Fund Quantum Mechanics Lect & HW Solutions 59

Fund Quantum Mechanics Lect & HW Solutions 59 -...

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Unformatted text preview: 4.1. ANGULAR MOMENTUM 41 4.1.3.1 Solution anguc-a Question: The general wave function of a state with azimuthal quantum number l and magnetic quantum number m is = R ( r ) Y m l ( , ), where R ( r ) is some further arbitrary function of r . Show that the condition for this wave function to be normalized, so that the total probability of finding the particle integrated over all possible positions is one, is that integraldisplay r =0 R ( r ) R ( r ) r 2 d r = 1 . Answer: You need to have ( | ) = integraltext d 3 vectorr = 1 for the wave function to be normalized. Now the volume element d 3 vectorr is in spherical coordinates given by r 2 sin d r d d , so you must have integraldisplay r =0 integraldisplay =0 integraldisplay 2 =0 R ( r ) Y m l ( , ) R ( r ) Y m l ( , ) r 2 sin d r d d = 1 . Taking this apart into two separate integrals: integraldisplay r =0 R ( r ) R ( r ) r 2 d r integraldisplay...
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This note was uploaded on 01/06/2012 for the course PHY 3604 taught by Professor Dr.danielarenas during the Fall '11 term at UNF.

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