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{{Quantum book backlink|Statistical mechanics and kinetic theory}}
'''Quantum partition function''' is the central quantity of [[Physics:Quantum Statistical mechanics|quantum statistical mechanics]], encoding how the energy eigenstates of a quantum system are thermally populated at temperature <math>T</math>.<ref name="Pathria">{{cite book
|last1=Pathria
|first1=R. K.
|last2=Beale
|first2=Paul D.
|title=Statistical Mechanics
|edition=3
|publisher=Elsevier
|year=2011
|isbn=9780123821881
}}</ref><ref name="Tolman">{{cite book
|last=Tolman
|first=Richard C.
|title=The Principles of Statistical Mechanics
|publisher=Dover Publications
|year=1979
|isbn=9780486638966
}}</ref> It provides the bridge between microscopic quantum states and macroscopic thermodynamic quantities such as internal energy, entropy, Helmholtz free energy, and heat capacity.<ref name="LandauLifshitz">{{cite book
|last1=Landau
|first1=L. D.
|last2=Lifshitz
|first2=E. M.
|title=Statistical Physics, Part 1
|edition=3
|publisher=Butterworth-Heinemann
|year=1980
|isbn=9780750633727
}}</ref>
[[File:Partition_function_visualization.jpg|thumb|400px|Partition function as a weighted sum over all quantum states, where each state's contribution depends exponentially on its energy.]]
==Definition==
For a quantum system in thermal equilibrium with Hamiltonian <math>H</math>, the canonical partition function is defined by

<math>
Z = \mathrm{Tr}\left(e^{-\beta H}\right),
</math>

where

<math>
\beta = \frac{1}{k_B T}.
</math>

Here <math>k_B</math> is Boltzmann’s constant, <math>T</math> is the absolute temperature, and the trace is taken over the system’s Hilbert space.<ref name="Pathria"/><ref name="Reif">{{cite book
|last=Reif
|first=Frederick
|title=Fundamentals of Statistical and Thermal Physics
|publisher=Waveland Press
|year=2009
|isbn=9781577666127
}}</ref>

If the Hamiltonian has discrete energy eigenstates <math>|n\rangle</math> with eigenvalues <math>E_n</math>, then the trace becomes

<math>
Z = \sum_n e^{-\beta E_n}.
</math>

This shows that each energy level contributes with a Boltzmann weight determined by its energy.<ref name="Tolman"/><ref name="Pathria"/>

==Physical meaning==
The partition function summarizes the thermal accessibility of all possible quantum states of the system.<ref name="LandauLifshitz"/> Low-energy states contribute most strongly at low temperature, while many higher-energy states become thermally populated as the temperature increases.<ref name="Reif"/>

Once <math>Z</math> is known, the equilibrium probability of finding the system in eigenstate <math>|n\rangle</math> is

<math>
P_n = \frac{e^{-\beta E_n}}{Z}.
</math>

Thus the partition function serves as the normalization factor for the canonical ensemble.<ref name="Pathria"/>

==Density operator==
In quantum statistical mechanics, the canonical ensemble is represented by the density operator

<math>
\rho = \frac{e^{-\beta H}}{Z}.
</math>

This operator gives expectation values of observables through

<math>
\langle A\rangle = \mathrm{Tr}(\rho A).
</math>

The partition function therefore appears directly in the normalization of the thermal density matrix.<ref name="LandauLifshitz"/><ref name="Pathria"/>

==Thermodynamic relations==
The quantum partition function generates the main thermodynamic quantities of the canonical ensemble.<ref name="Tolman"/><ref name="Reif"/>

The Helmholtz free energy is

<math>
F = -k_B T \ln Z.
</math>

The internal energy is

<math>
U = -\frac{\partial}{\partial \beta}\ln Z.
</math>

The entropy can be written as

<math>
S = -\left(\frac{\partial F}{\partial T}\right)_V,
</math>

and the heat capacity at constant volume is

<math>
C_V = \left(\frac{\partial U}{\partial T}\right)_V.
</math>

These relations show that all equilibrium thermodynamics can be derived from <math>Z</math>.<ref name="Pathria"/><ref name="LandauLifshitz"/>

==Role of degeneracy==
If an energy level <math>E_n</math> has degeneracy <math>g_n</math>, then the partition function becomes

<math>
Z = \sum_n g_n e^{-\beta E_n}.
</math>

Degeneracy increases the statistical weight of a level and can significantly affect thermodynamic behavior, especially at low temperatures where only a few low-lying states contribute appreciably.<ref name="Reif"/>

==Simple examples==
===Two-level system===
For a system with two energy levels <math>E_0</math> and <math>E_1</math>, the partition function is

<math>
Z = e^{-\beta E_0} + e^{-\beta E_1}.
</math>

This model is useful for spin systems, qubits, and other simple quantum systems.<ref name="Pathria"/>

===Quantum harmonic oscillator===
For a one-dimensional quantum harmonic oscillator with energies

<math>
E_n = \hbar\omega\left(n+\frac{1}{2}\right),
\qquad n=0,1,2,\dots,
</math>

the partition function is

<math>
Z = \sum_{n=0}^{\infty} e^{-\beta \hbar\omega(n+1/2)}
= \frac{e^{-\beta\hbar\omega/2}}{1-e^{-\beta\hbar\omega}}.
</math>

This is one of the standard exactly solvable examples in quantum statistical mechanics.<ref name="Pathria"/><ref name="Reif"/>

===Ideal quantum gas===
For systems of many identical particles, the partition function must reflect indistinguishability and quantum statistics.<ref name="LandauLifshitz"/> In that case one passes naturally to the grand canonical formalism and to Bose-Einstein or Fermi-Dirac occupation factors.<ref name="Pathria"/>

==Relation to the classical partition function==
The quantum partition function is the direct analogue of the classical partition function, but with the trace over Hilbert space replacing the phase-space integral.<ref name="Tolman"/><ref name="LandauLifshitz"/> In the semiclassical limit, where quantum level spacings become very small compared with <math>k_B T</math>, the quantum description approaches the classical one.<ref name="Reif"/>

This connection explains why statistical mechanics can often be formulated in a unified way, with quantum theory providing the more fundamental description.

==Connection with imaginary time==
In more advanced formulations, the partition function may be written as an imaginary-time evolution operator over a period <math>\beta\hbar</math>:

<math>
Z = \mathrm{Tr}\left(e^{-\beta H}\right).
</math>

This relation underlies the path-integral formulation of quantum statistical mechanics and connects thermal field theory with quantum dynamics in imaginary time.<ref name="FeynmanHibbs">{{cite book
|last1=Feynman
|first1=Richard P.
|last2=Hibbs
|first2=Albert R.
|title=Quantum Mechanics and Path Integrals
|publisher=McGraw-Hill
|year=1965
|isbn=9780486477220
}}</ref><ref name="NegeleOrland">{{cite book
|last1=Negele
|first1=John W.
|last2=Orland
|first2=Henri
|title=Quantum Many-Particle Systems
|publisher=Westview Press
|year=1998
|isbn=9780738200521
}}</ref>

==Grand partition function==
When both energy and particle number may fluctuate, the appropriate quantity is the grand partition function

<math>
\mathcal{Z} = \mathrm{Tr}\left(e^{-\beta(H-\mu N)}\right),
</math>

where <math>\mu</math> is the chemical potential and <math>N</math> is the particle-number operator.<ref name="Pathria"/><ref name="LandauLifshitz"/> This form is essential for describing quantum gases, photons, phonons, and many-body systems in contact with both a heat bath and a particle reservoir.

==Physical interpretation==
The quantum partition function explains how:

* discrete energy spectra determine thermal populations
* microscopic energy levels generate macroscopic thermodynamics
* degeneracy modifies statistical weights
* quantum statistics enters many-body equilibrium theory<ref name="Pathria"/><ref name="LandauLifshitz"/>

It is therefore one of the foundational objects of both [[Physics:Quantum Statistical mechanics|quantum statistical mechanics]] and modern many-body physics.

==Applications==
Quantum partition functions are used in:

* two-level spin systems
* harmonic oscillators and lattice vibrations
* ideal Bose and Fermi gases
* magnetic systems
* quantum field theory at finite temperature
* condensed-matter and many-body physics<ref name="Pathria"/><ref name="NegeleOrland"/>

==See also==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

==References==
{{reflist|3}}
{{Author|Harold Foppele}}
[[Category:Quantum mechanics]]
[[Category:Statistical mechanics]]

{{Sourceattribution|Quantum partition function|1}}

Physics:Quantum Partition function - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

imported>WikiHarold: Repair Quantum Collection B backlink template