Question

Please answer all parts of the question, show all work, and box your final answers. Thank you. Please do the best you can, this is all the information that is provided.

In this problem you will use first-order perturbation theory to determine the energy shift in the hydrogen ground state due to the finite size of the proton. (a) (6 pts) Write down the spatial wavefunction for the 1S state (ground state) of hydrogen. (b) (6 pts) Assume that the proton is a uniformly charged sphere of radius rp and total charge e. Then the Coulomb potential experienced by the electron in hydrogen is given by: (e2/4160 2re (3- (-)) (151) U(r) - { (e? / 41€0) (r > rp) r Write down the perturbation term for this problem. Note that this is the deviation of U (r) from the case when the proton is considered a point charge. (c) (8 pts) Calculate the energy shift for the 1S state of hydrogen due to the finite proton radius. Assume that the proton is a uniformly charged sphere with radius rp = 0.9 fm. Use first-order perturbation theory. Also note that rp < do, in order to simplify your calculation.

Answer 1

F Y 4 / 4 (ē906) (3 God 289 g poston is 280 redp ener _28/00 éter e 0 (350 I da Vp 2 L КТС 28/ =x 263 da 2r - 2 23(-e (a) Spatial wave for IS stali is - Hao Vita Where as is bohr radius U(v)- rco e т тр TITE them perturbing term due to the finite size of the Eye(3-6) (6.) Now the shift due to the finite- size of the proton és <41U (r<&p) (4) ชร 4th Trao 81 Corp Letx= rp e 13%8 fp r=az 2 zrelas 9 4 =xile 132 auz lyre 1689 ale az dz Now ->re then fiont'lermis neglected w.rut the selcond lerm Z e 2 - 2 ² - 42 -126 3283 -242-25 as re <<d then we neglect first four term and write 280/60 24xao leglo 32003 ?Xalq to flere a=.53 then we write 3 ê cu 8ITfor e=1.6810c N-m c 8 4.516410 T do p hodz. leon 28e/ao 3 Z -3 ifre 3286 aot 28e/a ха, ? & 24 dz 3286 6 da o Coo Golgo 40 8p=0.9fm 2 - =t ./8 46=9x105 =