solp - Physics 741 Graduate Quantum Mechanics 1 Solution...

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Physics 741 – Graduate Quantum Mechanics 1 Solution Set P 1. [30] In class we solved the hydrogen atom, which is a spherical 1/ r potential. Consider another spherically symmetric potential, namely, the spherical harmonic oscillator 2 22 1 Hm m ω =+ P Q This potential is most easily solved by separation of variables, but it is very helpful to take advantage of the spherical symmetry to find solutions. (a) [3] Factor eigenstates of this Hamiltonian into the form () ( ) ( ) ,, , m l rR r Y ψ θφ = . Find a differential equation satisfied by the radial wave function R r . We are attempting to find eigenstates of the Hamiltonian, that is, solutions of HE = . Since we have spherical symmetry, we expect their angular dependence to take the form of spherical harmonics, so that ( ) ( ) ( ) , m l r Y = . Plugging into Schrödinger’s equation, we see from the lecture notes (8.45) that we have () () () 2 2 2 2 2 2 1 , 2 21 dl l ERr rRr Rr VrRr m r dr r mE d l l m rdr r ⎧⎫ + =− + ⎡⎤ ⎨⎬ ⎣⎦ ⎩⎭ + + + = == (b) [5] At large r , which term besides the derivative term dominates? Show that for large r , we can satisfy the differential equation if ( ) 2 1 2 exp R rA r , and determine the factor A that will make this work. For Hydrogen, the potential term vanished at infinity, but in this case, the potential becomes large at infinity, and cannot be ignored. The leading terms will then be the derivative term acting both times on R ( r ) and the potential term, so we are approximately trying to satisfy the equation 2 2 dm R rr R r dr =
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solp - Physics 741 Graduate Quantum Mechanics 1 Solution...

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