This chapter addresses the quantum mechanics of identical particles, emphasizing the role of wavefunction symmetry in determining statistical behavior. Symmetric and antisymmetric wavefunctions are defined, and the implications of particle indistinguishability are examined, particularly regarding how it influences measurable outcomes and the concept of energy exchange degeneracy. The distinguishability of identical particles is discussed and shown to affect specific calculations. For systems where particle interactions can be neglected in the zero-order approximation, the total energy eigenfunction is expressed as a product of single-particle eigenfunctions. The Pauli exclusion principle is derived as a consequence of antisymmetrization. The relationship between symmetry type (bosonic or fermionic) and quantum statistics is elaborated. As an illustration, the ground state and the first excited
s-state of the helium atom are analyzed under a simplified approximation. Metadata/Keywords: identical particles, symmetric wavefunction, antisymmetric wavefunction, Pauli exclusion principle, quantum statistics, energy exchange degeneracy, indistinguishability, helium atom, symmetry and statistics
A feature of quantum mechanics is that identical particles cannot be labelled. This means that identical particles must be treated as indistinguishable and implies that if we exchange two particles in a many – particle system – e.g. the electrons in a molecule – then the wavefunction describing the system cannot change, apart from a phase factor. Besides some exotic possibilities in strictly two-dimensional systems, this phase factor has only two values: +1 and -1, which lead to symmetric and anti-symmetric wavefunctions, respectively. Particles with symmetric wavefunctions, such as photons, are called bosons, while those with anti-symmetric wavefunctions, such as electrons, are called fermions.