Question

In this optional assignment you will find the eigenfunctions and eigenenergies of the hydrogen atom using an operator method which involves using Supersymmetric Quantum Mechanics (SUSY QM). In the SUSY QM formalism, any smooth potential Vx) (or equivalently Vr)) can be rewritten in terms of a superpotential Wix)l (Based upon lecture notes for 8.05 Quantum Krishna Rajagopal at MIT Physics II as taught by Prof Recall that the Schroedinger radial equation for the radial wavefunction u(r)-r Rfr) can be rewritten as )2mrEr) We will nondimensionalize it and rewrite the Hamiltonian in the following dimensions: nfo":景+V1(r) where v,(r)sid+1)-- A± = + w(r) where w(r) is a smooth function known as the superpotential. Id We define the two ladder operators dr2 dr (a) Show that two Hamiltonians constructed from the above operators will have the following form: d2 d2 (b) Show that if Hp a, then the state A-p, is an eigenstate of H with the same eigenvalue E. (Hint: Operate A on the term Hh from the left and regroup.) The only exception to the above is the possibility that A-P.-0. (corresponding to a zero energy eigenstate which we will call фо- (c) Show that this implies Po expl-GdyW(y)] Similarly, a zero energy eigenstate of H (2) obeys A+Vo = 0 . (d) Show that this implies o expl dy wo)]

Answer 1

dr 2 dr2 s, A. it wave funchoms of Ht ih same uig envalue c)the only excep Hon is A.фо-o tinee E2.0

L+1 2 Lt, 2一つ d L41 充 사! ︵ 2(tI) dr2 4) 4)2

し41 Or %1 (r) integrating OL or

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