Question

The probablity finction PC the first excited Hydrogen ( 2p state... n 2) is given by: where r 0.53x10-10 m (Bohr radius) P( r)-- 24 a) Determine the tireoby ealuatings dr b) Regarding maxima and minima, deduce the most probable and least probable radial locations for the electron in this state. Interpret EACH critical point as it pertains to Bohr atomic structure and compare any results-interpretations to the Bohr Theory for the n 2 quantum state

Answer 1

Given That P(r) = r^4 e^-r/r_1/24r_1^5 where, r_1 = 0.53 times 10^-10 m (a) dP/dr = 0 Now, 1/24r_1^5 d/dr (r^4 e^-r/r_1) = 0 4r^3 e^-r/r_1 + r^4(-1/r_1)e^-r/r_1 = 0 r^3 e^-r/r_1 (4 - r/r_1) = 0 ThereFore The Three critical point's are r = 0 r = 4r_1 & r = infinity (b) dP/dr = 1/24r_1^5 [4r^3 e^-r/r_1 - r^4/r_1 e^-r/r_1] d^2 P/dr^2 = 1/24r_1^5 [4r^3(-1/r_1)e^-r/r_1 + 12r^2 e^-r/r_1 - 1/a (4r^3 e^-r/r_1 + r^4(-1/r_1) e^-r/r_1)] d^2 P/dr^2 = 1/24 r_1^5 [-8r^3/r_1 + r^4/r_1^2 + 12r^2]e^-r/r_1 r = 0, d^2 P/dr^2 = 0 \r\n r = 0 d^2 P/dr^2 = 1/24r_1^5 [-8(0)^3/r_1 + (0)^4/r_1^2 + 12(0)^2]e^-0/r_1 Therefore d^2 P/dr^2 = 0 r = 4r_1 d^2 P/dr^2 = 1/24r_1^5 [-8(4r_1)^3/r_1 + (4r_1)^4/r_1^2 + 12(4r_1)^2]e^-r_1/r_1 d^2 P/dr^2 = -8/3r_1^3 e^-1 < 0 r = infinity d^2 P/dr^2 > 0 There Fore, r = 4r_1 is The maximum hence it is The most probable radial location r rightarrow infinity is