PhysicsRock
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- Homework Statement
- Consider two identical particles that follow Fermi-Dirac statistics in a thermodynamical system with two energy levels, ##E_0## and ##E_1##, ##E_1 > E_0##, that do no interact and are in contact with a heat reservoir at temperature ##T##. Determine the canonical partition function.
- Relevant Equations
- ##Z = \sum_\text{all configurations} e^{-\beta E}##
My solution is
$$
Z = e^{-2\beta E_0} + 4 e^{-\beta (E_0 + E_1)} + e^{-2\beta E_1},
$$
since Fermions have non-zero spin and there are four options for distributing spins (assuming only ##\pm \frac{1}{2}##) among ##E_0## and ##E_1## such that ##E_\text{tot} = E_0 + E_1## and one option each that leads to a total energy of ##2E_{0,1}##. However, I'm not sure whether or not the terms ##e^{-2\beta E_{0,1}}## should also be weighted with a degeneracy factor (##2## in this case), because both up / down and down / up are possible. The quantum state remains identical under swapping the spins, but it also leads to an additional option to obtain the same total energy.
Help is appreciated, have a great day everyone.
$$
Z = e^{-2\beta E_0} + 4 e^{-\beta (E_0 + E_1)} + e^{-2\beta E_1},
$$
since Fermions have non-zero spin and there are four options for distributing spins (assuming only ##\pm \frac{1}{2}##) among ##E_0## and ##E_1## such that ##E_\text{tot} = E_0 + E_1## and one option each that leads to a total energy of ##2E_{0,1}##. However, I'm not sure whether or not the terms ##e^{-2\beta E_{0,1}}## should also be weighted with a degeneracy factor (##2## in this case), because both up / down and down / up are possible. The quantum state remains identical under swapping the spins, but it also leads to an additional option to obtain the same total energy.
Help is appreciated, have a great day everyone.