Wavefunction Symmetry and Antisymmetry
In quantum mechanics there is a fundamental distinction between identical particles and distinguishable particles. For example, any two electrons are fundamentally indistinguishable from each other; the laws of physics make it impossible to "stamp an identification number" on a certain electron to keep track of it. This translates to a requirement on the wavefunction: For example, if particles 1 and 2 are indistinguishable, then:
where s is the spin quantum number of the particle: integer for bosons (s = 1, 2, 3...) and half-integer for fermions (s = 1/2, 3/2...).
The wavefunction is said to be symmetric (no sign change) under boson interchange and antisymmetric (sign changes) under fermion interchange. This feature of the wavefunction is known as the Pauli principle.
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—George Gordon Noel Byron (1788–1824)