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

1. Jun 12, 2012

### wdednam

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 $\varphi$ = 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($\tau$1$\bullet$$\tau$2) for r < b
and V(r) = 0 for r > b

and $\tau$1$\bullet$$\tau$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:

$\tau$1$\bullet$$\tau$2|T=0> = -3|T=0>

$\tau$1$\bullet$$\tau$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($\tau$1$\bullet$$\tau$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.

Thanks