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How to evaluate the 2 partial derivatives from the expression for Z?

How to evaluate the 2 partial derivatives from the expression for Z?

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How to evaluate the 2 partial derivatives from the expression for Z?

The magnetic quantum number M for the z direction takes values S, S-1,...,-S. The energy for the state M is -MgμBB; thus the probability of the atom being in this state, given by the Boltzmann factor, is proportional to exp(Mu) where u = gμBBB. Put Then and S Z= £ exp(Mu)=sinh{(S+})u} M=-S = sinh(u) Σs M exp(Mu)_az/au Z Z 2 ((S)²)=sM² exp(Mu)_a²Z/au² Z Z Evaluate ǝZ/ǝu and a²Z/au² from the expression for Z and substi- tute.
Transcribed Image Text:The magnetic quantum number M for the z direction takes values S, S-1,...,-S. The energy for the state M is -MgμBB; thus the probability of the atom being in this state, given by the Boltzmann factor, is proportional to exp(Mu) where u = gμBBB. Put Then and S Z= £ exp(Mu)=sinh{(S+})u} M=-S = sinh(u) Σs M exp(Mu)_az/au Z Z 2 ((S)²)=sM² exp(Mu)_a²Z/au² Z Z Evaluate ǝZ/ǝu and a²Z/au² from the expression for Z and substi- tute.
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