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# sol5 - ACM 95b/100b Solutions for Problem Set 5 Sean Mauch...

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ACM 95b/100b Solutions for Problem Set 5 Sean Mauch [email protected] 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 ) ψ = where ψ is the wavefunction, E is the particle energy and ~ is Plank’s constant. When suitable boundary conditions are specified, ψ 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, θ, φ ), where r is radius, θ ; 0 θ π is a polar angle and φ ; 0 φ 2 π is an azimuthal angle, a solution can be sought using separation of variables (ACM95c), in which ψ ( r, θ, φ ) is written in the form ψ ( r, θ, φ ) = R ( r )Θ( θ )Φ( φ ) where R ( r ), Θ( θ ) and Φ( φ ) 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 ), Θ( θ ) and Φ( φ ) 1 r 2 d d r r 2 d R d r = λ r 2 + 2 m ~ 2 ( V ( r ) - E ) R ( r ) sin θ d d θ sin θ d θ = m 2 - λ sin 2 θ Θ d 2 Φ d φ 2 = - m 2 Φ where m = 0 , ± 1 , ± 2 , ... and λ is a ‘separation constant’ which appears in the separation-of-variables analysis. λ has the physical interpretation that λ ~ 2 are the allowable values of the square of the particle angular momentum. a. Show that with the transformation x = cos θ, y ( x ) = Θ( θ ) , and when m = 0, the ODE for Θ( θ ) can be written as ( 1 - x 2 ) d 2 y d x 2 - 2 x d y d x + λy = 0 . This called Legendre’s differential equation. 1

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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 θ π - θ , these solutions are either symmetric or axisymmetric about the plane θ = π/ 2.] c. Show that Legendre’s 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) - λ ( 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 λ must satisfy λ = 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 0 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 = 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 )]. h. Show that the first four P n ( x ), ordered by their degree and normalized so that P n ( x = 1) = 1, are P 0 = 1 P 1 = x P 2 = 1 2 ( 3 x 2 - 1 ) P 3 = 1 2 ( 5 x 3 - 3 x ) 2 i. Verify that these satisfy Rodrigue’s formula P n ( x ) = 1 2 n n !
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sol5 - ACM 95b/100b Solutions for Problem Set 5 Sean Mauch...

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