Question

The wave functions of the H-atom for the ground state and the first excited state are given by Yoo(θ, φ), 100(r' exp - , 200(r, θ, φ) a. Show that these 5 wave-functions are all mutually orthogonal to each other. b. Determine the expectation values(r2〉nl of the operator r2 defined as follows 〈r2〉10-rd3TV1,00(a) r2ψ1,0,0(2) and 〈r2〉20, 〈r2)21 defined analogously

Answer 1

As I_nlm = R_nl r_lm and as integral r*_lm r_l�m� d ohm = delta_ll� delta mm� And as for the given eigenster: l, m are different As different r_lm e are mutually orthogonal to each other, so the given wave function are mutually orthogonal to each other. R_10 = 2/squareroot r^3_0 e^-r/r^0 so, < r^2 >_10 = integral^infinity_0 R^2_10 r^2 r^2 dr = (theta^4/r^3_0) integral^infinity_0 r^4 e^-2x/r_0 dx. Let an replace (r + r_0) = x then < r^2 >_10 = 4 theta r^2_0 integral^infinity_0 x^4 e^-2x dx = 4 r^2_0 times 3/4 = 3 r^2_0 = > < r^2 >_10 = 3 r^2_0 And R_20 = 1/squareroot 2 r^3_0(1 - r/2 r_0)e^-r/2 r_0 \r\n so, < r^2 >_20 = 1/2 r^3_0 integral^infinity_0 (1 - r/2 r_0)^2 r^4 e^-r/r_0 dx = r^2_0/2 integral^infinity_0 (1 - x^2/2)^2 x^4 e^-x dx. < r^2 >_20 = 42 r^2_0 = r^2_0/2 times 84 = 42 r^2_0 And for R_21 = 1/squareroot 24 r^3_0 (r/r_0)e^-r/2 r_0 < r^2 >_21 = 1/24 r^3_0 integral^infinity_0 (r/r_0)^2 r^4 e^-r/r_0 dr = r^2_0/24 integral^infinity_0 x^6