Answer Happy

Consider the hydrogen atom. When spin is ignored, the Schrödinger equation of the electron with mass me and charge -e (e > 0) is h² 8² h² 20 L² 2m, dr² + 2mer ar 2mer² 4nEo As shown in the figure, we use spherical coordinates (r.8, 4), the origin of which is the nuclear position. Here, h, o. , and E are Planck constant divided by 2n, permittivity of vacuum, wave function, and energy, respectively. L is the angular momentum operator and the following formula holds. 10 L² = -h²| (sine)+ IsinᎾ ᏧᎾ 1 A4nto 7) 4(1.0.4) = B 1 2² sin²0 0² Then, solutions of the Schrödinger equation can be labeled by three quantum numbers n,1,m and written as (r,0,0) = R(r)Y (8,4) and E= En. Here, n, l, m are integers: n = 1,2,, l= 0,1,n-1, and m= -1,-1+1,-1,1. The normalized radial wave functions Ri(r) are written as R₁0(r) = (-1)³ 2₂exp(-7). R20 = Roule)-((--(-). 1 r = (--)²2/600 exp(-20). 2√6a R21(r): Y(0,0)=- 3 8n Y(0,4)= (r.0,0)=E(r.0.0). 1 using Bohr radius ao. The normalized angular wave functions Y (8,4) are written as 1 Y(0,4)= 4π 3 8π exp 3 411 -sinfexp(id), cose, 200 (5) 8 Y₁¹(0,0) = -sinfexp(-io). Answer the following questions. You may use the following integration formula for a > 0 and N = 0, 1, 2,...;
*x*exp(-x) dx = N! aN+1¹ (1) Obtain E₁. (2) Show that a = m₂e² Aneh² (3) Obtain the expectation value of r for the n = 1 state. (4) Obtain the value of r where the probability of finding the electron in the n = 1 state reaches a maximum. How many times larger is this value than the value obtained in (3)? (10) Next, uniform electric field E₂ is applied to the hydrogen atom in the positive z direction. The effect is treated as perturbation by assuming that the electric field is sufficiently weak. (5) Obtain the energy of the n = 1 state to the first order perturbation. (6) At E₂ = 0, 4 states with n= 2 are degenerate. Consider that the perturbed Hamiltonian is represented by a 4x4 matrix using 4 states with n = 2 as bases. Show that (R₂0Yolz R20Y). (R21Y|2|R21Y), and (R2oYolz/R21Y+¹) are 0. Here, m and m' are independent integers m = -1,0,1 and m' = -1,0,1. Note that it is enough to consider only angular parts of the wave function. X (7) Obtain (R₂0Yolz R21Y), and represent the perturbed Hamiltonian as a 4x4 matrix using 4 states with n = 2 as bases. (8) Obtain energies for the n = 2 states to the first order perturbation. Z (r,0,0)