Question

Problem 2. Being good sports let us consider the familiar (although mysterious!) hydrogen atom. The excited state wavefunction corresponding to a hydrogenic 2s orbital is given by where the Bohr radius ao 52.9 pm -1 (a) Find the normalized wavefunction. (b) Estimate the probability that an electron is in a volume t1.0 pm at the nucleus (r 0). (c) Estimate the probability that an electron is in a volume t -10 pm3 in an arbitrary direction at the Bohr radius (ra (d) Estimate the probability that an electron is in a volume t-10 pm3 at infinity (r

Answer 1

after solving this we get p (x) = -21/16 e + 8/16 p (r) = -0.1679 + 0.50 p (r) = 0.3320 p (r) = 33.20% b for r rightarrow infinity p (r) = integral^infinity_0 1 + 1^2 4 pi r^2 dr = 1/16 a^3_0 1/4 pi integral^infinity_0 (4 + r^2/a^2_0 - 4r/a_0) e^-r/a_0 times 4 pi r^2 dr = 1/16 a^3_0 integral^infinity_0 [4 r^2 e^-r/a_0 dr + r^4/a^2_0 + r^4/a^2_0 e^-r/a_0 dr - 4 r^3/a_0 e^-r/a_0] = 1/16 a^3_0 [4 squareroot 3 a^3_0 + 1/a^2_0 squareroot 5 a^3_0 - 4/a_0 squareroot 4 a^4_0] = 1/16 a^3_0 [4 times 2 times 1 a^3_0 + 4 times 3 times 2 a^5_0/a^2_0 - 4 times 3 times 2 a^4_0/a_0] = 8 a^3_0/16 a^3_0 p (r) = 50% \r\n Probability in q volume T = 1.0 pm^3 at r = 0 p (r) dr = integral^r_0 4 pi r^2 Vertical-bar psi Vertical-bar ^2 dr Here integral will go for 0 rightarrow 0 so p (r) dr = 0 probability =