Pauli principle

The Pauli principle (also Pauli prohibition or Pauli exclusion principle ) is a physical law that has an effect in quantum physics . It was formulated in 1925 by Wolfgang Pauli for the quantum theoretical explanation of the structure of atoms and stated that every two electrons in an atom cannot match all quantum numbers . In the modern formulation, the Pauli principle states that the wave function of a quantum system is antisymmetric with regard to the exchange of identical fermions . There tooQuarks, as building blocks of protons and neutrons, belong to the fermions, the Pauli principle applies to all matter in the generally understood sense: Fermions "exclude each other", so they cannot exist at the same time in the same place (spacetime). This is the only way to understand the differentiated structure of matter with atoms and molecules. The Pauli principle therefore not only determines the structure of the atom , but also that of larger structures. One consequence is the resistance that condensed matter offers to further compression.

The Pauli principle should not be confused with the Pauli effect .

Simplified representation

In quantum mechanics, identical particles are indistinguishable . This means that the course of an experiment or, more generally, the development of a physical system does not change if two identical particles are exchanged. In quantum theory, the measured values ​​that a system generates depend on the absolute square of the total wave function of the system. This square of the absolute value must therefore remain the same after the exchange of two identical particles - which in this case means that only the phase component of the wave function may change due to the exchange . In a world with three spatial dimensions, this phase factor can only be or . Particles in which the wave function remains the same under exchange are called bosons , particles with a change in sign in the wave function are called fermions . The change in sign is called the antisymmetry of the wave function with respect to particle interchange. ${\displaystyle +1}$ ${\displaystyle -1}$

In its special and first observed form, the Pauli principle states that no two electrons in an atom match in all four quantum numbers, which are necessary for its state description in the orbital model . For example, if two electrons have the same major, minor and magnetic quantum numbers, they must differ in the fourth quantum number, in this case the spin quantum number. Since this can only take the values and , a maximum of two electrons can be in a single atomic orbital . This fact largely determines the structure of the chemical elements (see periodic table ). ${\displaystyle -{\tfrac {1}{2}}}$ ${\displaystyle +{\tfrac {1}{2}}}$

The solution of the Schrödinger equation for the simplest atom “affected” by the Pauli principle, the helium atom , can serve as a calculation example .

General form (generalized Pauli principle)

formulation

The total wave function of a system of identical fermions must total anti-symmetric with respect to each permutation P be two particles: ${\displaystyle \psi ({\vec {r}}_{1},s_{1};{\vec {r}}_{2},s_{2};\dots )}$ ${\displaystyle n}$

${\displaystyle \psi ({\vec {r}}_{1},s_{1};\ldots ;{\vec {r}}_{n},s_{n})=-(P\psi )({\vec {r}}_{1},s_{1};\ldots ;{\vec {r}}_{n},s_{n})\,\,.}$

It is the place the spin of th fermions and each permutation , the respective effects the interchange of two particles, that is z. B. for the exchange of the first particle with the second: ${\displaystyle {\vec {r}}_{i}}$ ${\displaystyle s_{i}}$ ${\displaystyle i}$ ${\displaystyle P}$

${\displaystyle (P\psi )({\vec {r}}_{1},s_{1};{\vec {r}}_{2},s_{2};\dots ):=\psi ({\vec {r}}_{2},s_{2};{\vec {r}}_{1},s_{1};\dots )\,\,.}$

Clear interpretation

If one considers a system of two indistinguishable fermions, the total wave function applies because of the antisymmetry

${\displaystyle \psi ({\vec {r}}_{1},s_{1};{\vec {r}}_{2},s_{2})=-\psi ({\vec {r}}_{2},s_{2};{\vec {r}}_{1},s_{1}).}$

For it follows that d. H. . Thus the square of the magnitude of this wave function, i.e. the probability density that one finds both fermions at the same place with the same spin , must be zero. ${\displaystyle ({\vec {r}}_{1},s_{1})=({\vec {r}}_{2},s_{2})}$ ${\displaystyle \psi ({\vec {r}}_{1},s_{1};{\vec {r}}_{1},s_{1})=-\psi ({\vec {r}}_{1},s_{1};{\vec {r}}_{1},s_{1})}$ ${\displaystyle \psi ({\vec {r}}_{1},s_{1};{\vec {r}}_{1},s_{1})=0}$ ${\displaystyle {\vec {r}}_{1}}$ ${\displaystyle s_{1}}$

In many cases (such a case is always given , for example, for nondegenerate eigenfunctions of Hamilton operators without spin-orbit coupling ) the total wave function can be represented as the product of the spatial wave function and the spin wave function ${\displaystyle \psi }$ ${\displaystyle \phi }$ ${\displaystyle \chi }$

${\displaystyle \psi ({\vec {r}}_{1},s_{1};{\vec {r}}_{2},s_{2})=\phi ({\vec {r}}_{1},{\vec {r}}_{2})\chi (s_{1},s_{2}).}$

Because of the antisymmetry then . If, for example, the spin wave function is symmetrical, that is , then the antisymmetry of the spatial wave function follows . Accordingly, it is generally true that the symmetry of one of the functions or is equivalent to the antisymmetry of the other. So if the two fermions are in about the same spin state , then is symmetrical and therefore the antisymmetry follows the spatial wave function. ${\displaystyle \phi ({\vec {r}}_{2},{\vec {r}}_{1})\chi (s_{2},s_{1})=-\phi ({\vec {r}}_{1},{\vec {r}}_{2})\chi (s_{1},s_{2})}$ ${\displaystyle \chi (s_{1},s_{2})=\chi (s_{2},s_{1})}$ ${\displaystyle \phi }$ ${\displaystyle \phi }$ ${\displaystyle \chi }$ ${\displaystyle s}$ ${\displaystyle \chi (s_{1},s_{2})=\delta _{ss_{1}}\delta _{ss_{2}}}$

These relationships also apply if more than two indistinguishable fermions are involved.

validity

In nature, fermions only occur with half- integer spin and bosons only with integer spin, as described by the spin statistics theorem . Pauli's principle of exclusion applies to all particles with half-integer spin and only to these.

For bosons, however, Paul’s exclusion principle does not apply . These particles meet the Bose-Einstein statistics and can assume the same quantum states, in extreme cases up to Bose-Einstein condensate .

Permutation and rotation behavior

The different permutation behavior of fermions and bosons matches the different rotation behavior of the respective spinors . In both cases there is a factor of , with the (+) sign for bosons ( integer) and the (-) sign for fermions ( half- integer ), corresponding to a rotation of 360 °. The connection is obvious, among other things, because an exchange of particles 1 and 2 corresponds to a complementary rotation of the two particles by 180 ° (for example particle 1 to location 2 on the upper semicircle, particle 2 to location 1 on the lower semicircle). ${\displaystyle (-1)^{2s}=\mp 1}$ ${\displaystyle s}$ ${\displaystyle s}$

Consequences

The Pauli principle leads to an exchange interaction and explains the spin order in atoms ( Hund's rules ) and solids ( magnetism ).

In astrophysics, the Pauli principle explains that old stars with the exception of black holes - for example white dwarfs or neutron stars - do not collapse under their own gravity. The fermions generate a counter pressure, the degenerative pressure , which counteracts a further contraction. This counter-pressure can be so strong that a supernova occurs.

In the case of scattering processes of two identical particles, the pair of trajectories is always exchanged in two different, but not distinguishable from the outside, possibilities. This must be taken into account in the theoretical calculation of the cross section and the scattered wave function .

Individual evidence

1. ^ Entry on Pauli principle. In: Römpp Online . Georg Thieme Verlag, accessed on December 28, 2014.
2. ^ Pauli principle in the Lexicon of Physics, Spektrum.de, accessed on December 28, 2014.
3. ^ Peter W. Atkins, Quantum Terms and Concepts for Chemists, VCH, ISBN 3-527-28423-0