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) \,

Read more about this topic:  Pauli Exclusion Principle

Famous quotes containing the words connection to, connection, quantum, state and/or symmetry:

    “One must always maintain one’s connection to the past and yet ceaselessly pull away from it. To remain in touch with the past requires a love of memory. To remain in touch with the past requires a constant imaginative effort.”
    —Gaston Bachelard (1884–1962)

    “The connection between dress and war is not far to seek; your finest clothes are those you wear as soldiers.”
    —Virginia Woolf (1882–1941)

    “But how is one to make a scientist understand that there is something unalterably deranged about differential calculus, quantum theory, or the obscene and so inanely liturgical ordeals of the precession of the equinoxes.”
    —Antonin Artaud (1896–1948)

    “Death is not natural for a state as it is for a human being, for whom death is not only necessary, but frequently even desirable.”
    —Marcus Tullius Cicero (106–43 B.C.)

    “What makes a regiment of soldiers a more noble object of view than the same mass of mob? Their arms, their dresses, their banners, and the art and artificial symmetry of their position and movements.”
    —George Gordon Noel Byron (1788–1824)