By Altland A., Simons B.D.

Sleek experimental advancements in condensed subject and ultracold atom physics current bold demanding situations to theorists. This publication offers a pedagogical creation to quantum box thought in many-particle physics, emphasizing the applicability of the formalism to concrete difficulties. This moment variation includes new chapters constructing course quintessential techniques to classical and quantum nonequilibrium phenomena. different chapters conceal a variety of themes, from the advent of many-body suggestions and useful integration, to renormalization staff equipment, the speculation of reaction services, and topology. Conceptual elements and formal technique are emphasised, however the dialogue makes a speciality of functional experimental functions drawn mostly from condensed subject physics and neighboring fields. prolonged and tough issues of totally labored options offer a bridge among formal manipulations and research-oriented considering. geared toward raising graduate scholars to a degree the place they could interact in self reliant learn, this publication enhances graduate point classes on many-particle conception

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Sleek experimental advancements in condensed subject and ultracold atom physics current ambitious demanding situations to theorists. This e-book offers a pedagogical advent to quantum box conception in many-particle physics, emphasizing the applicability of the formalism to concrete difficulties. This moment variation includes new chapters constructing course essential methods to classical and quantum nonequilibrium phenomena.

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**Example text**

Assume that the atomic exchange constants remain the same in all directions). Show that the dispersion generalizes to ωk = v|k|, where k = (n1 , . . , nd )2π/L and ni ∈ Z. Show that the speciﬁc heat shows the temperature dependence cv ∼ T d . Answer: As discussed in the text, the quantum eigenstates of the system are given by |n1 , n2 , . . , where nm is the number of phonons of wavenumber km = 2πm/L, E|n1 ,n2 ,... = nm m ωkm (nm + 1/2) ≡ m m the eigenenergy, and ωm = v|km |. In the energy representation, the quantum partition function then takes the form Z = tr e−β H = e−βEstate = ˆ states ∞ m=1,2,...

Compute the energy density u = −(1/L)∂β ln Z of one-dimensional longitudinal phonons ˆ with dispersion ωk = v|k|, where Z = tr e−β H denotes the quantum partition function. 8 Problems 37 −1 where nB ( ) = eβ − 1 is the Bose–Einstein distribution function. Approximate the sum over k by an integral and show that cv ∼ T . At what temperature Tcl does the speciﬁc heat cross over to the classical result cv = const? e. assume that the atomic exchange constants remain the same in all directions). Show that the dispersion generalizes to ωk = v|k|, where k = (n1 , .

The next chapter is devoted to a more comprehensive discussion of both the formal aspects and applications of this formulation. 8 Problems Electrodynamics from a variational principle Choosing the Lorentz-gauged components of the vector potential as generalized coordinates, the aim of this problem is to show how the wave equations of electrodynamics can be obtained as a variational principle. Electrodynamics can be described by Maxwell’s equations or, equivalently, by wave-like equations for the vector potential.