Discussion11 - (a) Rewrite the radial wavefunction as R(r)...

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Physics 486 Fall 2007 Discussion Problems 11 1) Superposition of spherical harmonics. Consider a particle with the following wavefunction: ( ) ( ) 2 2 2 2 / ( , , ) x y z a x y z N x y z e ψ - + + = + + , where N is a normalization constant. (a) Rewrite the wavefunction in terms of spherical co-ordinates. (b) Express the angular dependence in terms of (as a superposition of) the appropriate spherical harmonics m l Y . (c) What are the possible results of measurements of L 2 and L z , and what are the probabilities of these measurements? (Remember that in a superposition, if the co-efficient preceding an eigenstate is complex, the probability of being in that state is proportional to the square of the magnitude of the complex number.)
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Physics 486 Fall 2007 2) Infinite spherical well. The radial part of Schrodinger’s equation for central potentials is given by 2 2 2 2 ( 1) ( ) , 2 2 R l l r R V r R ER m r mr + - - + = where the entire wavefunction in the product ψ = R(r)Y( θ , ϕ ), and l is the quantum number associated with the net angular momentum.
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Unformatted text preview: (a) Rewrite the radial wavefunction as R(r) = u(r)/r, and by substituting into the equation above, obtain a differential equation for u. What is the advantage of doing this transformation? (b) The infinite spherical well is give by the potential V(r) = 0 for r a, V(r) = for r > a. Draw the effective potential appropriate for the Schrodinger equation describing the function u for the quantum values l = 0, l = 1 and l =2. For the values l = 0 and l = 1, draw the corresponding eigenstate wavefunctions. How are these wavefunctions related to the quantum number n for the radial wavefunction? (c) Find the eigenstates and eigenenergies associated with u for l = 0 (the familiar case). Keep in mind that the actual wavefunction is R = u/r and that it should not blow up at r = 0. (The solutions for general l are known as Bessel functions See Griffiths 4.1.3)...
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Discussion11 - (a) Rewrite the radial wavefunction as R(r)...

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