In this optional assignment you will find the eigenfunctions andeigenenergies of the hydrogen atom using an operator method which involves using Supersymmetric Quantum Mechanics (SUSY QM). In the SUSY QM formalism, any smooth potential V(x) (or equivalently V(r) ) can be rewritten in terms of a superpotential W(x). (Based upon lecture notes for 8.05 Quantum Physics II as taught by Prof. Krishna Rajagopal at MIT.Recall that the Schroedinger radial equation for the radial wavefunction u(r)=r R(r) can be rewritten asħ� �� �(� + �)ħ�[−����� +�(�)+ ���� ]�(�)=��(�)We will nondimensionalize it and rewrite the Hamiltonian in the following dimensions: ����� = − �� + � (�) where � (�) = �(�+�) − � We define the two ladder operators � ��� � � �� � �± = ∓ � + �(�) where �(�) is a smooth function known as the superpotential. ��(a) ShowthattwoHamiltoniansconstructedfromtheaboveoperatorswillhavethefollowing form:���(�) ≡�+�− =−��� +��(�)−�′(�)���(�) ≡�−�+ =−��� +��(�)+�′(�)(b) Show that if �(�)� = �(�)� , then the state � � is an eigenstate of �(�) with the same ��� −�eigenvalue �(�). (Hint: Operate � on the term �(�)� from the left and regroup.) �−�The only exception to the above is the possibility that �−�� = �. (Corresponding to a zero energy eigenstate which we will call �� .(c ) Show that this implies �� = ���[− ∫� �� �(�)] �Similarly, a zero energy eigenstate of �(�) obeys �+�� = � .(d) Show that this implies �� = ���[∫� �� �(�)] � Now we see that if �� is normalizable, �� is not, and vice versa. Hence only one of the two Hamiltonians (�(�) or �(�) ) could have a zero energy eigenvalue.Hence, the spectra of �(�) and �(�) are identical and the corresponding eigenstates ��and �� are related via �� = �−�� and �� = �+��, except for possible zero eigenstates,which will be simple functions of the superpotential �(�).( e ) Show that using the superpotential ��(�) = − �+� + � implies the following set ladder operators� �(�+�)��± =∓ � −�+�+ ��� � �(�+�) ( f ) Show that using the above set of ladder operators yields the following set of partner Hamiltonians:��(�)=� � ==�����+ �� �+ �− � �(�+�)� and��(�)=� � =�����+ � where�����=−�� +�(�) where�(�)=�(�+�)−�� �−�+ �+� �(�+�)� � ��� �(g) Show that ��−���(�) = � implies ���(�) ∝ ��+��� � [− � ] �(�+�)(h) Since � (�) is a zero energy eigenfunction of ��(�), show this implies � �� � ���= −�� � Identifying�=�+�impliestheenergygivenvalues� ∝−�.And� (�)∝�����[−�],where ������ ��� is nondimensionalized in units of the Bohr radius, as is the case for the Hydrogen atom! Science Physics

# In this optional assignment you will find the eigenfunctions andeigenenergies of

by Panel | Oct 22, 2022 | Uncategorized