Question

Please give solutions with detailed explanations. I appreciate neat hand writing. Thanks a lot.

3. (a) Show that [e-21/a (1 + 2r/a + 2r2/a?)] = -4r²e-20/a. (b) Use this result to show that the ground state wavefunction of the hydrogen atom is properly normalized: S• P(r) dr = 1. In astronomy, analysis of the photons emitted from very hot plasmas (such as are ob- served at the surface of stars, or during supernova explosions) commonly reveals spec- tral lines from highly ionized iron (see e.g. the paper by Decaux et al., which is available on the Problem Set 9 Assignment web-page). (c) Treating 25-times ionized iron as a hydrogenic atom to which the usual Schrödinger [4] equation applies, calculate: i. the most-probable distance to find the electron from the nucleus (hint: see sample problem 39.06 in H&R); ii. the energy of photons emitted for a transition between the 2s and is states. Are these photons in the infra-red, visible, ultra-violet, x-ray or 7-ray part of the spec- trum? (d) If you were to treat the kinetic energy of the electron, in the 1s state, as classical (i.e. [2] K =(half of the binding energy), and K = mv2) what would be the speed of the electron? Should we be using special relativity to describe the electron in the ground state of 25-times ionized iron? (e) If we were to use special relativity, one obvious thing we would have to do (there are [1] others) is to replace m with ym, where y=1/11 - 02/c2. Would this change mean that the electron is closer on average, or farther, from the nucleus, than we found above?

Answer 1

9a*(1+84 +132)] - * (2 +13) **(1+2+ 3 *** (379-A--) -- nem 6) Hydrogen ground state care function . 1 as Bohr rative t = e la The probability distribution function to) - LUIS SH2 2* sie do dy The normalisation (ground state is shesically symetric Ya dr. sinodo=2 I df = 217 2T Ta3 -- HE*(1+2+28) : :- {* (148*2

we have her an zu* (1+24+*%) = 0, hay altopical rule. Štai pag ds. 07(2* (****) Hence ground state of hydrogen atom is properly nomat sed to one. (c) Most probable radius (7) ร6 To=% ***** #6372) (25-3)** O 8=80 : - → 1-170 : 7 = 2 The most probable radius of hydrogen's ground state = a, Bohr radius.

menores The Bohir vadive is 0.529 Ao (ii) we have energy of hylogen atom is En = - 13-6 eV. for is, n-i, F, = -136 eV for as, m=2, F2 = -13€ 73.4 eV Chorag in Ka>79 , SE - hv=15-Fl - | -3644186 = 10.2 ev:

but frequency 2=% hc = DE h = 4.136x16 els Ca 3x105 nls - DE -158 hc = 4.136X10 X 3x10 DE 10.2 = 1.2165x10m - 121.65 nm in the ur range in em spectrum This is