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## Introduction to superconductivity and superfluidity (lecture notes)

These notes have appeared as a result of a one-term course in superfluidity and superconductivity given by the author to fourth-year undergraduate students and first-year graduate students of the Department of Physics, Moscow State University of Education. The goal was not to give a detailed picture of these two macroscopic quantum phenomena with an extensive coverage of the experimental background and all the modern developments, but rather to show how the knowledge of undergraduate quantum mechanics and statistical physics could be used to discuss the basic concepts and simple problems, and draw parallels between superconductivity and superfluidity.

Superconductivity and superfluidity are two phenomena where quantum mechanics, typically constrained to the microscopic realm, shows itself on the macroscopic level. Conceptually and mathematically, these phenomena are related very closely, and some results obtained for one can, with a few modifications, be immediately carried over to the other. However, the student of these notes should be aware of important differences between superconductivity and superfluidity that stem mainly from two facts: (1) electrons in a superconductor carry a charge, therefore one has to take into account interaction with electromagnetic radiation; (2) electrons move in a lattice, therefore phonons play a role not only a mediators of attractive interaction between pairs of electrons, but also as scatterers of charge carriers.

Although these are notes on superfluidity *and *superconductivity, and there are a few cross-references, the two subjects can be studied independently with, perhaps, a little extra work by the student to fill in the gaps resulting from such study. The material of Chapter 1 introduces the method of second quantisation that is commonly used to discuss systems with many interacting particles. It is then applied in Chaper 2 to treat the uniform weakly interacting Bose gas within the approach by N. Bogoliubov, and in Chapter 4 to formulate the theory of the uniform superconducting state put forth by J. Bardeen, L. Cooper and R. Schrieffer. Chapter 3 presents the theory proposed independently by E. Gross and L. Pitaevskii of a non-uniform weakly interacting Bose gas, with a discussion of vortices, rotation of the condensate, and the Bogoliubov equations. In Chapter 5 we discuss the Ginzburd-Landau theory of a non-uniform superconductor near the critical temperature and apply it to a few simple problems such as the surface energy of the boundary between a normal metal and a superconductor, critical current and critical magnetic field, and vortices.

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