Quantum statistical mechanics: A spin in a magnetic field Consider a magnetic moment with spin-one-half in a magnetic field. Being a lazy theorist, I prefer not to write /2 repeatedly, so I will use units in which the spin σ = ±1. I will also choose the units of the magnetic field such that the energy of the system is just
E = -hσ

The system is in contact with a heat reservoir at temperature T.

1. Calculate the probability of the spin having the value +1.

2. Calculate the average magnetization m = σ. Express your answer in terms of hyperbolic functions.

3. Calculate the two leading terms in a high-temperature expansion of the magnetization in powers of β = 1/kBT.

4. Calculate the leading (non-constant) term in a low-temperature power series expansion of the magnetization in some variable at low temperatures. (Hint: it will be in the form of an exponential.)

5. Calculate the average energy. Plot it as a function of temperature.

6. Calculate the specific heat. Plot it as a function of temperature

7. Calculate the magnetic susceptibility.