Download Quantum Mechanics Exam 2: Solutions for Problems 1-4 - Prof. Roland Allen and more Exams Quantum Mechanics in PDF only on Docsity! Physics 606, Quantum Mechanics, Exam 2 NAME___________________________________ Please show all your work. (You are graded on your work, with partial credit where it is deserved.) All problems are, of course, nonrelativistic. __________________________________________________________ 1. In this problem, start with the following two equations for the scattering of a plane wave, which we derived in class: ! k ! r ! ( ) = eik " !" "r " !" + e ikr r f k ! ',k ! ( ) f k ! ',k ! ( ) = ! m 2""2 d 3# r 'e!ik # !# '$r # !# ' V r ! '( )% k! r ! '( ) (a) (10) In the first Born approximation, and for a central potential (i.e. V r ! ( )= V r( ) ), obtain a simple integral expression for f in the form f !( ) = constant " dr 0 # $ (function of q and r) V r( ) where q = q ! , q ! = k ! '! k ! , and ! is the angle between k ! and k ! ' (b) (10) Let us model a target by V r( ) = !V 0 for r < R V r( ) = 0 for r > R with V 0 > 0 . Calculate f !( ) as a function of qR (writing your final answer in a simple form). (c) (i) (3) Now obtain the differential cross section d! d" as a function of qR . (ii) (3) Solve for q in terms of k and sin ! / 2( ) . Then roughly sketch a graph of d! d" as a function of sin ! / 2( ) for fixed k and R . (iii) (2) Roughly sketch a graph of d! d" as a function of ! (again for fixed k ), labeling ! = " / 2 , ! = " , etc. (iv) (2) Explain qualitatively how you could use this graph of d! d" as a function of ! to estimate the size R of the target. 2. (20) The proton in a hydrogen atom is not really a point charge. To roughly estimate what might be the effect of its finite size, let us model it by a hollow shell of charge of radius R = 10 !5 a 0 where a 0 is the Bohr radius. Then the potential energy of an electron is V r( ) = !k e 2 R for r < R V r( ) = !k e 2 r for r > R. (In our CGS units, k = 1 .) For the 1s state (ground state) of hydrogen, with wavefunction 1 4! 2 a 0 3/2 e "r /a 0 , calculate the change !E 1s in the energy due to the finite size of the proton in this model. You may make the approximation R! a 0 where appropriate. Give your answer in terms of a 0 and e . Also give a roughly approximate numerical answer, in eV.
