Pauli Exclusion Principle - Connection To Quantum State Symmetry

Connection To Quantum State Symmetry

The Pauli exclusion principle with a single-valued many-particle wavefunction is equivalent to requiring the wavefunction to be antisymmetric. An antisymmetric two-particle state is represented as a sum of states in which one particle is in state and the other in state :

|\psi\rangle = \sum_{x,y} A(x,y) |x,y\rangle

and antisymmetry under exchange means that A(x,y) = −A(y,x). This implies that A(x,x) = 0, which is Pauli exclusion. It is true in any basis, since unitary changes of basis keep antisymmetric matrices antisymmetric, although strictly speaking, the quantity A(x,y) is not a matrix but an antisymmetric rank-two tensor.

Conversely, if the diagonal quantities A(x,x) are zero in every basis, then the wavefunction component:

A(x,y)=\langle \psi|x,y\rangle = \langle \psi | ( |x\rangle \otimes |y\rangle )

is necessarily antisymmetric. To prove it, consider the matrix element:

\langle\psi| ((|x\rangle + |y\rangle)\otimes(|x\rangle + |y\rangle)) \,

This is zero, because the two particles have zero probability to both be in the superposition state . But this is equal to

\langle \psi |x,x\rangle + \langle \psi |x,y\rangle + \langle \psi |y,x\rangle + \langle \psi | y,y \rangle \,

The first and last terms on the right hand side are diagonal elements and are zero, and the whole sum is equal to zero. So the wavefunction matrix elements obey:

\langle \psi|x,y\rangle + \langle\psi |y,x\rangle = 0 \,.

or

A(x,y)=-A(y,x) \,

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