Plane waves in a box with periodic boundary conditionsConsider plane waves (r) = a exp(ik. r) in a d-dimensional cubic region of linear size L ineach direction and "volume" V = Lª. There are p possible_wave polarizations. Assume thatthe waves satisfy "periodic boundary conditions": () = (r+ Lêj), j = 1, ···, d, where ê;is the unit vector along the j-th direction.(i) The boundary conditions restrict the possible values of the wavevector K. Show that thosepossible values can be written in the form k = (8k) Σj-1¹jêj, where nj, j = 1,…‚d areintegers, and calculate the quantity Sk.(ii) Assume that there are p possible polarizations, and that the angular frequency growslinearly with the wavevector: w = ck. Compute the density of modes g(w) in the generalcase of d dimension and list g(w) for d = 1,2,3. For this, note that the volume of a sphere ofradius R in d-dimension isVd-sphere=77d/2T(1+d/2)Rd,where I() is Euler's gamma function.(iii) Apply your result to the case of electromagnetic waves in a 3D cavity of volume V.

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