Consider a system of a single spin, in thermal equilibrium with a reservoir of temperature T....

 

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Consider a system of a single spin, in thermal equilibrium with a reservoir of temperature T. Once an external magnetic field is applied, the spin can be in one of two states: aligned with the field (energy -B) and antialigned with the field (energy +B). What is the ratio of the probabilities of being in each of these states? Hint: This is basically a rederivation of the Boltzmann factor from the canonical ensemble. Assume there is energy Eo in the reservoir when the spin is antialigned, and energy Eo+2B when the spin is aligned (the energy has just been transferred between systems). The logarithm of the number of states in the reservoir follows In R(Eo+2B) In R(EO) + Jln R E=E0 2B and the derivative can be related to the temperature. Then argue that the probabilities of the spin being in the given states depend on the ratio of R(Eo) and R(Eo + 2B). Consider a system of a single spin, in thermal equilibrium with a reservoir of temperature T. Once an external magnetic field is applied, the spin can be in one of two states: aligned with the field (energy -B) and antialigned with the field (energy +B). What is the ratio of the probabilities of being in each of these states? Hint: This is basically a rederivation of the Boltzmann factor from the canonical ensemble. Assume there is energy Eo in the reservoir when the spin is antialigned, and energy Eo+2B when the spin is aligned (the energy has just been transferred between systems). The logarithm of the number of states in the reservoir follows In R(Eo+2B) In R(EO) + Jln R E=E0 2B and the derivative can be related to the temperature. Then argue that the probabilities of the spin being in the given states depend on the ratio of R(Eo) and R(Eo + 2B).

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To determine the ratio of the probabilities of the spin being in the aligned and antialigned states ... View the full answer