phy 302

Since the spherical harmonics " ($2) are a complete set of functions for spherical angular coordinates # 0, 4, we should be able to rewrite any spherical-coordinate function in terms of a sum over spherical harmonics. Cartesian coordinates are related to spherical via (VC 51): r = r sin 0 cosy>, y = r sin0 sind. > = rcos 0, we can rewrite any function of I, y, = as a sum over spherical harmonics, with coefficients dependent on only r and some constants. That is, f(x, y.=) = > Cmn(r ) Ym(n). Carry out this procedure for each of the following functions, writing an expression for them entirely in terms of a sum over spherical harmonics with coefficients. 1. Only this first function will actually eam you points; the later functions are for practice so you can check your methods work in general. 2 22 12 = 15 32nt 2102 p, (5) X 3. 4. 5.x - 2iry + y' = 6. Ty= =