By A. Davydov
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Extra resources for Theory of Molecular Excitons
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.