PHYS 598 hw5 - 1 Missing State In Homework Set 4 you found...

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Physics 498/MMA Handout 5 Fall 2007 Mathematical Methods in Physics I m-stone5 Prof. M. Stone 2117 ESB University of Illinois 1) Missing State : In Homework Set 4 you found that the Schr¨ odinger equation - d 2 dx 2 - 2 sech 2 x ψ = E ψ has eigensolutions ψ k ( x ) = e ikx ( - ik + tanh x ) with eigenvalue E = k 2 . For x large and positive ψ k ( x ) A e ikx e ( k ) , while for x large and negative ψ k ( x ) A e ikx e - ( k ) , the (complex) constant A being the same in both cases. Express δ ( k ) as the inverse tangent of an algebraic expression in k . Impose periodic boundary conditions ψ ( - L/ 2) = ψ (+ L/ 2) where L 1. Find the allowed values of k and hence an explicit expression for the k -space density, ρ ( k ) = dn dk , of the eigenstates. Compare your formula for ρ ( k ) with the corresponding expression, ρ 0 ( k ) = L/ 2 π , for the eigenstate density of the zero-potential equation and compute the integral Δ N = Z -∞ { ρ ( k ) - ρ 0 ( k ) } dk.
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