# Quantum statistics

The quantum statistics apply to study macroscopic systems, methods and concepts of classical statistical physics , and also takes into account the quantum mechanical peculiarities of the behavior of the particles. It assumes that the system is in a state that is only determined by macroscopic quantities, but can be realized by a large number of different, not known, micro-states . However, the counting of the various possible microstates is modified in such a way that exchanging two identical particles does not produce a different microstate. This is the special character of theIndistinguishability of identical particles taken into account. In addition, only the quantum-mechanically possible values ​​are permitted for the energies of the states of individual particles.

Like quantum mechanics, quantum statistics also take into account the following double ignorance:

1. If the state of a system is known exactly - if there is a pure state - and if this is not an intrinsic state of the observables , then the measured value of an individual measurement cannot be predicted exactly
2. If you do not know the exact state of the system, a mixed state must be assumed.

## Explanation

If the system is in a state that is given by a vector of the Hilbert space or by a wave function , one speaks of a pure state . In analogy to the classical ensemble usually be ensembles of various pure states considered the state mixture e (semantically less precise than mixed conditions referred). These are described by the density operator (also called statistical operator , state operator or density matrix ): ${\ displaystyle | \ psi \ rangle}$ ${\ displaystyle \ psi)}$ ${\ displaystyle | \ psi _ {n} \ rangle}$ ${\ displaystyle \ rho = \ sum _ {n} \; \; p_ {n} | \ psi _ {n} \ rangle \ langle \ psi _ {n} |}$ .

It describes the real probabilities with which the system is in the individual pure states. ${\ displaystyle p_ {n} \! \,}$ The overlay is incoherent . This is expressed in the fact that the density operator is independent of phase relationships between the states . Any complex phase factors that would have an effect in the case of a coherent superposition stand out in the projection operators . ${\ displaystyle | \ psi _ {n} \ rangle}$ ${\ displaystyle {\ hat {P}} _ {n} = | \ psi _ {n} \ rangle \ langle \ psi _ {n} |}$ One consequence is that operations where coherence is important, e.g. B. quantum computing or quantum cryptography , cannot easily be described in the context of quantum statistics or are made more difficult by thermodynamic effects.

### Indistinguishable Particles

The existence of identical particles is important for quantum statistics . These are quantum objects that cannot be distinguished by any measurement ; d. That is, the Hamilton operator of the system , which is fundamental for quantum physics (see e.g. Mathematical Structure of Quantum Mechanics ), must be symmetrical in the particle variables, e.g. B. in the spatial and spin degrees of freedom of the individual particle . The many-body wave function has to remain the same with interchangeability up to a factor of magnitude 1, every operator of an observable commutes with every permutation of the identical particles: ${\ displaystyle \ psi (1,2, \ ldots, N)}$ ${\ displaystyle A}$ ${\ displaystyle P}$ ${\ displaystyle \ left [A, P \ right] = 0}$ Since every permutation can be composed of transpositions and is valid, it makes sense to only consider totally symmetric ( ) or totally antisymmetric ( ) many-body states: ${\ displaystyle \ tau _ {ij}}$ ${\ displaystyle \ tau _ {ij} ^ {2} = 1}$ ${\ displaystyle \ tau _ {ij} = 1}$ ${\ displaystyle \ tau _ {ij} = - 1}$ ${\ displaystyle \ tau _ {ij} | \ ldots, i, \ ldots, j \ ldots \ rangle = | \ ldots, j, \ ldots, i, \ ldots \ rangle = \ pm | \ ldots, i, \ ldots , j, \ ldots \ rangle}$ .

In other words: for symmetric many-particle states of identical particles, the sign of the total wave function is retained when two arbitrary particles are interchanged ; for antisymmetric many-particle states it changes.

The experiment shows that nature actually only realizes such states, which can be seen from the lack of exchange degeneracy. This fact is also called the symmetrization postulate .

## Bosons and fermions

### General

The probabilities with which a many-body system is distributed to its individual pure states is described by the Bose-Einstein statistics for bosons and the Fermi-Dirac statistics for fermions . ${\ displaystyle p \! \,}$ Bosons are particles with integer spin , fermions with half- integer spin, each measured in units of with the quantum of action . In addition, the wave function of the bosons is symmetric and that of the fermions is antisymmetric . ${\ displaystyle \ hbar = h / (2 \ pi)}$ ${\ displaystyle h \! \,}$ This connection of the particle spin with the symmetry of the wave function or the sign of the wave function when two particles are interchanged is called the spin statistics theorem . It was proved by Wolfgang Pauli from general principles of relativistic quantum field theory .

In two dimensions, a phase factor is also conceivable when interchanged; these particles are called anyons , but have not yet been observed. Anyone can have rational numbers for the spin. ${\ displaystyle e ^ {i \ phi}}$ Examples of quantum statistical effects, i. H. Effects in which the commutation properties of the overall wave function play a decisive role are:

### Connection with the rotation behavior of the wave function

The rotation behavior of the wave function is also interesting in this context: with a spatial rotation of 360 °, the wave function for fermions only changes by 180 °: ${\ displaystyle \ psi (1,2, \ ldots, N)}$ ${\ displaystyle \ psi (1,2, \ ldots, N) \ mapsto e ^ {i2 \ pi} \ psi (1,2, \ ldots, N) = - \ psi (1,2, \ ldots, N) }$ ,

while it reproduces for bosons:

${\ displaystyle \ psi (1,2, \ ldots, N) \ mapsto e ^ {i2 \ pi} \ psi (1,2, \ ldots, N) = + \ psi (1,2, \ ldots, N) }$ .

Such a 360 ° rotation allows two particles to be interchanged: Particle 1 moves to location 2, e.g. B. on the upper half of a circular line, while particle 2 moves to the empty location of 1 on the lower semicircular line in order to avoid a meeting. The result of the permutation equation fits the unusual rotation behavior of fermionic wave functions (mathematical structure: see double group SU (2) for the usual rotation group SO (3)).

### Statistics of ideal quantum gases

To derive the statistics of ideal quantum gases we consider a system in the grand canonical ensemble , i.e. H. the system under consideration is coupled to a heat bath and a particle reservoir. The grand-canonical partition function is then given by

${\ displaystyle Z_ {G} = \ mathrm {Tr} \, e ^ {- \ beta ({\ hat {H}} - \ mu {\ hat {N}})} \ ,,}$ where is the tracking , the Hamilton operator and the particle number operator . The easiest way to execute the trace with common eigen-states for both operators. This is what the so-called jib conditions meet . Here is the occupation number of the -th eigenstate. Then the state sum is written as ${\ displaystyle \ mathrm {Tr}}$ ${\ displaystyle {\ hat {H}}}$ ${\ displaystyle {\ hat {N}}}$ ${\ displaystyle | n_ {1}, n_ {2}, \ ldots, n _ {\ nu}, \ ldots \ rangle}$ ${\ displaystyle n _ {\ nu}}$ ${\ displaystyle \ nu}$ ${\ displaystyle Z_ {G} = \ sum _ {N} \ sum _ {E} e ^ {- \ beta (E- \ mu N)} \,}$ The energy depends on the total number of particles and the occupation of the respective eigenstates. The -th eigenstate has energy . Then a -fold occupation of the -th eigenstate means an energy contribution from and total energy from . Thus the state sum is ${\ displaystyle E}$ ${\ displaystyle N = \ Sigma _ {\ nu} n _ {\ nu}}$ ${\ displaystyle \ nu}$ ${\ displaystyle \ varepsilon _ {\ nu}}$ ${\ displaystyle n _ {\ nu}}$ ${\ displaystyle \ nu}$ ${\ displaystyle E _ {\ nu} = n _ {\ nu} \ varepsilon _ {\ nu}}$ ${\ displaystyle E}$ ${\ displaystyle E_ {N} = \ Sigma _ {\ nu} E _ {\ nu}}$ {\ displaystyle {\ begin {aligned} Z_ {G} & = \ sum _ {N = 0} ^ {\ infty} \ sum _ {\ {n _ {\ nu} \ in I | \ Sigma _ {\ nu} n _ {\ nu} = N \}} e ^ {- \ beta \ sum _ {\ nu} (\ varepsilon _ {\ nu} - \ mu) n _ {\ nu}} \, \\ & = \ sum _ {N = 0} ^ {\ infty} \ sum _ {\ {n _ {\ nu} \ in I | \ Sigma _ {\ nu} n _ {\ nu} = N \}} \ prod _ {\ nu} e ^ {- \ beta (\ varepsilon _ {\ nu} - \ mu) n _ {\ nu}} \, \ end {aligned}}} The second sum runs over all possible occupation numbers ( for fermions or for bosons), the sum of which always gives the total number of particles . Since all total numbers of particles are added together, both sums can be summarized by removing the restriction in the second sum: ${\ displaystyle n _ {\ nu} \ in I}$ ${\ displaystyle I = \ {0.1 \}}$ ${\ displaystyle I = \ mathbb {N} _ {0}}$ ${\ displaystyle N}$ ${\ displaystyle N}$ ${\ displaystyle Z_ {G} = \ prod _ {\ nu} \ sum _ {n _ {\ nu} \ in I} e ^ {- \ beta (\ varepsilon _ {\ nu} - \ mu) n _ {\ nu }} \,}$ The sum can be evaluated for the two types of particles. For fermions one gets

${\ displaystyle Z_ {G} = \ prod _ {\ nu} \ sum _ {n _ {\ nu} = 0} ^ {1} e ^ {- \ beta (\ varepsilon _ {\ nu} - \ mu) n_ {\ nu}} = \ prod _ {\ nu} \ left (1 + e ^ {- \ beta (\ varepsilon _ {\ nu} - \ mu)} \ right) \,}$ and for bosons

{\ displaystyle {\ begin {aligned} Z_ {G} & = \ prod _ {\ nu} \ sum _ {n _ {\ nu} = 0} ^ {\ infty} e ^ {- \ beta (\ varepsilon _ { \ nu} - \ mu) n _ {\ nu}} = \ prod _ {\ nu} \ sum _ {n _ {\ nu} = 0} ^ {\ infty} \ left (e ^ {- \ beta (\ varepsilon _ {\ nu} - \ mu)} \ right) ^ {n _ {\ nu}} \\ & = \ prod _ {\ nu} {\ frac {1} {1-e ^ {- \ beta (\ varepsilon _ {\ nu} - \ mu)}}} \ quad \ mathrm {if} \ quad \ beta (\ varepsilon _ {\ nu} - \ mu)> 0 \ end {aligned}} \,} In the last step, the convergence of the geometric series was required. With knowledge of the grand canonical partition function, the grand canonical potential

${\ displaystyle \ Phi (T, V, \ mu) \ equiv - {\ frac {1} {\ beta}} \ ln Z_ {G}}$ specify. The thermodynamic quantities entropy , pressure and number of particles (or the mean quantities in each case) can thus be obtained: ${\ displaystyle S}$ ${\ displaystyle p}$ ${\ displaystyle N}$ ${\ displaystyle {\ begin {pmatrix} S \\ p \\ N \ end {pmatrix}} = - {\ begin {pmatrix} \ partial _ {T} \\\ partial _ {V} \\\ partial _ { \ mu} \ end {pmatrix}} \ Phi (T, V, \ mu) \,}$ We are interested here in the mean occupation number of the -th state. Using the relation with the Kronecker delta one obtains: ${\ displaystyle \ langle n_ {j} \ rangle}$ ${\ displaystyle j}$ ${\ displaystyle \ partial \ varepsilon _ {\ nu} / \ partial \ varepsilon _ {j} = \ delta _ {\ nu, j}}$ ${\ displaystyle \ delta _ {\ nu, j}}$ {\ displaystyle {\ begin {aligned} \ langle n_ {j} \ rangle & = {\ frac {1} {Z_ {G}}} \ prod _ {\ nu} \ sum _ {n _ {\ nu} \ in I} n_ {j} e ^ {- \ beta (\ varepsilon _ {\ nu} - \ mu) n _ {\ nu}} \\ & = {\ frac {1} {Z_ {G}}} \ left ( - {\ frac {1} {\ beta}} {\ frac {\ partial} {\ partial \ varepsilon _ {j}}} \ right) \ underbrace {\ prod _ {\ nu} \ sum _ {n _ {\ nu} \ in I} e ^ {- \ beta (\ varepsilon _ {\ nu} - \ mu) n _ {\ nu}}} _ {= Z_ {G}} \\ & = - {\ frac {1} {\ beta}} {\ frac {\ partial} {\ partial \ varepsilon _ {j}}} \ ln Z_ {G} \ end {aligned}} \,} This gives the Fermi-Dirac distribution for fermions

{\ displaystyle {\ begin {aligned} \ langle n_ {j} \ rangle & = - {\ frac {1} {\ beta}} {\ frac {\ partial} {\ partial \ varepsilon _ {j}}} \ sum _ {\ nu} \ ln \ left (1 + e ^ {- \ beta (\ varepsilon _ {\ nu} - \ mu)} \ right) = {\ frac {e ^ {- \ beta (\ varepsilon _ {j} - \ mu)}} {1 + e ^ {- \ beta (\ varepsilon _ {j} - \ mu)}}} \\ & = {\ frac {1} {e ^ {\ beta (\ varepsilon _ {j} - \ mu)} + 1}} \ end {aligned}} \,} and for bosons the Bose-Einstein distribution

{\ displaystyle {\ begin {aligned} \ langle n_ {j} \ rangle & = - {\ frac {1} {\ beta}} {\ frac {\ partial} {\ partial \ varepsilon _ {j}}} \ sum _ {\ nu} \ ln {\ frac {1} {1-e ^ {- \ beta (\ varepsilon _ {\ nu} - \ mu)}}} = {\ frac {e ^ {- \ beta ( \ varepsilon _ {j} - \ mu)}} {1-e ^ {- \ beta (\ varepsilon _ {j} - \ mu)}}} \\ & = {\ frac {1} {e ^ {\ beta (\ varepsilon _ {j} - \ mu)} - 1}} \ end {aligned}} \,} ## Central applications

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

The difference between fermions and bosons just discussed is essential: B. the quantized sound waves, the so-called phonons , bosons , while the electrons are fermions . The elementary excitations in question make very different contributions to the specific heat in solid bodies : the phonon contribution has a characteristic temperature dependency while the electron contribution behaves, i.e. at sufficiently low temperatures in all solids in which both excitations occur (e.g. in metals) the dominant contribution is. ${\ displaystyle \ propto T ^ {3} \ ,,}$ ${\ displaystyle \ propto T ^ {1}}$ For these and similar problems one can often use methods of quantum field theory, e.g. B. Feynman diagrams . The theory of superconductivity can also be treated in this way.

2. Because of the conservation of probability, which is expressed by.${\ displaystyle | \ psi | ^ {2}}$ 