Question

please Solve part D and E!!!!! PLEASE AND THANK YOU

acc1 Our discussion of the Bohr model of the hydrogen atom was non-relativistic throughout, which was justified because the velocity of the electron in the nth state of Bohr's hydrogen atom was v= (1) 1377 where a = 1 is the fine-structure constant, and qe is the electron charge, ħ is (the reduced) Planck's constant, and c is the speed of light. Clearly, as n grows, the speed does become less relativistic, but if we increase the nuclear charge (Z) then the electron orbits faster and at some point relativistic effects will become important (or, if you prefer, if we make careful enough measurements, then the small relativistic effects should produce some small corrections. This quiz will investigate those relativistic corrections. 1. We now take the energy of the electron in the hydrogen atom to be given by the relativistic formula mc 1 4 = mc- e, 1 - 02/02 4TETE? 41EOT 192 E =- where the first term is the rest-mass energy and the kinetic energy, and the second term is the electrostatic potential energy between the electron and proton. The electrostatic potential energy doesn't change because the proton is taken to be standing still, and so is described by Coulomb's law. (a) Because the electron has a stable circular orbit, the electric force is providing the centripetal force. Starting with the non-relativistic expression for the centripetal force, replace the non-relativistic momentum, mv, with the relativistic expression, ymu, and set it equal to the electric force from Coulomb's law. Show that you get mu? =12 mo4160 (3) (b) Now recall Bohr's quantization condition, such that the orbital angular momen- tum, L, is an integer, n, multiple of Planck's constant, L = nh. (4) Show, using the relativistic expression for the momentum in Eq. (4) and also Eq. (3) that the velocity of the electron in the relativistic case is not changed! That is, show that you still get Eq. (1). (c) Determine the new Bohr radii of the atom, In. Show that you get (5) where ab = 4mment is the Bohr radius, and m is the electron mass. (d) Find the new energies of the relativistic atom, and show that you get En=me/1- (e) Finally, show that, since a/n <1, the energy becomes En – meæ -_mge Ry 87760ħ-n2 7 2 (7) where Ry is the Rydberg of energy. That is, show that we get the correct Bohr energy in the first-order approximation. Note: the algebra is quite a bit nicer if you define e? = , and express your answers in terms of e?!

Answer 1

E = mc (1 ) from bionomeal expansion ? 0(4) terms of order 4 order Now, & assumed n so o(4) and higher to to be zero terms can be sol En mê - mer 202 Now, a saate fine es stouchire constant defined as thes ) so, Enrme =- me? 901 tror 2nt 160 m9e1 -- Ray Ryle I (Standard definition, car check