In this paper, we give a pedagogical introduction to the ideas of quantum thermodynamics and work fluctuations, using only basic concepts from quantum and statistical mechanics. After reviewing the concept of work as usually taught in thermodynamics and statistical mechanics, we discuss the framework of non-equilibrium processes in quantum systems together with some modern developments, such as the Jarzynski equality and its connection to the second law of thermodynamics. We then apply these results to the problem of magnetic resonance, where all calculations can be done exactly. It is shown in detail how to build the statistics of the work, both for a single particle and for a collection of non-interacting particles. We hope that this paper will serve as a tool to bring the new student up to date on the recent developments in non-equilibrium thermodynamics of quantum systems.

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For completeness, we mention that it is very easy to demonstrate Eq. (19) in the case of infinitesimal processes as discussed in Sec. II B. But it is important to clarify that this case is physically of no interest at all: The importance of the Jarzynski equality is that it holds for non-equilibrium processes. For infinitesimal processes, we do not need such an equality. In any case, the demonstration is as follows. Based on Eq. (8), we can define the random work performed in an infinitesimal process as δW=dEn=En(λ+dλ)En(λ). Equation (8) can then be interpreted instead as the definition of δW. In other words, Pn is the probability distribution of the random variable δW. We can then compute any type of average we want. In particular, eβδW=neβδWPn=neβδW(eβEn(λ)/Z(λ))=1Z(λ)neβEn(λ+dλ)=((Z(λ+dλ))/(Z(λ)))=eβdF, where we have used the fact that Z(λ+dλ)=eβF(λ+dλ).

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