Physics Friday 79

Part 4 of 4

In the previous part, we introduced the spin-statistics theorem, which tells us that identical bosons, particles with integer spin, have joint wavefunctions symmetric with respect to exchange of the particles; while fermions, with half-integer spin, have wavefunctions antisymmetric under exchange. In our first part, we noted that in terms of the tensor product Hilbert space HH of the joint wavefunction, the two-particle wave functions for one particle in state a and one in state b is
for bosons, and
for fermions (the Cs are normalization constants).
What happens, then, if we put the two particles into the same state, here state a? Our boson wavefunction becomes
(assuming the single particle state |a\rangle is properly normalized), so that two (or more) bosons may be placed in the same quantum state.
However, our fermion wavefunction becomes
which means it is a state with zero probability; it is impossible for any two indistinguishable fermions to exist in the same quantum state simultaneously. This is the Pauli exclusion principle.


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