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:

    “It may comfort you to know that if your child reaches the age of eleven or twelve and you have a good bond or relationship, no matter how dramatic adolescence becomes, you children will probably turn out all right and want some form of connection to you in adulthood.”
    —Charlotte Davis Kasl (20th century)

    “What is the vanity of the vainest man compared with the vanity which the most modest person possesses when, in connection with nature and the world, he experiences himself as “man”!”
    —Friedrich Nietzsche (1844–1900)

    “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)

    “From this elevation, just on the skirts of the clouds, we could overlook the country, west and south, for a hundred miles. There it was, the State of Maine, which we had seen on the map, but not much like that,—immeasurable forest for the sun to shine on, the eastern stuff we hear of in Massachusetts. No clearing, no house. It did not look as if a solitary traveler had cut so much as a walking-stick there.”
    —Henry David Thoreau (1817–1862)

    “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)