The requirement for bosons and fermions to be described by symmetric and antisymmetric wave functions, respectively, can be demonstrated using the principles of quantum mechanics and the concept of particle exchange symmetry. This is known as the Pauli exclusion principle.
Let's start with fermions. Fermions are particles that obey Fermi-Dirac statistics, which means they follow the Pauli exclusion principle. This principle states that no two identical fermions can occupy the exact same quantum state simultaneously. To see why fermions require antisymmetric wave functions, consider a system of two identical fermions, labeled as particle A and particle B.
Let ψ_A be the wave function describing particle A and ψ_B be the wave function describing particle B. If we interchange the labels of the particles, the wave function should remain unchanged up to a phase factor, because the particles are indistinguishable. Mathematically, this can be written as:
ψ(A ↔ B) = ±ψ,
where ψ(A ↔ B) represents the wave function after exchanging the labels of particles A and B, and ±ψ represents the original wave function. The ± sign represents the phase factor and can be either +1 or -1.
Now, let's consider two possible scenarios: symmetric and antisymmetric wave functions.
Symmetric Wave Function (Bosons): If the wave function ψ is symmetric under particle exchange, it means that ψ(A ↔ B) = ψ. This implies that the phase factor ±1 must be +1. For bosons, which obey Bose-Einstein statistics, multiple particles can occupy the same quantum state simultaneously. Therefore, the wave function of bosons must be symmetric to satisfy this property.
Antisymmetric Wave Function (Fermions): If the wave function ψ is antisymmetric under particle exchange, it means that ψ(A ↔ B) = -ψ. This implies that the phase factor ±1 must be -1. For fermions, the Pauli exclusion principle prohibits multiple particles from occupying the same quantum state. Therefore, the wave function of fermions must be antisymmetric to ensure that the wave function vanishes when two identical fermions occupy the same state.
By requiring the wave function to be either symmetric or antisymmetric under particle exchange, we can account for the different statistical behaviors of bosons and fermions and satisfy the Pauli exclusion principle.