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Unformatted text preview: Chem 3890 Physical Chemistry I Fall 2009 Chemistry 3890 Problem Set 9 Fall 2009 Due: in class, Friday, November 20 Last revised: November 13, 2009 Problem 1 Consider a free particle, mass μ , inside a spherical cavity of radius L . This could be a model for an electron inside a quantum dot or trapped inside a cavity in a fluid, or an atom rattling around inside a buckyball molecule. The particle moves in 3D subject to the potential V ( r ) = braceleftBigg ≤ r ≤ L, ∞ r > L. (1) 1. Setting ψ nlscriptm = R nlscript ( r ) Y lscriptm ( θ,φ ), write down the time-independent Schr¨odinger equation for the radial function R nlscript . 2. Define the function u nlscript ≡ rR nlscript ( r ). Derive an equation for u nlscript . What are the boundary conditions on u nlscript at r = 0 and r = L appropriate for a bound state? 3. Consider the case of zero angular momentum ( lscript = 0). Solve the equation for u nlscript =0 . Determine the eigenvalues E nlscript =0 and the normalized wavefunctions R nlscript =0 ( r )....
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