Physics:Quantum Distribution functions

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Quantum distribution functions describe the average occupation of energy states in a many-particle system at thermal equilibrium. They distinguish classical from quantum statistical behavior.

For a state of energy E, the occupation depends on particle type.[1]

Comparison of Maxwell–Boltzmann, Bose–Einstein, and Fermi–Dirac distribution functions showing how equilibrium occupation depends on energy.

Maxwell–Boltzmann distribution

In the classical limit:

f(E)=e−β(E−μ)[2]

Valid when quantum degeneracy is negligible.[2]

Bose–Einstein distribution

For bosons:

f(E)=1eβ(E−μ)−1[1]

Bosons can accumulate in low-energy states, leading to Bose–Einstein condensation.[3]

Fermi–Dirac distribution

For fermions:

f(E)=1eβ(E−μ)+1[1]

The Pauli exclusion principle limits occupation to one particle per state.[4]

At low temperature, the distribution approaches a step function at the Fermi energy.

Classical limit

When eβ(E−μ)≫1, both quantum distributions reduce to:

f(E)≈e−β(E−μ)[2]

Chemical potential

The chemical potential μ controls particle number.

  • For fermions: μ→EF at low temperature
  • For bosons: μ≤E0

These constraints determine quantum gas behavior.[1]

Physical interpretation

The three distributions reflect different statistics:

  • Maxwell–Boltzmann → classical limit
  • Bose–Einstein → state clustering
  • Fermi–Dirac → exclusion principle

These differences produce distinct macroscopic phenomena.[1]

Applications

Quantum distribution functions are essential in:

  • classical gases and kinetic theory[2]
  • electron behavior in solids[4]
  • photons and phonons[3]
  • quantum many-body systems[1]

See also

Table of contents (138 articles)

Index

Full contents

1. Foundations (7)
  1. Physics:Quantum basics
  2. Physics:Quantum Postulates
  3. Physics:Quantum Hilbert space
  4. Physics:Quantum Observables and operators
  5. Physics:Quantum mechanics
  6. Physics:Quantum mechanics measurements
  7. Physics:Quantum Mathematical Foundations of Quantum Theory

2. Conceptual and interpretations (10)
  1. Physics:Quantum Interpretations of quantum mechanics
  2. Physics:Quantum Wave–particle duality
  3. Physics:Quantum Complementarity principle
  4. Physics:Quantum Uncertainty principle
  5. Physics:Quantum Measurement problem
  6. Physics:Quantum Bell's theorem
  7. Physics:Quantum Hidden variable theory
  8. Physics:Quantum A Spooky Action at a Distance
  9. Physics:Quantum A Walk Through the Universe
  10. Physics:Quantum The Secret of Cohesion and How Waves Hold Matter Together

3. Mathematical structure and systems (11)
  1. Physics:Quantum Density matrix
  2. Physics:Quantum Exactly solvable quantum systems
  3. Physics:Quantum Formulas Collection
  4. Physics:Quantum A Matter Of Size
  5. Physics:Quantum Symmetry in quantum mechanics
  6. Physics:Quantum Angular momentum operator
  7. Physics:Quantum Runge–Lenz vector
  8. Physics:Quantum Approximation Methods
  9. Physics:Quantum Matter Elements and Particles
  10. Physics:Quantum Dirac equation
  11. Physics:Quantum Klein–Gordon equation

4. Atomic and spectroscopy (11)
    Quantum atomic structure and spectroscopy: orbitals, energy levels, and emission and absorption spectra.
  1. Physics:Quantum Atomic structure and spectroscopy
  2. Physics:Quantum Hydrogen atom
  3. Physics:Quantum Multi-electron atoms
  4. Physics:Quantum Fine structure
  5. Physics:Quantum Hyperfine structure
  6. Physics:Quantum Isotopic shift
  7. Physics:Quantum Zeeman effect
  8. Physics:Quantum Stark effect
  9. Physics:Quantum Spectral lines and series
  10. Physics:Quantum Selection rules
  11. Physics:Quantum Fermi's golden rule

5. Wavefunctions and modes (8)
    A quantum wavefunction showing probability amplitude in space; the square of its magnitude gives the probability density.
  1. Physics:Quantum Wavefunction
  2. Physics:Quantum Superposition principle
  3. Physics:Quantum Eigenstates and eigenvalues
  4. Physics:Quantum Boundary conditions and quantization
  5. Physics:Quantum Standing waves and modes
  6. Physics:Quantum Normal modes and field quantization
  7. Physics:Number of independent spatial modes in a spherical volume
  8. Physics:Quantum Density of states

6. Quantum dynamics and evolution (9)
  1. Physics:Quantum Time evolution
  2. Physics:Quantum Schrödinger equation
  3. Physics:Quantum Time-dependent Schrödinger equation
  4. Physics:Quantum Stationary states
  5. Physics:Quantum Perturbation theory
  6. Physics:Quantum Time-dependent perturbation theory
  7. Physics:Quantum Adiabatic theorem
  8. Physics:Quantum Scattering theory
  9. Physics:Quantum S-matrix

7. Measurement and information (6)
  1. Physics:Quantum Measurement theory
  2. Physics:Quantum Measurement operators
  3. Physics:Quantum Projective measurement
  4. Physics:Quantum POVM
  5. Physics:Quantum Weak measurement
  6. Physics:Quantum Measurement collapse

8. Quantum information and computing (6)
  1. Physics:Quantum information theory
  2. Physics:Quantum Qubit
  3. Physics:Quantum Entanglement
  4. Physics:Quantum Gates and circuits
  5. Physics:Quantum Computing Algorithms in the NISQ Era
  6. Physics:Quantum Noisy Qubits

9. Quantum optics and experiments (4)
    Experimental quantum physics: qubits, dilution refrigerators, quantum communication, and laboratory systems.
  1. Physics:Quantum Nonlinear King plot anomaly in calcium isotope spectroscopy
  2. Physics:Quantum optics beam splitter experiments
  3. Physics:Quantum Ultra fast lasers
  4. Physics:Quantum Experimental quantum physics
    5. Template:Quantum optics operators

10. Open quantum systems (7)
  1. Physics:Quantum Open systems
  2. Physics:Quantum Master equation
  3. Physics:Quantum Lindblad equation
  4. Physics:Quantum Decoherence
  5. Physics:Quantum Markovian dynamics
  6. Physics:Quantum Non-Markovian dynamics
  7. Physics:Quantum Trajectories

11. Quantum field theory (18)
    Structural dependency map of quantum field theory.
  1. Physics:Quantum field theory (QFT) basics
  2. Physics:Quantum field theory (QFT) core
  3. Physics:Quantum Fields and Particles
  4. Physics:Quantum Second quantization
  5. Physics:Quantum Harmonic Oscillator field modes
  6. Physics:Quantum Creation and annihilation operators
  7. Physics:Quantum vacuum fluctuations
  8. Physics:Quantum Propagators in quantum field theory
  9. Physics:Quantum Feynman diagrams
  10. Physics:Quantum Path integral formulation
  11. Physics:Quantum Renormalization in field theory
  12. Physics:Quantum Renormalization group
  13. Physics:Quantum Field Theory Gauge symmetry
  14. Physics:Quantum Non-Abelian gauge theory
  15. Physics:Quantum Electrodynamics (QED)
  16. Physics:Quantum chromodynamics (QCD)
  17. Physics:Quantum Electroweak theory
  18. Physics:Quantum Standard Model

12. Statistical mechanics and kinetic theory (10)
  1. Physics:Quantum Statistical mechanics
  2. Physics:Quantum Partition function
  3. Physics:Quantum Distribution functions
  4. Physics:Quantum Liouville equation
  5. Physics:Quantum Kinetic theory
  6. Physics:Quantum Boltzmann equation
  7. Physics:Quantum BBGKY hierarchy
  8. Physics:Quantum Transport theory
  9. Physics:Quantum Relaxation and thermalization
  10. Physics:Quantum Thermodynamics

13. Condensed matter and solid-state physics (7)
  1. Physics:Quantum Band structure
  2. Physics:Quantum Fermi surfaces
  3. Physics:Quantum Semiconductor physics
  4. Physics:Quantum Phonons
  5. Physics:Quantum Electron-phonon interaction
  6. Physics:Quantum Superconductivity
  7. Physics:Quantum Topological phases of matter

14. Plasma and fusion physics (9)
    Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.
  1. Physics:Quantum Plasma (fusion context)
  2. Physics:Quantum Fusion reactions and Lawson criterion
  3. Physics:Quantum Magnetic confinement fusion
  4. Physics:Quantum Inertial confinement fusion
  5. Physics:Quantum Plasma instabilities and turbulence
  6. Physics:Quantum Tokamak
  7. Physics:Quantum Tokamak core plasma
  8. Physics:Quantum Tokamak edge physics and recycling asymmetries
  9. Physics:Quantum Stellarator

15. Timeline (8)
  1. Physics:Quantum mechanics/Timeline
  2. Physics:Quantum mechanics/Timeline/Pre-quantum era
  3. Physics:Quantum mechanics/Timeline/Old quantum theory
  4. Physics:Quantum mechanics/Timeline/Modern quantum mechanics
  5. Physics:Quantum mechanics/Timeline/Quantum field theory era
  6. Physics:Quantum mechanics/Timeline/Quantum information era
  7. Physics:Quantum mechanics/Timeline/Quantum technology era
  8. Physics:Quantum mechanics/Timeline/Quiz/

16. Advanced and frontier topics (6)
  1. Physics:Quantum Supersymmetry
  2. Physics:Quantum Black hole thermodynamics
  3. Physics:Quantum Holographic principle
  4. Physics:Quantum gravity
  5. Physics:Quantum De Sitter invariant special relativity
  6. Physics:Quantum Doubly special relativity

References

  1. ↑ 1.0 1.1 1.2 1.3 1.4 1.5 https://www.damtp.cam.ac.uk/user/tong/statphys/statmechhtml/S3.html
  2. ↑ 2.0 2.1 2.2 2.3 https://ocw.mit.edu/courses/5-62-physical-chemistry-ii-spring-2008/2351f20e4727ae0a7e03ccaca02452d7_08_562ln08.pdf
  3. ↑ 3.0 3.1 https://ocw.mit.edu/courses/8-08-statistical-physics-ii-spring-2005/resources/the_bose_gas/
  4. ↑ 4.0 4.1 https://ocw.mit.edu/courses/8-08-statistical-physics-ii-spring-2005/3d0cf2cb43a2b62f92089db14e8e2904_the_fermi_gas.pdf
Author: Harold Foppele




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