Mathematical Methods for Physics and Engineering - Matematica.NET

PDES: SEPARATION OF VARIABLES AND OTHER METHODS21.2 A cube, made of material whose conductivity is k, has as its six faces the planesx = ±a, y = ±a and z = ±a, and contains no internal heat sources. Verify thatthe temperature distributionu(x, y, z, t) =A cos πx ( )πzsina a exp − 2κπ2 ta 2obeys the appropriate diffusion equation. Across which faces is there heat flow?What is the direction and rate of heat flow at the point (3a/4,a/4,a)attimet = a 2 /(κπ 2 )?21.3 The wave equation describing the transverse vibrations of a stretched membraneunder tension T and having a uniform surface density ρ is( )∂ 2 uT∂x + ∂2 u= ρ ∂2 u2 ∂y 2 ∂t . 2Find a separable solution appropriate to a membrane stretched on a frame oflength a and width b, showing that the natural angular frequencies of such amembrane are given by( )ω 2 = π2 T n2ρ a + m2,2 b 2where n and m are any positive integers.21.4 Schrödinger’s equation for a non-relativistic particle in a constant potential regioncan be taken as( )− 2 ∂ 2 u2m ∂x + ∂2 u2 ∂y + ∂2 u= i ∂u2 ∂z 2 ∂t .(a) Find a solution, separable in the four independent variables, that can bewritten in the form of a plane wave,ψ(x, y, z, t) =A exp[i(k · r − ωt)].Using the relationships associated with de Broglie (p = k) and Einstein(E = ω), show that the separation constants must be such thatp 2 x + p 2 y + p 2 z =2mE.(b) Obtain a different separable solution describing a particle confined to a boxof side a (ψ must vanish at the walls of the box). Show that the energy ofthe particle can only take the quantised valuesE = 2 π 22ma 2 (n2 x + n 2 y + n 2 z),where n x , n y and n z are integers.21.5 Denoting the three terms of ∇ 2 in spherical polars by ∇ 2 r, ∇ 2 θ , ∇2 φin an obviousway, evaluate ∇ 2 r u, etc. for the two functions given below and verify that, in eachcase, although the individual terms are not necessarily zero their sum ∇ 2 u is zero.Identify the corresponding values of l and m.((a) u(r, θ, φ) = Ar 2 + B ) 3cos 2 θ − 1.r 3 2(b) u(r, θ, φ) =(Ar + B )sin θ exp iφ.r 221.6 Prove that the expression given in equation (21.47) for the associated Legendrefunction Pl m (µ) satisfies the appropriate equation, (21.45), as follows.768