Fermions

Fermions are identical particles whose physical state is antisymmetric under exchange of particle slots. A two-fermion state changes sign when the slots are swapped, and an $N$-fermion state picks up the sign of the permutation:

$$U(\pi)\lvert\Psi\rangle = \operatorname{sgn}(\pi)\lvert\Psi\rangle.$$

This is the fermionic half of the symmetrization postulate. It is the structural origin of the Pauli exclusion principle, Slater determinants, fermionic occupation bitstrings, and anticommutation relations.

Antisymmetric Exchange Rule

For two identical fermions built from a one-particle Hilbert space $\mathcal h$, the formal slot-labeled space is

$$\mathcal h\otimes\mathcal h.$$

The exchange operator $P_{12}$ swaps the two slots. A two-fermion state satisfies

$$P_{12}\lvert\Psi\rangle = - \lvert\Psi\rangle.$$

In a coordinate-spin representation, the complete one-particle label is

$$q=(\mathbf x,s,\ldots),$$

and the two-fermion wavefunction obeys

$$\Psi(q_1,q_2) = - \Psi(q_2,q_1).$$

The exchange acts on all one-particle degrees of freedom. For electrons, $q$ includes spin as well as position. For atoms, $q$ can include internal hyperfine labels when those labels are part of the one-particle state.

For $N$ identical fermions, the physical fixed-particle-number Hilbert space is the antisymmetric subspace

$$\wedge^N\mathcal h \subset \mathcal h^{\otimes N}.$$

Equivalently,

$$U(\pi)\lvert\Psi\rangle = \operatorname{sgn}(\pi) \lvert\Psi\rangle \qquad \text{for all }\pi\in S_N.$$

Two-Fermion Examples

Let $\lvert a\rangle$ and $\lvert b\rangle$ be orthonormal one-particle states. The antisymmetric two-fermion state with one fermion in $a$ and one in $b$ is

$$\lvert a,b\rangle_A = \frac{1}{\sqrt2} \bigl( \lvert a\rangle_1\lvert b\rangle_2 - \lvert b\rangle_1\lvert a\rangle_2 \bigr).$$

In wavefunction notation this is

$$\Psi_A(q_1,q_2) = \frac{1}{\sqrt2} \bigl[ \varphi_a(q_1)\varphi_b(q_2) - \varphi_b(q_1)\varphi_a(q_2) \bigr].$$

Exchanging $q_1$ and $q_2$ changes the sign:

$$\Psi_A(q_2,q_1) = - \Psi_A(q_1,q_2).$$

If the two one-particle states are the same, the antisymmetric combination vanishes:

$$\frac{1}{\sqrt2} \bigl( \lvert a\rangle_1\lvert a\rangle_2 - \lvert a\rangle_1\lvert a\rangle_2 \bigr) = 0.$$

The zero vector is not a physical state. This is the two-particle version of Pauli exclusion.

No Double Occupation of a Complete One-Particle State

The phrase “complete one-particle state” is essential. A one-particle state includes all quantum numbers needed to specify a mode: spatial wavefunction, spin state, internal state, band index, trap level, or any other relevant one-particle degree of freedom.

For electrons in atomic physics, a complete one-particle state is often called a spin-orbital:

$$\varphi(\mathbf x)\chi_s.$$

Two electrons can share the same spatial orbital only if their spin states differ and the total two-electron state remains antisymmetric. In the common helium ground-state approximation, the spatial wavefunction is symmetric while the spin state is the antisymmetric singlet. The electrons are still identical fermions; opposite spin labels do not make them distinguishable particles.

The exclusion principle says that two identical fermions cannot occupy the same complete state, not that they can never be found near the same position.

Fermionic Occupation Numbers

Because identical-particle labels are not observable, many-fermion states are usually described by mode occupation. In a chosen orthonormal mode basis, a fermionic occupation-number state is written

$$\lvert n_1,n_2,\ldots\rangle_F.$$

For fermions,

$$n_i\in\{0,1\}.$$

The state

$$\lvert1_a,1_b\rangle_F$$

means modes $a$ and $b$ are occupied once. There is no physical state

$$\lvert2_a\rangle_F$$

for two identical fermions in the same complete mode $a$.

In creation-operator language, this is encoded by

$$(c_a^\dagger)^2=0.$$

The signs of many-fermion occupation states depend on a fixed ordering convention for modes. For example,

$$c_a^\dagger c_b^\dagger\lvert0\rangle = - c_b^\dagger c_a^\dagger\lvert0\rangle$$

when $a\ne b$. The detailed operator algebra belongs to Fermionic Anticommutation Relations.

Examples of Fermionic Systems

Fermions include many of the particles that build ordinary matter.

Electrons. Electrons are spin-$1/2$ fermions. Atomic shell structure, chemical valence, band filling, and much of condensed-matter physics rely on electron antisymmetry.

Protons and neutrons. Protons and neutrons are spin-$1/2$ composite fermions in ordinary nuclear physics. Their internal quark structure is not usually resolved in nonrelativistic nuclear models, where each nucleon is treated as a fermionic particle.

Atoms and molecules. Composite atoms can behave as fermions when their total spin is half-integer and the relevant internal state is fixed. Helium-3 atoms are standard fermionic atoms in low-energy atomic physics.

Quasiparticles. In many-body systems, effective fermionic quasiparticles can appear even when the underlying microscopic model has more structure. The fermionic algebra describes the effective excitation modes in the regime where the quasiparticle description is valid.

Fermions and Spin

In relativistic quantum field theory, the spin-statistics theorem relates half-integer spin to fermionic statistics and integer spin to bosonic statistics under standard assumptions. Nonrelativistic quantum mechanics usually takes this assignment as an input when choosing the physical exchange sector for a particle species.

Thus:

• electrons, protons, and neutrons are fermions;
• many atoms with half-integer total spin behave as fermions;
• the same chemical element can have bosonic and fermionic isotopes;
• the exchange rule applies only among identical particles of the same species in the same relevant internal state.

Two different fermion species, such as an electron and a proton, are distinguishable subsystems. Their joint state is not required to be antisymmetric under exchanging the electron with the proton.

Physical Consequences

Fermionic antisymmetry changes state counting and interference. It is responsible for:

• Pauli exclusion in atoms and solids;
• shell structure and chemical periodicity;
• filled Fermi seas and Fermi surfaces;
• degeneracy pressure in dense fermionic matter;
• signs from exchanging fermionic creation and annihilation operators;
• determinant structures in many-electron wavefunctions.

It is still not an ordinary repulsive force. The Hamiltonian supplies interactions. Antisymmetry supplies the allowed state space and the interference signs inside that state space.

Common Mistakes

• Saying two fermions can never be at the same position.
• Forgetting that “same state” means same complete one-particle state.
• Treating opposite-spin electrons as distinguishable particles.
• Treating Pauli exclusion as an extra force rather than a consequence of antisymmetry.
• Ignoring the fixed mode ordering behind fermionic occupation signs.
• Assuming a Slater determinant is optional decoration rather than the compact form of antisymmetry.
• Applying fermionic antisymmetry to particles of different species.
• Confusing the nonrelativistic exchange rule with a proof of the spin-statistics theorem.

Cross-Links

• Indistinguishability
• Symmetrization Postulate
• Symmetric Group
• Bosons
• Exchange Operators
• Pauli Exclusion Principle
• Symmetric and Antisymmetric Wavefunctions
• Identical-Particle Entanglement Cautions
• Spin and Spatial Wavefunctions
• Slater Determinants
• Occupation-Number Basis
• Fermionic Fock Space
• Number States
• Fermionic Anticommutation Relations
• Fermi–Dirac Statistics
• Two Spin-1/2 Particles
• Reference Bridge: Second Quantization

References

• A. Messiah and O. W. Greenberg, “Symmetrization Postulate and Its Experimental Foundation,” Physical Review 136, B248-B267, 1964.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics, Wiley, 1977.
• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover, 2003.

Exercises

1. Exchange check. Verify that

$$\lvert a,b\rangle_A = \frac{1}{\sqrt2} \bigl( \lvert a\rangle_1\lvert b\rangle_2 - \lvert b\rangle_1\lvert a\rangle_2 \bigr)$$

is antisymmetric under $P_{12}$.

Solution

Applying $P_{12}$ swaps the slots:

$$P_{12}\lvert a,b\rangle_A = \frac{1}{\sqrt2} \bigl( \lvert b\rangle_1\lvert a\rangle_2 - \lvert a\rangle_1\lvert b\rangle_2 \bigr).$$

Factoring out $-1$ gives

$$P_{12}\lvert a,b\rangle_A = - \lvert a,b\rangle_A.$$

2. Same-state cancellation. Show that the antisymmetric two-fermion state vanishes if both fermions are assigned the same normalized one-particle state.

Solution

Set $\lvert b\rangle=\lvert a\rangle$ in the antisymmetric combination:

$$\frac{1}{\sqrt2} \bigl( \lvert a\rangle_1\lvert a\rangle_2 - \lvert a\rangle_1\lvert a\rangle_2 \bigr) = 0.$$

The zero vector cannot be normalized, so it is not a physical state.

3. Same spatial orbital. Can two electrons occupy the same spatial orbital in the helium ground-state approximation?

Solution

Yes, provided their complete one-particle states differ by spin and the total two-electron state is antisymmetric. In the usual approximation, the spatial part is symmetric because both electrons occupy the same spatial orbital, while the spin part is the antisymmetric singlet. The product is antisymmetric overall.

4. Occupation bitstrings. Which of the following are allowed fermionic occupation patterns?

$$\lvert2_a\rangle_F, \qquad \lvert1_a,1_b\rangle_F, \qquad \lvert0_a,1_b,1_c\rangle_F.$$

Solution

The state $\lvert2_a\rangle_F$ is not allowed because one fermionic mode cannot have occupation $2$. The other two patterns are allowed if $a$, $b$, and $c$ are distinct modes, because each occupation number is either $0$ or $1$.

5. Ordering sign. If $a\ne b$, why does changing the order of fermionic creation operators change the sign?

Solution

Fermionic creation operators anticommute:

$$\{c_a^\dagger,c_b^\dagger\}=0.$$

Therefore

$$c_a^\dagger c_b^\dagger = - c_b^\dagger c_a^\dagger.$$

Acting on the vacuum gives the same occupied modes but with the opposite sign.

6. Fermions and forces. Does Pauli exclusion mean there is a new repulsive force between identical fermions?

Solution

No. Pauli exclusion follows from antisymmetry of the state. It restricts the allowed state space and changes the kinetic and interaction-energy possibilities, but the forces themselves come from the Hamiltonian.