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QM scattering - Section 19.5 in Shankar

  1. Apr 13, 2004 #1
    The question is:

    Show that the s-wave phase shift for a square well of depth V0 and range r0 is

    δ0 = -k r0 + tan-1{(k/k') tan(k'r0)}

    where k' and k are the wave numbers inside and outside the well. For k small, kr0 is some small number and we ignore it. Let us see what happens to δ0 as we vary the depth of the well, i.e., change k'. Show that whenever k' ~ k'n = (2n+1)π/2r0, δ0 takes on the resonant form Eq. (19.5.30) with Γ/2 = (hbar)2kn/μr0, where kn is the value of k when k' = k'n. Starting with a well that is too shallow to have any bound state, show k'1 corresponds to the well developing its first bound state, at zero energy. (See Exercise 12.6.9.) (Note: A zero-energy bound state corresponds to k = 0.) As the well is deepended further, this level moves down, and soon, at k'2, another zero-energy bound state is formed, and so on.

    Supporting items:

    Eq. (19.5.30):
    δl = δb + tan-1{(Γ/2)(E0-E)}

    Excercise 12.6.9:
    Show that the quantization condition for l = 0 bound states in a spherical well of depth -V0 and radius r0 is

    k'/κ = -tan(k'r0)

    where k' is the wave number inside the well and iκ is the complex wave number for the exponential tail outside. Show that there are no bound states for V0 < π2(hbar)2/8μr02.

    Can anyone please tell me what this question is asking. I don't have a clue.
  2. jcsd
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