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Negative absolute temperatures

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The probability that some state r with energy E_r is occupied is:

P_r = Exp(- beta E_r)/Z (1)

Suppose that an electron in an atom can be in two states (r = 1, and r = 2 and E_2 > E_1). If there are N atoms, then the total number of atoms with an electron in state r is nr = N P_r. The ratio of n2 and n1 is thus:

n2/n1 = N P_2/(NP_1) = P_2/P_1 = (using (1) ) = [Exp(- beta E_2)/Z] / [ Exp(- beta E_1)/Z ] =

Exp[-(E_2 - E_1)/(k T) ] (2)

We can see from this equation that n2/n1 < 1 if E_2 > E_1. Here we assume ...

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