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.

Answered: Image /qna-images/answer/0d3710c0-a4a2-4a99-9fae-0fefeeaef54e.jpg

Plane waves in a box with periodic boundary conditions Consider plane waves &(r) = a exp(ik. r) in a d-dimensional cubic region of linear size L in each direction and “volume” V = Lª. There are p possible_wave polarizations. Assume that the waves satisfy "periodic boundary conditions": (r) = (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 those possible values can be written in the form k = (dk) Σ;=1 njêj, where nj, j = 1,...,d are integers, and calculate the quantity ok. (ii) Assume that there are p possible polarizations, and that the angular frequency grows linearly with the wavevector: w = ck. Compute the density of modes g(w) in the general case of d dimension and list g(w) for d = 1,2,3. For this, note that the volume of a sphere of radius R in d-dimension is 774/2 Vd-sphere = D Rd.

Plane waves in a box with periodic boundary conditions Consider plane waves &(r) = a exp(ik. r) in a d-dimensional cubic region of linear size L in each direction and “volume” V = Lª. There are p possible_wave polarizations. Assume that the waves satisfy "periodic boundary conditions": (r) = (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 those possible values can be written in the form k = (dk) Σ;=1 njêj, where nj, j = 1,...,d are integers, and calculate the quantity ok. (ii) Assume that there are p possible polarizations, and that the angular frequency grows linearly with the wavevector: w = ck. Compute the density of modes g(w) in the general case of d dimension and list g(w) for d = 1,2,3. For this, note that the volume of a sphere of radius R in d-dimension is 774/2 Vd-sphere = D Rd.