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Homework Help: Triplet S-wave (l = 0) phase shift in n-p scattering at low energy

  1. Jun 12, 2012 #1
    1. The problem statement, all variables and given/known data

    Evaluate the triplet S-wave (l = 0) phase shift in n-p scattering at low energy assuming the interaction potential is given by a square well of depth V0 = 38.5 MeV and width b = 1.73 fm. Deduce also the value of the triplet scattering length.

    2. Relevant equations

    Normalized S-wave triplet wavefunction [itex]\varphi[/itex] = 1/√(4∏)×u(r)/r×|S=1>|T=0>

    where u(r) = A1sinKr and K2 = M(E-V(r))/hbar2 for r < b
    and u(r) = A2sin(kr + δ) and k2 = ME/hbar2

    and M = mass of proton = mass of neutron and δ = phase shift

    V(r) = -V0 - V0([itex]\tau[/itex]1[itex]\bullet[/itex][itex]\tau[/itex]2) for r < b
    and V(r) = 0 for r > b

    and [itex]\tau[/itex]1[itex]\bullet[/itex][itex]\tau[/itex]2 is the isospin operator that distinguishes between the singlet |T=0> and triplet |T=1> isospin components of the neutron-proton wavefunction and have eigenvalues:

    [itex]\tau[/itex]1[itex]\bullet[/itex][itex]\tau[/itex]2|T=0> = -3|T=0>

    [itex]\tau[/itex]1[itex]\bullet[/itex][itex]\tau[/itex]2|T=1> = 1|T=1>

    Matching condition for radial components of wave function at r = b:

    KcotKb = kcot(kb + δ)

    3. The attempt at a solution

    I can't use the matching condition directly to give me the triplet phase shift because that hasn't worked. So I got a hint from my lecturer to include the isopin component of the triplet wave function and solve the schrodinger equation for that. When I apply the potential operator V(r) = -V0 - V0([itex]\tau[/itex]1[itex]\bullet[/itex][itex]\tau[/itex]2) to the wave function however it gives me the dispersion relation

    K2 = M(E-2V0)/hbar2 which when I take the limit E --> 0 has a negative number under the square root, so I can't even use it in the matching condition and solve for δ.

    I am really stumped on this one and would greatly appreciate any help.

  2. jcsd
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