Quantum statistical mechanics: Another two-level system
Although a two-level system might seem very simple, it is very important and occurs frequently in various guises. Here is another form that we will see often. A system only has two states, which are both non-degenerate. The energies of these two states are E = 0 and E = ? > 0. The system is in contact with a heat reservoir at temperature T.
1. Calculate the probability of being in the excited state.
2. Calculate the average energy. Sketch it as a function of temperature.
3. Calculate the average specific heat. Sketch it as a function of temperature.
4. Calculate the two leading terms in a high-temperature expansion of the energy in powers of β = 1/kBT.
5. Calculate the leading (non-constant) term in a low-temperature power series expansion of the energy in some variable at low temperatures. (Hint: the variable will be in the form of an exponential.)