Question

Expectation values. Calculate the expectation value of the distance of an electron in a hydrogen atom from its nucleus when the electron is in its ground state. Let the wave function of the electron be: 1/2 rao) exp(-r/a.) where: ao is a constant 0.529 A, and r is the separation of the point of observation from the point nucleus. Hint: to solve this problem, remember that the "expectation integral" is done over the volume of all space! So you must write the volume infinitesimal as you would for any problem in spherical coordinates! That is, don't forget the Ф and coordinates!

Answer 1

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Given that psi_1 0 = (1/(pi a_0^3)^1/2 e^-r/a_0 Now, is r is the distance then < r > = integral^infinity_0 integral^pi_0 integral^2 pi_0 r Vertical-bar psi_1 0 Vertical-bar ^2 r^2 sin theta d theta d phi dr = integral^infinity_0 r^3 Vertical-bar psi_2 0 Vertical-bar ^2 dr integral^pi_0 sin theta d theta integral^2 pi_0 d phi = 4 pi/pi a_0^3 integral^infinity_0 r^3 e^-2r/a_0 dr = 4 pi/pi a_0^3 [e^-2r/a_0 (-a_0 r^3/2 - 3a_0^2 r^2/4 - 3a_0^3 r/4 - 3a_0^4/8] = 4/a_0^3 [3a_0^4/8] = 3a_0/2 < r > = 3a_0/2.