If the angular frequencies of waves in a three-dimensional boxof sides L general

Question

If the angular frequencies of waves in a three-dimensional boxof sides L generalize to: =((c)/L)(nx2+ny2+nz2)1/2 where all n are integers, show that the number of distinctstates in the frequency interval f to f+f (f = /(2)) is givenby (where f is large): dN =4(L3/c3)f2df If the angular frequencies of waves in a three-dimensional boxof sides L generalize to: =((c)/L)(nx2+ny2+nz2)1/2 where all n are integers, show that the number of distinctstates in the frequency interval f to f+f (f = /(2)) is givenby (where f is large): dN =4(L3/c3)f2df where all n are integers, show that the number of distinctstates in the frequency interval f to f+f (f = /(2)) is givenby (where f is large): dN =4(L3/c3)f2df

Explanation / Answer

f is related to the integers as f = /(2) = (c/2L)(Nx^2+Ny^2+Nz^2) In the (Nx,Ny,Nz) space, each f represents a distinct wave (orwe can call it a state). The total number of states from f = 0 to fis the volume of the sphere in (Nx,Ny,Nz) space of radius R =)(Nx^2+Ny^2+Nz^2); actually we should consider only oneoctant of the sphere where Nx, Ny and Nz are all positive: N = (1/8)(4R^3)/3 =(/6)R^3=(/6)((Nx^2+Ny^2+Nz^2))^3 = (/6)(2Lf/c)^3 dN/df = (/2)(2L/c)^3 f^2 = (4L^3/c^3) f^2

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