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I am writing an appendix to my thesis where i try to explain how nmr works but i am having trouble understanding something.

Suppose that we have a sample with many nuclei with 1/2 spin precessing around an external magnetic field [itex]\vec{B_0}[/itex] aligned with the z axis. At Boltzmann equilibrium the net magnetization [itex]\vec{M}[/itex] of the sample would be along the positive direction of z.

Since the rate of change of individual magnetic moments is given by

[itex]\frac{d\vec\mu}{dt}=\gamma\vec\mu\times \vec{B_0}[/itex]

the generalization to the net magnetization vector [itex]\vec{M}[/itex] would be

[itex]\frac{d\vec{M}}{dt}=\gamma\vec{M}\times \vec{B_0}[/itex]

suppose that we now introduce a constant, non oscillating, magnetic field [itex]\vec{B_1}[/itex] orthogonally to [itex]\vec{B_0}[/itex]. I guess it is safe to write that:

[itex]\frac{d\vec{M}}{dt}=\gamma\vec{M}\times (\vec{B_0}+\vec{B_1})=\gamma\vec{M}\times (\vec{B_{eff}})[/itex]

This suggests that the magnetization vector would move in a circular path around [itex]\vec{B_{eff}}[/itex].

If everything i said until now is correct, i can't understand what happens at quantum level. I mean, although the macroscopic equation suggests that [itex]\vec{M}[/itex] would precess around [itex]\vec{B_{eff}}[/itex] what happens to the individual magnetic moments? I would think that a new Boltzmann equilibrium where they are randomly precessing around [itex]\vec{B_{eff}}[/itex] would be establish, however, that implies that [itex]\vec{M}[/itex] is static vector along that same direction!

I can't understand this conflict. There must be something wrong or something missing here. The only thing i can remember is that somehow the individual moments are no longer randomly distributed along the 2 cones but are now rotating in phase from where the excess of nuclei in the lower energy state would coincide with [itex]\vec{M}[/itex] but how did that happen?

My formation is quantum physics is not that advanced since i graduated in informatics and this is really killing me. Can someone help me understand what happens to the individual magnetic moments as soon as the second magnetic field is introduced?

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# NMR: what happens at quantum level when the orientation of the magnetic filed changes

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