) gives zero.

Pauli Exclusion Principle#

Pauli Exclusion Principle

No two identical fermions can occupy the same single-particle quantum state. The fermionic occupation number satisfies \(n_\alpha \in \{0, 1\}\).

(Bosons have no such restriction: \(n_\alpha \in \{0, 1, 2, \ldots\}\))

This follows directly from antisymmetry: if \(\vert\alpha_i\rangle = \vert\alpha_j\rangle\), the antisymmetrized state has two identical columns in the Slater determinant and vanishes. The consequences are far-reaching: atomic shell structure, chemical bonding, degeneracy pressure in white dwarfs and neutron stars, and the stability of matter itself.

Insertion and Deletion Rules#

To add or remove a particle while preserving symmetry, we define insertion (\(\rhd_\pm\)) and deletion (\(\lhd_\pm\)) operations on first-quantized states. Let \(\vert\alpha\rangle\) be a single-particle state, let \(\mathbb{1}\) be the tensor identity (generator of the zero-particle space \(\mathbb{C}\), satisfying \(\vert\alpha\rangle \equiv \mathbb{1} \otimes \vert\alpha\rangle \equiv \vert\alpha\rangle \otimes \mathbb{1}\)), and let \(\vert\Psi\rangle = \vert\alpha_1\rangle \otimes \vert\alpha_2\rangle \otimes \cdots\) be a generic tensor product state. The subscript \(\pm\) denotes symmetrization (+, bosons) or antisymmetrization (−, fermions).

Insertion and Deletion Rules

Insertion \(\rhd_\pm\) (adds a particle symmetrically/antisymmetrically):

\[\begin{split} \begin{split} \vert\alpha\rangle \rhd_\pm \mathbb{1} &= \vert\alpha\rangle, \\ \vert\alpha\rangle \rhd_\pm (\vert\beta\rangle \otimes \vert\Psi\rangle) &= \vert\alpha\rangle \otimes \vert\beta\rangle \otimes \vert\Psi\rangle \pm \vert\beta\rangle \otimes (\vert\alpha\rangle \rhd_\pm \vert\Psi\rangle) \end{split} \end{split}\]

Deletion \(\lhd_\pm\) (removes a particle symmetrically/antisymmetrically):

\[\begin{split} \begin{split} \vert\alpha\rangle \lhd_\pm \mathbb{1} &= 0, \\ \vert\alpha\rangle \lhd_\pm (\vert\beta\rangle \otimes \vert\Psi\rangle) &= \delta_{\alpha\beta}\,\vert\Psi\rangle \pm \vert\beta\rangle \otimes (\vert\alpha\rangle \lhd_\pm \vert\Psi\rangle) \end{split} \end{split}\]

Both operations are linear in the state being acted on.

The recursive structure means insertion places \(\vert\alpha\rangle\) at every possible position with alternating signs (for fermions) or all-positive signs (for bosons). Deletion removes \(\vert\alpha\rangle\) from every position where it appears, with the same sign pattern.

Example: Bosonic Insertion into a Two-Particle State

Problem. Start from the symmetric two-particle state

\[ \vert\beta,\gamma\rangle_+ = \frac{1}{\sqrt{2}}(\vert\beta\gamma\rangle+\vert\gamma\beta\rangle), \]

and compute \(\vert\alpha\rangle \rhd_+ \vert\beta,\gamma\rangle_+\).

Solution. By linearity,

\[ \vert\alpha\rangle \rhd_+ \vert\beta,\gamma\rangle_+ =\frac{1}{\sqrt{2}}\left(\vert\alpha\rangle \rhd_+ \vert\beta\gamma\rangle +\vert\alpha\rangle \rhd_+ \vert\gamma\beta\rangle\right). \]

Applying insertion to each ordered term gives the symmetric three-particle combination

\[ \vert\alpha\rangle \rhd_+ \vert\beta,\gamma\rangle_+ =\frac{1}{\sqrt{2}}(\vert\alpha\beta\gamma\rangle+\vert\beta\alpha\gamma\rangle+\vert\beta\gamma\alpha\rangle +\vert\alpha\gamma\beta\rangle+\vert\gamma\alpha\beta\rangle+\vert\gamma\beta\alpha\rangle). \]

Example: Fermionic Insertion into a Two-Particle State

Problem. Start from the antisymmetric two-particle state

\[ \vert\beta,\gamma\rangle_- = \frac{1}{\sqrt{2}}(\vert\beta\gamma\rangle-\vert\gamma\beta\rangle), \]

and compute \(\vert\alpha\rangle \rhd_- \vert\beta,\gamma\rangle_-\).

Solution. By linearity,

\[ \vert\alpha\rangle \rhd_- \vert\beta,\gamma\rangle_- =\frac{1}{\sqrt{2}}\left(\vert\alpha\rangle \rhd_- \vert\beta\gamma\rangle -\vert\alpha\rangle \rhd_- \vert\gamma\beta\rangle\right). \]

Evaluating both insertions yields the alternating-sign antisymmetric sum

\[ \vert\alpha\rangle \rhd_- \vert\beta,\gamma\rangle_- =\frac{1}{\sqrt{2}}(\vert\alpha\beta\gamma\rangle-\vert\beta\alpha\gamma\rangle+\vert\beta\gamma\alpha\rangle -\vert\alpha\gamma\beta\rangle+\vert\gamma\alpha\beta\rangle-\vert\gamma\beta\alpha\rangle). \]

The insertion and deletion rules \(\rhd_\pm\) and \(\lhd_\pm\) are the first-quantized building blocks for adding and removing particles. When properly normalized, they yield creation (\(\hat{a}^\dagger_\alpha\)) and annihilation (\(\hat{a}_\alpha\)) operators—the central objects of second quantization, developed in §2.1.3 Second Quantization.

Poll: Effect of particle exchange

A symmetric two-qubit state is \(\vert\psi_S\rangle = \frac{1}{\sqrt{2}}(\vert 01\rangle + \vert 10\rangle)\). If you apply the exchange operator \(\hat{P}\) (swap the two qubits), what do you get?

(A) A completely different state.

(B) The same state: \(\hat{P}\vert\psi_S\rangle = \vert\psi_S\rangle\) (eigenvalue \(+1\)).

(D) An entangled state.

Summary#

Identical particles require the many-body state to be symmetric (bosons, \(\lambda=+1\)) or antisymmetric (fermions, \(\lambda=-1\)) under exchange, determined by spin via the spin-statistics theorem.
Symmetric states (permanent form) allow unlimited occupation; antisymmetric states (Slater determinant form) automatically enforce the Pauli exclusion principle: \(n_\alpha \in \{0,1\}\) for fermions.
Insertion (\(\rhd_\pm\)) and deletion (\(\lhd_\pm\)) rules recursively add/remove particles while preserving symmetry and form the first-quantized foundation for creation and annihilation operators.
When normalized, insertion/deletion rules become the second-quantization operators (\(\hat{a}^\dagger_\alpha\), \(\hat{a}_\alpha\)) that govern fermionic and bosonic quantum fields (§2.1.3).

Homework#

1. Three-particle statistics. For three identical particles, consider the transpositions \(\hat{\mathcal{P}}_{12}, \hat{\mathcal{P}}_{23}, \hat{\mathcal{P}}_{13}\).

(a) Verify the braid-like relation \(\hat{\mathcal{P}}_{12}\hat{\mathcal{P}}_{23}\hat{\mathcal{P}}_{12} = \hat{\mathcal{P}}_{13}\) by acting on a generic tensor basis state \(\vert\alpha\rangle\otimes\vert\beta\rangle\otimes\vert\gamma\rangle\).

(b) Suppose a three-particle state \(\vert\Psi\rangle\) is a simultaneous eigenstate of every transposition, \(\hat{\mathcal{P}}_{ij}\vert\Psi\rangle = \lambda_{ij}\vert\Psi\rangle\) with \(\lambda_{ij} \in \{\pm 1\}\). Use the identity in (a) to show that \(\lambda_{13} = \lambda_{23}\), and analogous relations to conclude \(\lambda_{12} = \lambda_{23} = \lambda_{13}\).

(c) Conclude that the only consistent exchange statistics for three or more identical particles in three dimensions are bosonic (\(\lambda = +1\)) or fermionic (\(\lambda = -1\))—there is no third option.

2. Exchange projectors. For two particles, define \(\hat{\Sigma}_\pm = \tfrac{1}{2}(\hat{I} \pm \hat{\mathcal{P}}_{12})\). Show that \(\hat{\Sigma}_\pm^2 = \hat{\Sigma}_\pm\), \(\hat{\Sigma}_+ \hat{\Sigma}_- = 0\), and \(\hat{\Sigma}_+ + \hat{\Sigma}_- = \hat{I}\), and verify that \(\hat{\Sigma}_+\vert\Psi\rangle\) is symmetric and \(\hat{\Sigma}_-\vert\Psi\rangle\) is antisymmetric under \(\hat{\mathcal{P}}_{12}\). Conclude that every two-particle state decomposes uniquely as \(\vert\Psi\rangle = \hat{\Sigma}_+\vert\Psi\rangle + \hat{\Sigma}_-\vert\Psi\rangle\).

3. Generalized Pauli exclusion. The lecture shows the two-fermion state vanishes when \(\vert\alpha_1\rangle = \vert\alpha_2\rangle\). Here we extend this vanishing to linear dependence.

(a) Show that \(\vert\alpha_1, \alpha_2\rangle_- = 0\) whenever \(\vert\alpha_2\rangle = c\,\vert\alpha_1\rangle\) for any \(c \in \mathbb{C}\).

(b) For three fermions, show that \(\vert\alpha_1, \alpha_2, \alpha_3\rangle_-\) vanishes whenever \(\vert\alpha_3\rangle \in \mathrm{span}\{\vert\alpha_1\rangle, \vert\alpha_2\rangle\}\). (Hint: use linearity in the third slot and the two-fermion result.)

(c) Argue from multilinearity and antisymmetry that \(\vert\alpha_1, \ldots, \alpha_N\rangle_-\) depends only on the \(N\)-dimensional subspace spanned by \(\{\vert\alpha_i\rangle\}\), up to an overall scalar, and vanishes whenever these states are linearly dependent. This is the geometric form of Pauli exclusion: the physical fermionic state is determined by the filled subspace, not by a choice of basis inside it.

(d) Contrast with bosons. Show by explicit counterexample that \(\vert\alpha, \alpha\rangle_+ \ne 0\) for any nonzero \(\vert\alpha\rangle\), and that \(\vert\alpha_1, c\,\alpha_1\rangle_+ \ne 0\) for any \(c \ne 0\). Bosons admit no analogous linear-dependence vanishing.

4. Bosonic normalization factor. Work at the state level, with single-particle modes drawn from an orthonormal basis.

(a) For two bosons in distinct modes \(\vert\alpha\rangle, \vert\beta\rangle\) with \(\alpha \ne \beta\), verify that the lecture’s prefactor \(1/\sqrt{2!}\) gives a unit-norm state \(\vert\alpha, \beta\rangle_+\). For two bosons in the same mode, verify that the normalized state is \(\vert\alpha, \alpha\rangle_+ = \vert\alpha\rangle \otimes \vert\alpha\rangle\), whose prefactor differs from the naive \(1/\sqrt{2!}\) by an extra factor of \(\sqrt{2!}\).

(b) Three bosons with occupations \(n_\alpha = 2,\ n_\beta = 1\). Enumerate the six terms in the unnormalized sum \(\sum_{\pi \in S_3} \vert\alpha_{\pi(1)} \alpha_{\pi(2)} \alpha_{\pi(3)}\rangle\) (with \(\alpha_1 = \alpha_2 = \alpha\), \(\alpha_3 = \beta\)), observe that only three distinct orthonormal orderings appear (each with multiplicity \(2!\)), and compute the norm squared of the full sum: \((2!)^2 \cdot 3 = 3!\cdot 2!\).

(c) General formula. For \(N\) bosons with occupation numbers \(\{n_\alpha\}\) on an orthonormal basis, count the number of distinct orderings (\(N!/\prod_\alpha n_\alpha!\)) and the multiplicity of each (\(\prod_\alpha n_\alpha!\)). Deduce that the unnormalized sum has norm squared \(N!\,\prod_\alpha n_\alpha!\), hence

\[ \mathcal{N}_+ = \frac{1}{\sqrt{N!\,\prod_\alpha n_\alpha!}}. \]

(d) Explain in one sentence why fermions require no analogous \(n_\alpha!\) correction.

5. Insertion and deletion as inverses. Work in a single-mode Fock space throughout.

(a) Fermionic case. The fermionic Fock space for one mode contains only the vacuum \(\mathbb{1}\) and the occupied state \(\vert\alpha\rangle\). Apply the insertion and deletion rules to compute all four of \(\vert\alpha\rangle \rhd_- \mathbb{1}\), \(\vert\alpha\rangle \rhd_- \vert\alpha\rangle\), \(\vert\alpha\rangle \lhd_- \mathbb{1}\), and \(\vert\alpha\rangle \lhd_- \vert\alpha\rangle\). Verify that \(\vert\alpha\rangle \rhd_- \vert\alpha\rangle = 0\) (Pauli) and \(\vert\alpha\rangle \lhd_- \mathbb{1} = 0\).

(b) Verify that deletion undoes insertion on the vacuum, and insertion undoes deletion on the occupied state:

\[ \vert\alpha\rangle \lhd_- \bigl(\vert\alpha\rangle \rhd_- \mathbb{1}\bigr) = \mathbb{1}, \qquad \vert\alpha\rangle \rhd_- \bigl(\vert\alpha\rangle \lhd_- \vert\alpha\rangle\bigr) = \vert\alpha\rangle. \]

Combining both, conclude that \(\vert\alpha\rangle \lhd_- \bigl(\vert\alpha\rangle \rhd_- \vert\Psi\rangle\bigr) + \vert\alpha\rangle \rhd_- \bigl(\vert\alpha\rangle \lhd_- \vert\Psi\rangle\bigr) = \vert\Psi\rangle\) for every \(\vert\Psi\rangle\) in the single-mode fermionic Fock space.

(c) Bosonic case. Let \(\vert\alpha^n\rangle\) denote the \(n\)-boson state in a single mode (with \(\vert\alpha^0\rangle = \mathbb{1}\) and \(\vert\alpha^1\rangle = \vert\alpha\rangle\)). Using the recursive definition, prove by induction on \(n\):

\[ \vert\alpha\rangle \rhd_+ \vert\alpha^n\rangle = (n+1)\,\vert\alpha^{n+1}\rangle, \qquad \vert\alpha\rangle \lhd_+ \vert\alpha^n\rangle = n\,\vert\alpha^{n-1}\rangle. \]

(d) Compute \(\vert\alpha\rangle \lhd_+ \bigl(\vert\alpha\rangle \rhd_+ \vert\alpha^n\rangle\bigr) - \vert\alpha\rangle \rhd_+ \bigl(\vert\alpha\rangle \lhd_+ \vert\alpha^n\rangle\bigr)\) and show the result is \((2n+1)\,\vert\alpha^n\rangle\), not \(\vert\alpha^n\rangle\). Conclude that the bare bosonic insertion and deletion rules are mutual inverses only up to a particle-number-dependent multiplicity, whereas the fermionic rules are already exact inverses. The bosonic normalization is addressed in §2.1.3.

6. Slater determinant overlap. Derive the central identity relating many-fermion inner products to determinants of single-particle overlaps.

(a) For arbitrary (possibly non-orthogonal) single-particle states \(\vert\alpha_1\rangle, \vert\alpha_2\rangle, \vert\beta_1\rangle, \vert\beta_2\rangle\), compute \(\langle\alpha_1, \alpha_2 \vert \beta_1, \beta_2\rangle_-\) directly from the two-fermion antisymmetric definition, and show it equals \(\det M\) with \(M_{ij} = \langle\alpha_i \vert \beta_j\rangle\).

(b) Specialize to orthonormal \(\{\vert\alpha_i\rangle\}\) and verify \(\langle\alpha_1, \alpha_2 \vert \alpha_1, \alpha_2\rangle_- = 1\).

(c) Hydrogen molecule. Let \(\vert L\rangle, \vert R\rangle\) be unit-norm but non-orthogonal left and right atomic orbitals with real overlap \(\langle L \vert R\rangle = s \in [0, 1]\). Compute \(\langle L, R \vert L, R\rangle_-\) and interpret the limits \(s \to 0\) (atoms far apart) and \(s \to 1\) (atoms merge; linear dependence forces the antisymmetric state to vanish, a generalized form of Pauli exclusion).

(d) State without proof the \(N\)-fermion generalization \(\langle\alpha_1, \ldots, \alpha_N \vert \beta_1, \ldots, \beta_N\rangle_- = \det\langle\alpha_i \vert \beta_j\rangle\). Briefly note the bosonic analog, which replaces the determinant with the permanent of the same matrix.

7. Fermion counting. How many independent antisymmetric states can be formed by placing \(N\) fermions into \(D\) single-particle modes (with \(D \geq N\))? Give the combinatorial formula as a binomial coefficient. What happens when \(N > D\), and how does this connect to the Pauli exclusion principle?

8. Boson counting. How many independent symmetric states can be formed by placing \(N\) bosons into \(D\) single-particle modes? Derive the combinatorial formula (stars and bars) as a binomial coefficient, and contrast with the fermionic count from the previous problem. Explain why there is no upper bound on \(N\) at fixed \(D\) for bosons.