hw05_solutions.pdf - Physics 5110 Spring 2020 Homework#5 Due in class Friday February 14th 1 As discussed in class solutions to the Schr\u00a8odinger

hw05_solutions.pdf - Physics 5110 Spring 2020 Homework#5...

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Physics 5110, Spring 2020 Homework #5 Due in class Friday February 14 th 1. As discussed in class, solutions to the Schr¨ odinger Equation for central potentials can be separated into the form Ψ( r, θ, φ ) = R nl ( r ) Y lm ( θ, φ ) where Y lm are the spherical harmonics and R nl satisfies the equation d 2 R nl dr 2 + 2 r dR nl dr + bracketleftbigg 2 m planckover2pi1 2 ( E nl - V ( r )) - l ( l + 1) r 2 bracketrightbigg R nl = 0 In the case of the infinite square well potential V ( r ) = 0 ( r < a ) = ( r > a ) for r < a the radial equation reduces to the Spherical Bessel Equation , whose solutions are the Spherical Bessel Functions R nl ( r ) = j l parenleftBigg radicalbigg 2 mE nl planckover2pi1 2 r parenrightBigg The energy levels E nl are determined by finding the zeros of the j l for a given square well radius a . (a) Plot functions j 0 ( x ) through j 5 ( x ). Determine the first three zeros of each function to three significant figures. (b) Use these zeros to determine the energy levels of nucleons with energies less than or equal to that of the 3 s state. Assume m nucleon = 931 MeV/c 2 and a = 4 . 6 fm (roughly the radius of an iron nucleus). (c) Make a plot, similar to the one you made for the harmonic oscillator potential, of energy level versus angular momentum for states up to 3 s . Include a column of multiplicities. Do any of your multiplicities correspond to nuclear magic numbers? continued on the following page...
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2. Find the average time t that a nucleus will persist, if it belongs to a population obeying the exponential decay law. t
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