4. Adopting the spherical system of coordinates (see Problem 2), write down explicit expressions for (a) the Lame coefficients, (b) Vip, the gradient of a function p, given the unit vectors e, e and ex; (c) V . a, the divergence of a vector a, given the components a,, de and a. (d) Al, the Laplacian of a function 'P. In parts (c) and (d) make sure to simplify your results as much as possible.2. The angular momentum operator of a single particle (measured in units of h) is given by the expression 1 = -i[r x V]. (1) (a) Write down its components /; (i=x,yz) in rectangular coordinates. For the rest of this problem you will be guided through a calculation of the operators Ipi = x, y, z, in spherical coordinates defined in a standard fashion: x = r sin 0 cos. y = r sin 0 sin p. z = r cos 0. (2) (b) Take an arbitrary function I and compute the partial derivatives d'P/d0 and o'/dip expressing final results in terms of rectangular coordinates, partial derivatives o'l'/dx, d'/dy, d'P/dz and p = r sin0 = 1 (x2 + y2 (3) introduced for convenience; it will not be present in final answers. Comparing the expression for o'l'/dip with you result in part (a) determine Iz. (c) Multiply your results in part (b) for d'T/30 by x/p and for d'P/dd by -yz/p and add the two resulting equation. Now you should be able to extract an expression for /. What is it? (d) Using a suggestion in part (c) as a template of how to proceed, determine