Quantum mechanics

Many-body systems and identical particles

Exchange symmetry is part of a many-particle state: bosons have symmetric wavefunctions, fermions antisymmetric ones, yielding the Pauli principle.

Identical particles have no physically meaningful individual labels. Exchanging two particles changes the many-body state at most by a phase; in three dimensions the familiar possibilities are $+1$ for bosons and $-1$ for fermions.

\Psi_{B}(1,2)=+\Psi_{B}(2,1),\qquad \Psi_{F}(1,2)=-\Psi_{F}(2,1)

Definition: Two-fermion state

For orthonormal spin-orbitals $a,b$ , the normalized fermion state is the Slater determinant $|a,b\rangle_F=(|a(1)b(2)\rangle-|b(1)a(2)\rangle)/\sqrt2$ . If $a=b$ it vanishes: two fermions cannot occupy the same spin-orbital. Two electrons can share a spatial orbital when their spins differ.

Compare bosonic and fermionic probability densities as two wave packets approach; the antisymmetric fermion node becomes visible.

Bosons, fermions, and many-body systems

Exchange symmetry applies to coordinates and spin together. For two electrons, the total state must be antisymmetric: the antisymmetric spin singlet pairs with a symmetric spatial state, while a symmetric triplet pairs with an antisymmetric spatial state. Occupation numbers and creation–annihilation operators are common many-body tools.

Example: Example: helium ground state

In an independent-particle approximation, helium's two electrons share the spatial $1s$ orbital. To make the total state antisymmetric, their spin is the singlet $S=0$ : $(|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle)/\sqrt2$ .

Solution

The spatial part is symmetric under exchange, so the spin part must be antisymmetric for their product to change sign.

For interacting particles, the Hamiltonian typically combines one-particle kinetic energy with pair potentials $V(\mathbf r_i-\mathbf r_j)$ . The ground state need not factor into independent states: correlations produce observables not determined by one-particle density alone. In a mean-field approximation, each particle moves in an effective potential generated by the others, while residual correlations require further methods. For fermions, antisymmetry creates degeneracy pressure even at low temperature. These ideas connect quantum mechanics to condensed matter, ultracold atoms, and nuclear physics.

Quick check

What exchange property does the wavefunction of two identical fermions have?

When can two electrons share one spatial orbital?

References

J. J. Sakurai, Jim Napolitano (2020). Modern Quantum Mechanics