solutionhw5 - Solutions to Homework Set 5 1 Missing state...

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Solutions to Homework Set 5 1) Missing state : The continuum eigenfunctions are are ψ k = ( - ik + tanh x ) e ikx , so ψ k ( x ) = braceleftBigg ( k - i ) e ikx /i x lessmuch 0, ( k + i ) e - ikx /i x greatermuch 0. It now helps to draw a phasor diagram δ k Re Im i from which we see that δ ( k ) = tan - 1 (1 /k ) and A = - i 1 + k 2 . The periodic boundary conditions rquire Ae - e - ikL/ 2 = Ae e ikL/ 2 , so 2 πn = 2 δ ( k ) + kL. The density of states is therefore given by ρ ( k ) = dn dk = 1 2 π parenleftBigg L + 2 ∂δ ∂k parenrightBigg . The free density of states is ρ 0 = L/ 2 π , so N = integraldisplay -∞ { ρ ( k ) - ρ 0 ( k ) } dk = integraldisplay -∞ braceleftBigg 2 ∂δ ∂k bracerightBigg dk 2 π = 1 π [ δ ( k )] -∞ . Now when k → -∞ we have δ + π , and when k + we have δ 0. Thus N = - 1 , and there is one fewer state in the continuum than there was when there was no potential. The lowest energy continuum state was peeled away from the others, and has become localized as the bound state ψ 0 = 1 2 sech x . 1
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2) Continuum completeness : For positive κ ψ 0 ( x ) = 2 κe - κx is a normalized eigenstate of the given operator with eigenvalue E = - κ 2 . It obeys the boundary condition ψ/ψ prime | x =0 = tan θ = - κ - 1 .
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