# Thermodynamic variables for a two level system

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Consider a two level system (such as the orientation of the magnetic moments of protons in a magnetic field, e.g. NMR). Let the spacing between the two levels be delta that is, the lower level has an energy of 0 and the other an energy of delta. Evaluate the following quantities in terms of delta and T.

a) the partition function

b) the internal energy

c) the entropy

In the limit of large T, does the system in the previous problem conform with the equipartition principle? Should it?

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The partition function for a general system is:

Z = Sum over all states of Exp(-beta Energy of the state)

where beta = 1/(kT)

The probability of the system to be in some particular state is:

P(state) = Exp[-beta E(state)]/Z

In this case we have two states, one with energy 0 and another with energy delta. This means that:

Z = Exp(0) + Exp(-beta delta) = 1 + exp(-beta delta) (1)

The probability for the system to be in the state with energy 0 is:

P(0) = Exp(0)/[1 + exp(-beta delta)]

The probability for the ...

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