Question

The wavefunction of a system of identical fermions must be:

• A. Symmetric under particle exchange
• B. Antisymmetric under particle exchange
• C. Symmetric for spin, antisymmetric for space
• D. Unrelated to exchange symmetry

Hint:

Symmetry Summary Table: - Bosons: Integer Spin \(\rightarrow\) Symmetric Wavefunction \(\rightarrow\) \(\Psi(1,2) = \Psi(2,1)\) - Fermions: Half-Integer Spin \(\rightarrow\) Antisymmetric Wavefunction \(\rightarrow\) \(\Psi(1,2) = -\Psi(2,1)\)

Answer 1

The Correct Option is B

Concept: In quantum mechanics, identical particles are fundamentally indistinguishable from one another. Interchanging the coordinates of two identical particles in a multi-particle system must not alter any observable physical properties or probability densities of that system. This constraint leads to specific symmetry conditions for the total wavefunction under particle exchange.

Step 1: Define the mathematical formulation of the exchange operator.
Let \( \Psi(1, 2) \) represent the total collective wavefunction of two identical particles, where the labels ‘1‘ and ‘2‘ encompass all spatial and spin coordinates of particle 1 and particle 2. If we apply an exchange operator \( P_{12} \) that swaps the labels of the two particles, the new state is \( \Psi(2, 1) \). Because the particles are indistinguishable, the probability density must remain unchanged: \[ |\Psi(1, 2)|^2 = |\Psi(2, 1)|^2 \] This allows for two mathematical possibilities for the wavefunction itself: \[ \Psi(1, 2) = +\Psi(2, 1) \quad \text{(Symmetric)} \] \[ \Psi(1, 2) = -\Psi(2, 1) \quad \text{(Antisymmetric)} \]

Step 2: Apply the Spin-Statistics Theorem.
The connection between a particle's intrinsic spin and its exchange symmetry is established by the Spin-Statistics Theorem:
• Particles with integer spin (\( 0, 1, 2, \ldots \)) are classified as Bosons and possess symmetric wavefunctions under exchange.
• Particles with half-integer spin (\( \frac{1}{2}, \frac{3}{2}, \ldots \)) are classified as Fermions (such as electrons, protons, and neutrons) and possess antisymmetric wavefunctions under exchange.

Step 3: Connect with the Pauli Exclusion Principle.
The antisymmetric property of fermions requires that: \[ \Psi(1, 2) = -\Psi(2, 1) \] If two fermions occupy the exact same state, swapping them means their coordinates are identical, so \( \Psi(1, 1) = -\Psi(1, 1) \). This implies: \[ 2\Psi(1, 1) = 0 \quad \Rightarrow \quad \Psi(1, 1) = 0 \] This explains the Pauli Exclusion Principle: two identical fermions cannot occupy the exact same quantum state simultaneously. Therefore, the total wavefunction of a system of identical fermions must be antisymmetric under particle exchange.