Question

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Answer 1

1st page

Radial Schrodinger equation d^2 u(r)/dr^2 + 2m/hstrok^2[E - V_eff(r)] u(r) = 0 rightarrow (1) where V_eff(r) = V(r) + hstrok^2/2m l(l + 1) and u(r) = R_nv (r) middot r for 3d state n = 3, d = 2 therefore R_32 = 2 squareroot 2/27 squareroot 5 (1/3 a_0)^3/2 (r/a_0)^2 e^- r/3 a_0 therefore u(r) = 2 squareroot 2/27 squareroot 5 (1/3 a_0)^3/2 r^/a_0^2 e^- r/3 a_0 therefore du/dr = 2 squareroot 2/27 squareroot 5 1/(3 a_0)^3/2 middot 1/a_0^2[3 r^2 - r^3/3 a_0]e^- r/3a_0 d^2 u/dr^2 = 2 squareroot 2/27 squareroot 5 (1/3 a_0)^3/2 middot 1/a_0^2 [6r - 3 r^2/3a_0 - 3 r^2/3 a_0 + r^3/9a_0^2] e^- r/a_0 therefore from (1) (6r - 2r^2/a_0 + r^3/9a_0^2) + 2m/hstrok^2(- m e^4/32 pi^2 epsilon_0^2 hstrok^2 middot 9 + e^2/4 pi epsilon_0 r - hstrok^2 6/2 m r^2]r^3 implies (6r - 6r) + r^2 (- 2/a_0 + 2m e^2/4 pi epsilon_0 hstrok^2) + r^3[1/9 a_0^2 - 2 m^2 e^4/9(32 pi^2 epsilon_0^2 hstrok^4)].

2nd page