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Unformatted text preview: ACM 95b/100b Solutions for Problem Set 5 Sean Mauch firstname.lastname@example.org March 1, 2002 1 Problem 1 Problem. In the quantum mechanical theory of a particle in a central force field V ( r ), it is necessary to solve the time-independent Schr odinger equation- ~ 2 2 m 2 + V ( r ) = E where is the wavefunction, E is the particle energy and ~ is Planks constant. When suitable boundary conditions are specified, and E can be viewed as an eigenvector and eigenvalue re- spectively, of the Schr odinger equation. If V is a function only of r (e.g. the hydrogen atom; V ( r ) =- e 2 /r ), then in spherical polar coordinates ( r,, ), where r is radius, ; 0 is a polar angle and ; 0 2 is an azimuthal angle, a solution can be sought using separation of variables (ACM95c), in which ( r,, ) is written in the form ( r,, ) = R ( r )( )( ) where R ( r ), ( ) and ( ) are functions to be determined. When this form is substituted into the Schr odinger equation, using the proper form for 2 in spherical coordinates, the result is separate ODEs for R ( r ), ( ) and ( ) 1 r 2 d d r r 2 d R d r = r 2 + 2 m ~ 2 ( V ( r )- E ) R ( r ) sin d d sin d d = m 2- sin 2 d 2 d 2 =- m 2 where m = 0 , 1 , 2 ,... and is a separation constant which appears in the separation-of-variables analysis. has the physical interpretation that ~ 2 are the allowable values of the square of the particle angular momentum. a. Show that with the transformation x = cos , y ( x ) = ( ) , and when m = 0, the ODE for ( ) can be written as ( 1- x 2 ) d 2 y d x 2- 2 x d y d x + y = 0 . This called Legendres differential equation. 1 b. Show that this equation does not change its form under the transformation x - x [This shows that for the hydrogen atom problem we need only seek solutions which are odd or even functions of x . Since x - x implies - , these solutions are either symmetric or axisymmetric about the plane = / 2.] c. Show that Legendres equation has an ordinary point at x = 0 and regular singular points at x = 1. d. Using a power series near x = 0 of the form y ( x ) = X k =0 a k x k show that the a k obey the recursion relationship a k +2 = k ( k + 1)- ( k + 1)( k + 2) a k e. If the series does not terminate at some finite k , show that it must diverge at either x = 1 or at x =- 1. f. Hence show that must satisfy = l ( l + 1) , l = 0 , 1 , 2 , 3 ... [This leads to the famous law that the eigenvalues of the square of the angular momentum must be 0 , 2 ~ 2 , 6 ~ 2 , 12 ~ 2 ... ] g. Show that with the choice a 6 = 0, a 1 = 0, these eigenvalues lead to solutions which are a set of even polynomials in x and with the choice a = 0, a 1 6 = 0, they lead to solutions which are a set of odd polynomials in x [These are the Legendre Polynomials P n ( x )]....
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This note was uploaded on 01/08/2011 for the course ACM 95b taught by Professor Nilesa.pierce during the Winter '09 term at Caltech.
- Winter '09