Particle statistics

In quantum statistics the behavior of macroscopic systems is investigated using the methods of quantum mechanics. Similarly as in classical statistical physics, it is assumed that the system is in a state which is determined only by macroscopic quantities, but can be implemented by a large number of different, unspecified known, micro-states. However, the counting of the different possible microstates is amended so that the swapping of two identical particles produces no different microstate. Thus, the specific character of the indistinguishability of identical particles is taken into account.

Using the quantum statistics considering the following double ignorance:

  • 2.1 General
  • 2.2 associated with the rotational behavior of the wave function
  • 2.3 Statistics ideal quantum gases


If the system is in a state of the Hilbert space in front ( with the wave function ), it is called a pure state. In analogy to the classical ensemble usually overlays of different pure states are considered, the so-called mixed states ( semantically precise: State mixtures). These are described by the so-called density operator (also statistical operator state density matrix operator or given):

He describes, with which probabilities the system is in the individual pure states.

The overlay is incoherent. This is reflected in the fact that not the states themselves, but the associated projection operators are weighted by their probabilities.

One consequence is that methods in which coherence is required, such as quantum computing and quantum cryptography, easily within the framework of quantum statistics can not be described or will be complicated by thermodynamic effects.

Indistinguishable particles

For the quantum statistics of the existence of identical particles is important. These are quantum objects, which can be distinguished by any measurement; that is, the fundamental for quantum physics Hamiltonian of the system ( see, eg, Mathematical structure of quantum mechanics ) must be symmetric in the Teilchenvariablen, eg in the space and spin degrees of freedom of the individual particle. Thus, the many-body wave function must remain the same under the exchange of up to a factor of the amount 1, each operator commutes with a permutation P of the particles:

Since any permutation can be composed of transpositions and applies, it is useful to consider only totally symmetric () or totally antisymmetric () many-body states:


In other words, for symmetric many-body states of identical particles remains the sign of the total wave function for swapping any two particles in antisymmetric Vielteilchenzuständen it changes.

The experiment shows that nature actually realized only such states, which is indicated by the absence of exchange degeneracy. One refers to this fact as symmetrization.

Bosons and fermions


The probabilities with which a many-body system is distributed to its individual pure states, describes the Fermi -Dirac statistics for bosons, the Bose -Einstein statistics and for fermions.

In this case, bosons are particles with integer, fermions with half-integer spin, measured in units of the quantum of action. In addition, the wave function of bosons is symmetric and that of the fermions antisymmetric.

This linking of the particle spins with the symmetry of the wave function or the sign of the wave function in exchange of two particles is called the spin- statistics theorem. It has been proved by Wolfgang Pauli from general principles of relativistic quantum field theory.

In two dimensions, a phase factor in permutation imaginable, these particles are called anyons, but so far not observed. For anyons rational numbers for the spin may occur.

Examples of quantum statistical effects, ie effects, in which the Vertauschungseigenschaften the total wave function play a decisive role are:

  • For bosons Bose -Einstein condensation
  • Superconductivity ( Cooper pairs as bosons )
  • Superfluidity
  • Cavity blackbody radiation.
  • Heat capacity of solids
  • Band structure of metals and semiconductors,
  • Cohesion of white dwarfs and neutron stars against the self-gravity.

Related to the rotational behavior of the wave function

Also, the rotational behavior of the wave function is interesting in this context: at a spatial rotation of 360 °, the wave function for fermions only changes by 180 °:

While they are reproduced for bosons:

Such a 360 ° rotation of the exchange of two particles can be produced: particle 1 moves to site 2, for example, on the upper half of a circle, while particle 2 moves to the vacated place of one on the lower semi-circle line to to avoid meeting. The result of the Permutationsgleichung so watch for unusual rotational behavior of fermionic wave functions ( mathematical structure: see double group SU (2) to the ordinary rotation group SO (3) ).

Stats ideal quantum gases

For the derivation of the statistics ideal quantum gases, we consider a system in the grand canonical ensemble, that is, the system under consideration is coupled to a heat bath and a particle reservoir. The grand canonical partition function is then given by

Where Tr rutting, the Hamiltonian and the particle number is. The track is easiest to perform with common eigenstates of both operators. This fulfill the so-called Fockzustände. It is the occupation number of the -th eigenstate. Then the partition function writes as

The energy depends on the Gesamtteilchenanzahl and the occupation of the respective eigenstates. The -th eigenstate have the energy. Then that means one -fold occupation of the -th eigenstate of an energy contribution and total energy. Thus reads the partition function

The second sum runs over all possible occupation numbers ( for fermions or bosons ), their sum always gives the total number of particles. In addition, since summed over all Gesamtteilchenzahlen, one can combine the two sums by the restriction is lifted in the second sum:

The sum can be used for the two types of particles analyzed. For fermions one obtains

And for bosons

Which was required in the last step, the convergence of the geometric series. With knowledge of the grand canonical partition function can also be the grand potential

Specify. Thus, the thermodynamic quantities, entropy, pressure and particle number can be (or, respectively, the mean values) obtained:

Here we are interested for the mean occupation number of the -th state. Taking advantage of the relation with the Kronecker delta are obtained:

The results for fermions the Fermi - Dirac distribution

And for bosons, the Bose - Einstein distribution

Centralizing applications

The formalism takes into account both the thermodynamic and quantum mechanical phenomena.

The straight -treated difference between fermions and bosons is essential: for example, the quantized sound waves, called phonons, bosons, while the electrons are fermions. These two elementary excitations provide in solid bodies quite different contributions to the specific heat: the phonon has a characteristic temperature dependence while the electron contribution behaves, ie at sufficiently low temperatures in all solids, in which both excitations occur (eg metals ), always the dominant contribution is.

For this and similar problems, you can often also use methods of quantum field theory, such as Feynman diagrams. The theory of superconductivity can be treated so.