PX284 - E2 - two-level system (revisited)

the energy levels are:

\begin{aligned} __________ & + \frac{Δ}{2} \\ ↑ E & ↕ Δ \\ __________ & - \frac{Δ}{2} \end{aligned}

the partition function:

\begin{aligned} Z & = \sum_{i} \exp (- β E_{i}) \\ = \exp (β \frac{Δ}{2}) + \exp (- β \frac{Δ}{2}) \\ = 2 \cosh (β \frac{Δ}{2}) \\ ⟹ \ln Z & = \ln [2 \cosh (β \frac{Δ}{2})] \end{aligned}

the internal energy:

U = - \frac{d \ln Z}{d β} = - \frac{Δ}{2} \frac{2 \sinh (β \frac{Δ}{2})}{2 \cosh (β \frac{Δ}{2})} = - \frac{Δ}{2} \tanh (β \frac{Δ}{2})

image: wolfram

at very low temperatures, $(β \frac{Δ}{2})$ is very large $⟹ U \to - \frac{Δ}{2}$
at very high temperatures, $(β \frac{Δ}{2})$ is very small $⟹ U \to 0$
also, the helmholtz free energy:

F = - k_{B} T \ln Z = - k_{B} T \ln (2 \cosh (β \frac{Δ}{2}))

the entropy:

S = \frac{U - F}{T} = - \frac{Δ}{2 T} \tanh (β \frac{Δ}{2}) + k_{B} T \ln [2 \cosh (β \frac{Δ}{2})]

so, at very low temperatures, $(β \frac{Δ}{2})$ is large:

\ln (2 \cosh (β \frac{Δ}{2})) = \ln [\exp (β \frac{Δ}{2}) + \underset{t i n y}{\underset{⏟}{\exp (- β \frac{Δ}{2})}}] \to (β \frac{Δ}{2})

∴ S = - \frac{Δ}{2 T} + k_{B} β \frac{Δ}{2} = 0

at very high temperatures, $(β \frac{Δ}{2})$ is very small: $S \to k_{B} \ln 2$
this agrees with the third law of thermodynamics, ie: entropy of an isolated system approaches a constant value at thermal equilibrium
finally,

C_{V} = {(\frac{\partial U}{\partial T})}_{V} = k_{B} {(β \frac{Δ}{2})}^{2} {sech}^{2} (β \frac{Δ}{2})

this has a peak at approximately $\frac{k_{B} T}{Δ} \sim 0.42$
this peak is called the schottky anomaly