A hydrogen atom consists of a proton and an electron. According to the Bohr theory, the electron revolves about the proton in a circle of radius a (a=5⋅10−9 cm for the ground state). According to quantum mechanics, the electron may be at any distance r (from 0 to ∞) from the proton; for the ground state, the probability that the electron is in a volume element dV, at a distance r to r+dr from the proton, is proportional to e−2r/adV, where a is the Bohr radius. Write dV in spherical coordinates (see Chapter 5, Section 4) and find the density function f(r) so that f(r)dr is the probability that the electron is at a distance between r and r+dr from the proton. (Remember that the probability for the electron to be somewhere must be 1.) Sketch f(r) and show that its maximum value is at r=a; we then say that the most probable value of r is a. Also show that the average value of r−1 is a−1.