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):

\[\vert\alpha\rangle \rhd_\pm \mathbb{1} = \vert\alpha\rangle, \qquad \vert\alpha\rangle \rhd_\pm (\vert\beta\rangle \otimes \vert\Psi\rangle) = \vert\alpha\rangle \otimes \vert\beta\rangle \otimes \vert\Psi\rangle \pm \vert\beta\rangle \otimes (\vert\alpha\rangle \rhd_\pm \vert\Psi\rangle)\]

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

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

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.

Summary#

Identical particles have no labels; the many-body state must be symmetric (bosons, \(\lambda=+1\)) or antisymmetric (fermions, \(\lambda=-1\)) under particle exchange.
Symmetric states (permanent): sum over all permutations with positive signs; bosons allow unlimited occupation.
Antisymmetric states (Slater determinant): alternating-sign sum; the Pauli exclusion principle (\(n_\alpha \in \{0,1\}\)) follows automatically.
Insertion/deletion rules \(\rhd_\pm\), \(\lhd_\pm\) recursively add/remove particles while preserving symmetry. When normalized, they become the creation and annihilation operators of second quantization (§2.1.3).

Homework#

1. Let \(\hat{\mathcal{P}}_{12}\) be the operator that swaps particles 1 and 2. Show that \(\hat{\mathcal{P}}_{12}^2 = \hat{I}\), and deduce that the eigenvalues of \(\hat{\mathcal{P}}_{12}\) are \(\pm 1\).

2. Two identical bosons occupy orthogonal single-particle states \(\psi_1, \psi_2\). Write the normalized symmetric wavefunction. What changes if \(\psi_1 = \psi_2\)? Pay attention to normalization.

3. Two identical fermions occupy orthonormal states \(\psi_1, \psi_2\). Write the normalized antisymmetric wavefunction. Verify it changes sign under \(\boldsymbol{r}_1 \leftrightarrow \boldsymbol{r}_2\).

4. Prove the Pauli exclusion principle from antisymmetry: if \(\psi_1 = \psi_2\), show the fermionic wavefunction vanishes identically.

5. Use the bosonic insertion rule to compute \(\psi_\alpha \rhd_+ (\psi_\beta \otimes \psi_\gamma \otimes \psi_\delta)\). How many terms appear?

6. Use the fermionic insertion rule to compute \(\psi_1 \rhd_- (\psi_1 \otimes \psi_2)\). Show that the result vanishes, demonstrating Pauli exclusion at the first-quantized level.

7. Use the bosonic insertion rule \(\rhd_+\) to construct the three-boson symmetric state \(\vert\alpha,\alpha,\beta\rangle_+\) by inserting \(\vert\beta\rangle\) into \(\vert\alpha\rangle \otimes \vert\alpha\rangle\). Verify that the result is symmetric under all permutations and correctly normalized.

8. In a two-mode fermionic system \((\alpha, \beta)\), verify explicitly that \(\hat{c}^\dagger_\alpha \hat{c}^\dagger_\beta \vert 0,0\rangle = -\hat{c}^\dagger_\beta \hat{c}^\dagger_\alpha \vert 0,0\rangle\) by applying the antisymmetric insertion rule \(\rhd_-\) twice and comparing the two orderings.

9. How many independent antisymmetric states can be formed by placing \(N\) fermions into \(D\) single-particle states? Give the combinatorial formula. What happens when \(N > D\)?

10. Use the fermionic deletion rule \(\lhd_-\) to compute \(\hat{c}_2 \vert 1_1, 1_2, 1_3\rangle\), where \(\vert 1_1, 1_2, 1_3\rangle\) is the fully antisymmetrized three-particle state. Track every sign change and state the final sign of the result.