Editing Pauli exclusion principle (section)

== Connection to quantum state symmetry ==

In his Nobel lecture, Pauli clarified the importance of quantum state symmetry to the exclusion principle:<ref>{{cite web| url = https://www.nobelprize.org/uploads/2018/06/pauli-lecture.pdf| title = Wolfgang Pauli, Nobel lecture (December 13, 1946)}}</ref>
<blockquote>Among the different classes of symmetry, the most important ones (which moreover for two particles are the only ones) are the [[Boson|symmetrical class]], in which the wave function does not change its value when the space and spin coordinates of two particles are permuted, and the [[fermion|antisymmetrical class]], in which for such a permutation the wave function changes its sign...[The antisymmetrical class is] the correct and general wave mechanical formulation of the exclusion principle.</blockquote>

The Pauli exclusion principle with a single-valued many-particle wavefunction is equivalent to requiring the wavefunction to be [[Identical particles#Symmetrical and antisymmetrical states|antisymmetric with respect to exchange]]. If <math>|x\rangle</math> and <math>|y\rangle</math> range over the basis vectors of the [[Hilbert space]] describing a one-particle system, then the tensor product produces the basis vectors <math>|x,y\rangle=|x\rangle\otimes|y\rangle</math> of the Hilbert space describing a system of two such particles. Any two-particle state can be represented as a [[superposition principle|superposition]] (i.e. sum) of these basis vectors:
: <math>
|\psi\rangle = \sum_{x,y} A(x,y) |x,y\rangle,
</math>
where each {{nowrap|1=''A''(''x'', ''y'')}} is a (complex) scalar coefficient. Antisymmetry under exchange means that {{nowrap|1=''A''(''x'', ''y'') = −''A''(''y'', ''x'')}}. This implies {{nowrap|1=''A''(''x'', ''y'') = 0}} when {{nowrap|1=''x'' = ''y''}}, which is Pauli exclusion. It is true in any basis since local changes of basis keep antisymmetric matrices antisymmetric.

Conversely, if the diagonal quantities {{nowrap|1=''A''(''x'', ''x'')}} are zero ''in every basis'', then the wavefunction component
: <math>
A(x,y)=\langle\psi|x,y\rangle=\langle\psi|\Big(|x\rangle\otimes|y\rangle\Big)
</math>
is necessarily antisymmetric. To prove it, consider the matrix element
: <math>
\langle\psi| \Big((|x\rangle + |y\rangle)\otimes(|x\rangle + |y\rangle)\Big).
</math>

This is zero, because the two particles have zero probability to both be in the superposition state <math>|x\rangle + |y\rangle</math>. But this is equal to
: <math>
\langle \psi |x,x\rangle + \langle \psi |x,y\rangle + \langle \psi |y,x\rangle + \langle \psi | y,y \rangle.
</math>

The first and last terms are diagonal elements and are zero, and the whole sum is equal to zero. So the wavefunction matrix elements obey:
: <math>
\langle \psi|x,y\rangle + \langle\psi |y,x\rangle = 0,
</math>
or
: <math>
A(x,y) = -A(y,x).
</math>

For a system with {{nowrap|1=''n'' > 2}} particles, the multi-particle basis states become ''n''-fold tensor products of one-particle basis states, and the coefficients of the wavefunction <math>A(x_1,x_2,\ldots,x_n)</math> are identified by ''n'' one-particle states. The condition of antisymmetry states that the coefficients must flip sign whenever any two states are exchanged: <math>A(\ldots,x_i,\ldots,x_j,\ldots)=-A(\ldots,x_j,\ldots,x_i,\ldots)</math> for any <math>i\ne j</math>. The exclusion principle is the consequence that, if <math>x_i=x_j</math> for any <math>i\ne j,</math> then <math>A(\ldots,x_i,\ldots,x_j,\ldots)=0.</math> This shows that none of the ''n'' particles may be in the same state.

=== Advanced quantum theory ===
According to the [[spin–statistics theorem]], particles with integer spin occupy symmetric quantum states, and particles with half-integer spin occupy antisymmetric states; furthermore, only integer or half-integer values of spin are allowed by the principles of quantum mechanics.
In relativistic [[quantum field theory]], the Pauli principle follows from applying a [[Rotation operator (quantum mechanics)|rotation operator]] in [[imaginary time]] to particles of half-integer spin.

In one dimension, bosons, as well as fermions, can obey the exclusion principle. A one-dimensional Bose gas with delta-function repulsive interactions of infinite strength is equivalent to a gas of free fermions. The reason for this is that, in one dimension, the exchange of particles requires that they pass through each other; for infinitely strong repulsion this cannot happen. This model is described by a quantum [[nonlinear Schrödinger equation]]. In momentum space, the exclusion principle is valid also for finite repulsion in a Bose gas with delta-function interactions,<ref>{{Cite journal|url=http://insti.physics.sunysb.edu/~korepin/pauli.pdf|title=Pauli principle for one-dimensional bosons and the algebraic Bethe ansatz|author1=A. G. Izergin|author2=V. E. Korepin|journal=Letters in Mathematical Physics|volume=6|issue=4|pages=283–288|date=July 1982|doi=10.1007/BF00400323|bibcode=1982LMaPh...6..283I|s2cid=121829553|access-date=2009-12-02|archive-date=2018-11-25|archive-url=https://web.archive.org/web/20181125205409/http://insti.physics.sunysb.edu/~korepin/pauli.pdf|url-status=dead}}</ref> as well as for [[Heisenberg model (quantum)|interacting spins]] and [[Hubbard model]] in one dimension, and for other models solvable by [[Bethe ansatz]]. The [[Stationary state|ground state]] in models solvable by Bethe ansatz is a [[Fermi energy|Fermi sphere]].