Question

. Problem! #1: A hydrogen atom waveunction is given by Calculate the probability of finding the atom in the 1s state Useful expressions: n! ed

Answer 1

Calculation of the probability of finding the atom in the 1^st state: - rightwardsdoublearrow P(r = a_0, 0 lessthanorequalto theta lessthanorequalto pi, 0 lessthanorequalto phi lessthanorequalto 2 pi) = integral_0^a_0 integra_0^a_0 integral_0^pi integral_0^2 pi Psi* Psi r sin theta dr d theta d phi rightwardsdoublearrow Given that, Wave function is the function of r only: So Integration becomes rightwardsdoublearrow P = 4 pi integral_0^a_0 r^2 Psi* (r, theta, phi) Psi (r, theta, phi) dr = 4 pi integral_0^a_0 r^2 (2/45 pi) (1/a_0)^3 (r/a_0)^4 e^-2r/a_0 dr = 8/45 a_0 integral_0^a_0 r^6 e^-2r/a_0 dr rightwardsdoublearrow By solving the above equation integral_0^a_0 r^6 e^-2r/a_0 dr rightwardsdoublearrow Lets u = r/a dr = a du x^6 = a^6 u^6 = integral a^4 u^6 e^-24 du \r\n rightwardsdoublearrow a^y [-u^6 e^-2u/2 -integral - 3 u^5 e^-2u du] rightwardsdoublearrow a^y [u^6 e^-2u/2 - 3 u^5 e^-2u/2 - 15 u^4 e^-2u/2 - 15 u^3 e^-2u/2 - 15 u^3 e^-2u/2 - 15 u^3 e^-2u/2 - 45 u^2 e^-2u/4 - 45 u e^-2u/4 - 45 e^-2u/8]_limits