Partition function for 3 energy levels (Statistical physics)
The energy levels of a localized partcile are 0, $\epsilon$ ,$2\epsilon$ . the middle level is doubly degenerate and the other levels are nondegenerate.
a) write and simplify the partition function.
b) find the total energy, the heat capacity, and the entropy of a system of these particles. Sketch these properties as a function of the temperature and compare them with the corresponding properties of the spin-1/2 system.
There are possible 4 states, one with energy 0, two with energies $ε$ and one state with energy $2ε$ . The probability of one state having the energy $ε$ is $=1/Z*exp(-ε/kT)$ . Therefore the partition function Z is found from the condition that the sum of all probabilities is 1:
$Z=∑_iexp(-ε_i/kT)=∑_i exp(-β*ε_i)$
$Z=exp(-0/kT)+exp(-ε/kT)+exp(-ε/kT)+exp(-2ε/kT)=$
$=1+2*exp(-ε/kT)+exp(-2ε/kT)$
$Z=(1+exp(-ε/kT) )^2=[1+exp(-βε) ]^2$
The value of the average total energy is:
$E =<ε>=∑_i ε_i*P_i= (∑_i ε_i*exp(-βε_i ) )/Z=-1/Z ∑_i d/dβ*exp(-βε_i )$
$E=-1/Z*d/dβ*∑_i exp(βε_i )=-1/Z*dZ/dβ=-d(lnZ )/dβ=kT^2*(d(lnZ))/dT$
For entropy one has
$S=-k*∑_i P_i*ln(P_i )=-k*∑_i P_i*ln(1/Z)+k*∑_i P_i*ε_i/kT=k*lnZ+1/T*<ε>$
$S=k*lnZ+E/T=k*ln (Z+kE/β)$
The heat capacity is found from the equation:
$dE=d<U>=C*dT$ so that $C=dE/dT$
The results are
$E=(-2*exp(1-βε))/(exp(-βε)+1)$
$S=2k*ln[ 1+exp(-βε) ]-2k/β*(exp(1-βε))/(1+exp(-βε))$
$C=(2*exp(2-2ε/kT))/(kT^2*[exp(-ε/kT)+1]^2 )-$
$-(2*exp(2-2ε/kT))/(kT^2*[exp{-ε/kT}+1] )=$
$=(2*e^{2-2βε})/(βT*(e^{-βε}+1)^2 )-(2*e^{2-2βε})/(βT*(e^{-βε}+1))$
The computations of the derivatives have been done online.
