soll(1) - Physics 742 Graduate Quantum Mechanics 2 Solution...

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Physics 742 – Graduate Quantum Mechanics 2 Solution Set L 1. [25] A particle of mass m scatters from a potential ( ) ( ) Vr F r a δ = so that the potential exists only at the surface of a narrow sphere of radius a (a) [4] What equation must the radial wave functions ( ) l R r satisfy? Solve this equation in the regions r < a and r > a . The radial wave functions must satisfy () () () ( ) 22 2 2 2 12 ll l dl l l l m F rR r U r k R r r a k R r rdr r r ⎡⎤ ++ =+ ⎢⎥ ⎣⎦ = Away from the point r = a , the problem is simply that of a free particle, and the solution was worked out in class. The answer is spherical Bessel functions, and take the form ( ) ( ) l Aj kr Bn kr r a Rr Cj kr Dn kr r a < = > The constants will generally be different in the different regions. (b) [6] Apply appropriate boundary at r = 0. Deduce appropriate boundary conditions at r = a . We want the radial function to be well-behaved at r = 0, which implies we only want the well-behaved j l ( r ). Hence we demand B = 0. At the boundary r = a , we must take our radial Schrödinger equation and integrate it across the boundary.
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soll(1) - Physics 742 Graduate Quantum Mechanics 2 Solution...

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