In this problem you will derive the commutator [l̂x, l̂y] = ihl̂z.

a. The angular momentum vector in three dimensions has the form l = il_x + jl_y + kl_z, where the unit vectors in the x, y, and z directions are denoted by i, j, and k. Determine l_x , l_y , and l_z by expanding the 3 × 3 cross product l = r × p. The vectors r and p are given by r = i x + j y + k z and p = i px + j py + k pz .

b. Substitute the operators for position and momentum in your expressions for l_x and l_y. Always write the position operator to the left of the momentum operator in a simple product of the two.

c. Show that [l̂x, l̂y ] = ihl̂_z .