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I have a system with spin 1/2 particles (protons) in a magnetic field. The Zeeman effect leads to two energy levels (Spin up, spin down). Thus I have a state $$ \left|\Psi\right> = a\left|\uparrow\right> + b\left|\downarrow\right> $$

At room temperature, the probability to find a spin in one of the two states is determined by the Boltzmann distribution (high temp limit of Fermi). Hence, the probabilities are given by $p_\uparrow = e^{-E_\uparrow/k_B T}$ und $p_\downarrow = e^{-E_\downarrow/k_B T}$

Now my question: How does $a,b$ relate to $p_\uparrow$ and $p_\downarrow$. I have heard that you simply need to use density matrices to understand it, but I don't get it actually...

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There is no relation. $|\Psi\rangle$ is a pure state, while the thermal state $\rho$ is a statistical mixture. Since they describe different states, there is no relation.

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