A single molecular ion trapped in a solid has four non-degenerate states
PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Statistical foundations and the equilibrium ensembles · Statistical foundations and the equilibrium ensembles
Thermal Physics, Ch The Boltzmann factor, the canonical ensemble and the partition function
Problem
A single molecular ion trapped in a solid has four non-degenerate states, of energies 0, ε, 2ε and 3ε. The ion is weakly coupled to the surrounding solid, which acts as a reservoir at temperature T, and the temperature is such that kBT=1.5ε.
((a)) Write down the partition function Z as a number and find the probability of each of the four states.
((b)) Using U=−(∂lnZ/∂β)V with β=1/kBT, find the mean energy U in units of ε.
((c)) A student says "the ground state is the most probable state, so the mean energy of the ion must be 0". State the value of U your calculation gives and say in one sentence why the most probable state does not fix the mean.