The following integral will be of immense use in analyzing hydrogen wave functions z"e=1/ada = n! an+1 (a) consider the hydrogen state in, l, m). show that the expectation value (f(n)) for any operator that just depends on the radius may be given by (f(n)) = 60° 72 f(r) rne(r)l' dr. hint (highlight to reveal): [ite the wave function as a product of a radial wave function and integra ] (b) find (r)100, the expectation value of the radius for an electron in the ground state (1,0,0) of hydrogen. also find (72)100- (c) extra part (not for credit) find (r) 200 for the state 2,0,0) (n = 2, k = 0, m = 0). you should notice a general trend emerging for (r) for the states in, 0, 0). what is this trend? predict based on this trend what (r) will be for the state 3, 0, 0) (you do not need to explicitly compute this expectation value, though). (d) find (x) and (22) for an electron in the ground state of hydrogen. hint (highlight to reveal): [ his part requires no actual integration! for first write the wave next consider the state at time t = 0 given by la) = + (12,1,1)+2, 1, - (e) find the wave function va (1) at time t = 0. first just write a simplified expression in terms of the radial wave functions rner) and the spherical harmonics y" (0,4), simplifying where possible. then plug in the actual radial wave functions and spherical harmonics. again, simplify as much as possible. (note: for example, if the state was 1,0,0), then the first expression i am looking for is just r10(r)y(0,) and the second expression is het/a.] (f) find (y) for this state ( [supplementary part (not for credit): also find (ya)! ] hint (highlight to reveal): [recall that in spherical coordinates y er sim 9 sin ] (g) find the expectation value of the potential energy, (v) for this state. give both the formula and the actual number, in electron volts. [note: this is another demonstration of the virial theorem. however, you do not get to use the virial theorem to solve this problem!