Question

-15 61. The radius of a proton is about Ro 10 m. The probability that the hydrogen-atom electron is inside the proton is Ro P = | P(r) dr where P(r) is the radial probability density. Calculate this probability for the ground-state of hydrogen. Hint: Show that e /1 for r S Ro is valid for this calculation.

Answer 1

The probability of ground state of hydrogen is P = integral^P_0_0 P (r) dr = integral^P_0_0 Vertical-bar 2/a_0^3/2 e^r/a_0 Vertical-bar ^2 r^2 dr = integral^P_0_0 Vertical-bar 2/a_0^3/2 e^-r/a_0 Vertical-bar ^2 r^2 dr = 4/a^3_0 integral^P_0_0 e^-2 r/a_0 e^-2 r/a_0 r^2 dr = 4/a^3_0 (1/4 a_0 (a^2_0 - e^-2 R_0/a_0 (2 a_0 R_0 + a^2_0 + 2 R^2_0))