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One bosonic and one fermionic mode at a specified temperature

Physical scope

An exact hypothetical canonical calculation for one bosonic oscillator mode and one fermionic two-level mode with beta times excitation energy equal to log 2; it is not a measurement or a thermalization claim.

Example

Assume AC and Countable Choice as required by Finite bosonic and fermionic mode partition and occupation formulas. Choose $T > 0$ kelvins and excitation energy $ε = k_{B} T \log 2$ joules, so $q = e^{- β ε} = 1 / 2$ . Consider separately one bosonic mode with occupations $n = 0, 1, 2, \dots$ and one fermionic mode with occupations $n = 0, 1$ .

Facts & Assumptions

Given: The declared one-mode systems, their positive excitation energy, and the adopted Gibbs preparation.

[F1]

Finite bosonic and fermionic mode partition and occupation formulas: for $q = e^{- β ε}$ , the bosonic partition trace is $(1 - q)^{- 1}$ with mean $q / (1 - q)$ , and the fermionic partition sum is $1 + q$ with excited occupation $q / (1 + q)$ ; these are conditional on its stated quantum preparation and suppliers.

[F2]

SI quantities and unit conventions: $T$ is in kelvins, $k_{B}$ in $J K^{- 1}$ , and $β = (k_{B} T)^{- 1}$ in $J^{- 1}$ .

[A1]

The selected energy is $ε = k_{B} T \log 2 > 0$ joules, and $q = e^{- β ε} = 1 / 2$ .

[A2]

AC and Countable Choice are assumed through [F1] for the quantum Gibbs assignment and the supplied one-mode calculations.

Verification

Given: $q = 1 / 2$ and the finite-mode identities in [F1].

1.1givenF1F2A1algebra

The bosonic trace is $Z_{B} = (1 - 1 / 2)^{- 1} = 2$ , and the probability of occupation $n$ is $2^{- (n + 1)}$ . Its mean occupation is $(1 / 2) / (1 - 1 / 2) = 1$ .

2.1F1F2A1algebra∎

The fermionic partition sum is $Z_{F} = 1 + 1 / 2 = 3 / 2$ , so the ground and excited probabilities are $2 / 3$ and $1 / 3$ , respectively, and the mean occupation is $1 / 3$ . The bosonic and fermionic values belong to distinct declared models.

One bosonic and one fermionic mode at a specified temperature

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources