Problem

Problems are listed in approximate order of difficulty. A single dot (•) indicates straigh...

Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.

••• If you haven’t already done so, do parts (a) and (b) of Problem 1, and then do part (c), but for the five spherical harmonics with l = 2.

Problem 1

•••The normalization condition for a three-dimensional wave function is . (a) Show that in spherical polar coordinates, the element of volume is dV = r2dr sin θ dθ dϕ.[Hint: Think about the infinitesimal volume between r and r + dr, between θ and θ + dθ, and between ϕ and ϕ + dϕ.] (b) Show that if ψ = R(r)Y(θ, ϕ), the normalization integral is the product of two terms

(c) It is usually convenient to normalize the functions R(r) and Y(θ, ϕ) separately, so that each of the factors in this middle expression is equal to 1. Verify that all of the spherical harmonics Ylm(θ, ϕ) with l = 0 or 1 do satisfy