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Quantum statistical physics: fermions, bosons and quantum matter

Derive density operators, quantum occupation functions, degeneracy pressure and Bose condensation, with classical limits made explicit.

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Matter pathway: atom → solid → liquid → gas → plasma. Quantum mechanics and quantum field theory provide foundations across the pathway; they are not additional phases. This is a connected modeling route, not a universal heating curve. Actual phases depend on pressure, composition, and kinetics.

1. Thermal density operator

Definitions & inputs. ρhat density operator,H Hamiltonian,Z partition function,β1/(kBT).

  1. Replace classical probabilities with an operator whose energy eigenvalues are Bolt z mann weights.

    ρ^=e−βH/Z,Z=Tr⁡e−βH\hat\rho=e^{-\beta H}/Z,\quad Z=\operatorname{Tr}e^{-\beta H}

  2. Observable s and entropy follow from the thermal state.

    ⟨A⟩=Tr⁡(ρ^A),S=−kBTr⁡(ρ^ln⁡ρ^)\langle A\rangle=\operatorname{Tr}(\hat\rho A),\quad S=-k_B\operatorname{Tr}(\hat\rho\ln\hat\rho)

  3. Sum the oscillator geometric series,including zero-point energy.

    H=ℏω(n+1/2)⇒Z=e−βℏω/2/(1−e−βℏω)H=\hbar\omega(n+1/2)\Rightarrow Z=e^{-\beta\hbar\omega/2}/(1-e^{-\beta\hbar\omega})

Interpretation. Quantum thermal mixing and coherent superposition are different;an energy diagonal thermal state has no off-diagonal coherence in that basis.

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2. Derive Bose and Fermi occupations

Definitions & inputs. ε single-particle energy,μ chemical potential,zexpβμ,xβ(ε−μ),Ξ grand partition function.

  1. Sum the allowed occupation numbers for each mode.

    ΞF=1+e−x,ΞB=∑n=0∞e−nx=1/(1−e−x)\Xi_F=1+e^{-x},\quad\Xi_B=\sum_{n=0}^\infty e^{-nx}=1/(1-e^{-x})

  2. Differentiate to get mean occupation.

    nˉ=−∂xln⁡Ξ⇒nˉF=1/(ex+1),nˉB=1/(ex−1)\bar n=-\partial_x\ln\Xi\Rightarrow\bar n_F=1/(e^x+1),\quad\bar n_B=1/(e^x-1)

  3. Both recover Maxwell–Bolt z mann statistics when occ up an cie s are small.

    x≫1⇒nˉF≃nˉB≃e−xx\gg1\Rightarrow\bar n_F\simeq\bar n_B\simeq e^{-x}

Interpretation. Photons in equilibrium haveμ0;particle-conserving gases generally do not. Spin and other de genera cie s enter the count of modes.

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3. Fermi gas and degeneracy pressure

Definitions & inputs. n number density,kF Fermi wavevector,EF Fermi energy,m particle mass,gspin2.

  1. Fill each momentum state up to the Fermi sphere with two spin orientations.

    n=24πkF3/3(2π)3⇒kF=(3π2n)1/3n=2\frac{4\pi k_F^3/3}{(2\pi)^3}\Rightarrow k_F=(3\pi^2n)^{1/3}

  2. Integrate quadratic energy over the filled sphere.

    EF=ℏ2kF2/(2m),U/N=3EF/5E_F=\hbar^2k_F^2/(2m),\quad U/N=3E_F/5

  3. The energy-volume relation gives degeneracy pressure and a temperature scale.

    P=2nEF/5,TF=EF/kBP=2nE_F/5,\quad T_F=E_F/k_B

Interpretation. Electron bands in solids use effective parameters;relativistic dense matter needs a different dispersion relation.

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4. Bose gas and photons

Definitions & inputs. λth thermal wavelength,n number density,ζRiemann zeta function,Tc condensation temperature.

  1. Excited states have a finite capacity at fixed temperature in three dimensions.

    nex=λth−3g3/2(z),g3/2(1)=ζ(3/2)≃2.612n_{ex}=\lambda_{th}^{-3}g_{3/2}(z),\quad g_{3/2}(1)=\zeta(3/2)\simeq2.612

  2. Excess particles occupy the ground state below the ideal critical temperature.

    Tc=2πℏ2mkB[n/ζ(3/2)]2/3,N0/N=1−(T/Tc)3/2T_c=\frac{2\pi\hbar^2}{mk_B}[n/\zeta(3/2)]^{2/3},\quad N_0/N=1-(T/T_c)^{3/2}

  3. Multiply Bose photon occupation by the electromagnetic density of states and photon energy.

    uν=8πhν3c3[ehν/(kBT)−1]−1u_\nu=\frac{8\pi h\nu^3}{c^3}[e^{h\nu/(k_BT)}-1]^{-1}

Interpretation. Traps and interactions alter condensate behavior;Planck radiation connects quantum statistics with radiative transfer.

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Graphical worked example

Ideal Fermi–Dirac occupation per single-particle state at nonzero temperature. X axis: Energy offset (ε−μ)/kBT (dimensionless). Y axis: Fermi occupation (dimensionless).
Ideal Fermi–Dirac occupation per single-particle state at nonzero temperature. Related worked calculation · Download SVG · Plot data

Twenty worked examples

Open a problem to see its defined inputs, assumptions, equation, numerical substitution, result, and interpretation. Values are illustrative analytical exercises.

Example 01. Fermi occupation atμ

Definitions & inputs. x0.

  1. Choose the governing model and isolate the requested quantity.

    f=1/(ex+1)f=1/(e^x+1)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1/21/2

  3. Evaluate the expression; the result uses the units shown.

    Result=0.5 \mathrm{Result}=0.5\ {}

Interpretation. Finite-temperature half occupation.

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Example 02. Fermi occupation aboveμ

Definitions & inputs. x1.

  1. Choose the governing model and isolate the requested quantity.

    f=1/(e+1)f=1/(e+1)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1/(e+1)1/(e+1)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.2689414 \mathrm{Result}=0.2689414\ {}

Interpretation. Dimensionless energy offset.

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Example 03. Fermi occupation belowμ

Definitions & inputs. x−1.

  1. Choose the governing model and isolate the requested quantity.

    f=1/(e−1+1)f=1/(e^{-1}+1)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1/(e−1+1)1/(e^{-1}+1)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.7310586 \mathrm{Result}=0.7310586\ {}

Interpretation. Complements occupation at x1.

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Example 04. Bose occupation

Definitions & inputs. x1.

  1. Choose the governing model and isolate the requested quantity.

    n=1/(e−1)n=1/(e-1)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1/(e−1)1/(e-1)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.5819767 \mathrm{Result}=0.5819767\ {}

Interpretation. Positive boson energy offset.

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Example 05. Classical occupation

Definitions & inputs. x5.

  1. Choose the governing model and isolate the requested quantity.

    nMB=e−5n_{MB}=e^{-5}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    e−5e^{-5}

  3. Evaluate the expression; the result uses the units shown.

    Result=0.006737947 \mathrm{Result}=0.006737947\ {}

Interpretation. Low occupancy approximation.

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Example 06. Fermi variance

Definitions & inputs. f.5.

  1. Choose the governing model and isolate the requested quantity.

    V(n)=f(1−f)V(n)=f(1-f)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .5(.5).5(.5)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.25 \mathrm{Result}=0.25\ {}

Interpretation. Single mode Bernoulli occupation.

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Example 07. Bose variance

Definitions & inputs. n1.

  1. Choose the governing model and isolate the requested quantity.

    V(n)=n(1+n)V(n)=n(1+n)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1(2)1(2)

  3. Evaluate the expression; the result uses the units shown.

    Result=2 \mathrm{Result}=2\ {}

Interpretation. Enhanced single mode fluctuation.

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Example 08. Fermion mode partition

Definitions & inputs. x1.

  1. Choose the governing model and isolate the requested quantity.

    Ξ=1+e−x\Xi=1+e^{-x}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1+e−11+e^{-1}

  3. Evaluate the expression; the result uses the units shown.

    Result=1.367879 \mathrm{Result}=1.367879\ {}

Interpretation. Mode occupancy zero or one.

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Example 09. Boson mode partition

Definitions & inputs. x1.

  1. Choose the governing model and isolate the requested quantity.

    Ξ=1/(1−e−x)\Xi=1/(1-e^{-x})

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1/(1−e−1)1/(1-e^{-1})

  3. Evaluate the expression; the result uses the units shown.

    Result=1.581977 \mathrm{Result}=1.581977\ {}

Interpretation. Convergent geometric series.

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Example 10. Oscillator energy

Definitions & inputs. ℏω1energy unit,kBT1same unit.

  1. Choose the governing model and isolate the requested quantity.

    U=ℏω[1/2+1/(eβℏω−1)]U=\hbar\omega[1/2+1/(e^{\beta\hbar\omega}-1)]

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .5+1/(e−1).5+1/(e-1)

  3. Evaluate the expression; the result uses the units shown.

    Result=1.081977 \mathrm{Result}=1.081977\ {}

Interpretation. Includes zero-point energy.

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Example 11. Oscillator heat capacity

Definitions & inputs. xℏω/kBT1.

  1. Choose the governing model and isolate the requested quantity.

    C/kB=x2ex/(ex−1)2C/k_B=x^2e^x/(e^x-1)^2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    e/(e−1)2e/(e-1)^2

  3. Evaluate the expression; the result uses the units shown.

    Result=0.9206736 \mathrm{Result}=0.9206736\ {}

Interpretation. Quantum heat capacity.

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Example 12. Fermi wavevector

Definitions & inputs. n10²⁸/m³,gspin2.

  1. Choose the governing model and isolate the requested quantity.

    kF=(3π2n)1/3k_F=(3\pi^2n)^{1/3}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    (3π21028)1/3(3\pi^2 10^{28})^{1/3}

  3. Evaluate the expression; the result uses the units shown.

    Result=6.665105×109 m−1\mathrm{Result}=6.665105\times10^{9}\ \mathrm{m^{-1}}

Interpretation. Homogeneous3Dgas.

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Example 13. Fermi energy

Definitions & inputs. n10²⁸/m³,me9.1093837×10⁻³¹kg,ℏ1.054571817×10⁻³⁴SI.

  1. Choose the governing model and isolate the requested quantity.

    EF=ℏ2(3π2n)2/3/(2m)E_F=\hbar^2(3\pi^2n)^{2/3}/(2m)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    [1.054571817×10−34]2(3π21028)2/3/[2(9.1093837×10−31)][1.054571817\times10^{-34}]^2(3\pi^210^{28})^{2/3}/[2(9.1093837\times10^{-31})]

  3. Evaluate the expression; the result uses the units shown.

    Result=1.692532 eV\mathrm{Result}=1.692532\ \mathrm{eV}

Interpretation. SI joules converted to eV.

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Example 14. Fermi temperature

Definitions & inputs. EF2eV,kB8.617333262×10⁻⁵eV/K.

  1. Choose the governing model and isolate the requested quantity.

    TF=EF/kBT_F=E_F/k_B

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    2/(8.617333262×10−5)2/(8.617333262\times10^{-5})

  3. Evaluate the expression; the result uses the units shown.

    Result=23209.04 K\mathrm{Result}=23209.04\ \mathrm K

Interpretation. Energy scale not actual temperature.

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Example 15. Mean zero-temperature energy

Definitions & inputs. EF2eV.

  1. Choose the governing model and isolate the requested quantity.

    U/N=3EF/5U/N=3E_F/5

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    3(2)/53(2)/5

  3. Evaluate the expression; the result uses the units shown.

    Result=1.2 eV\mathrm{Result}=1.2\ \mathrm{eV}

Interpretation. Ideal non relativistic gas.

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Example 16. Degeneracy pressure

Definitions & inputs. n10²⁸/m³,EF2eV.

  1. Choose the governing model and isolate the requested quantity.

    P=2nEF/5P=2nE_F/5

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .4(1028)(2)(1.602176634×10−19).4(10^{28})(2)(1.602176634\times10^{-19})

  3. Evaluate the expression; the result uses the units shown.

    Result=1.281741×109 Pa\mathrm{Result}=1.281741\times10^{9}\ \mathrm{Pa}

Interpretation. Chosen EF illustrates formula independently of the previous density calculation.

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Example 17. Density scaling ofEF

Definitions & inputs. Doublen.

  1. Choose the governing model and isolate the requested quantity.

    EF′/EF=22/3E'_F/E_F=2^{2/3}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    22/32^{2/3}

  3. Evaluate the expression; the result uses the units shown.

    Result=1.587401 \mathrm{Result}=1.587401\ {}

Interpretation. Fixed mass and degeneracy.

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Example 18. Condensate fraction

Definitions & inputs. T/Tc.5.

  1. Choose the governing model and isolate the requested quantity.

    N0/N=1−(T/Tc)3/2N_0/N=1-(T/T_c)^{3/2}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1−.53/21-.5^{3/2}

  3. Evaluate the expression; the result uses the units shown.

    Result=0.6464466 \mathrm{Result}=0.6464466\ {}

Interpretation. Uniform ideal3D Bose gas.

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Example 19. Condensate threshold phase-space density

Definitions & inputs. z1.

  1. Choose the governing model and isolate the requested quantity.

    nλth3=ζ(3/2)n\lambda_{th}^3=\zeta(3/2)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    2.61237534872.6123753487

  3. Evaluate the expression; the result uses the units shown.

    Result=2.612375 \mathrm{Result}=2.612375\ {}

Interpretation. Thermodynamic ideal-gas limit.

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Example 20. Photon occupation

Definitions & inputs. hν/kBT2,μ0.

  1. Choose the governing model and isolate the requested quantity.

    n=1/(e2−1)n=1/(e^2-1)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1/(e2−1)1/(e^2-1)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.1565176 \mathrm{Result}=0.1565176\ {}

Interpretation. Per photon mode.

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Symbols and units

Each derivation and problem defines its own symbols and inputs. Symbols may be reused with different meanings in other subjects. Keep units consistent, retain sufficient precision during calculation, and apply the stated validity limits.