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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Physics 498/MMA Handout 5 Fall 2007 Mathematical Methods in Physics I http://w3.physics.uiuc.edu/ 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 ² 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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This homework help was uploaded on 01/29/2008 for the course PHYS 598 taught by Professor Stone during the Fall '07 term at University of Illinois at Urbana–Champaign.

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online