Green’s Function for a Neutron in Interaction with a Straight Current Carrying Wire

We construct an integral representation for the momentum space Green’s function for a Neutron in interaction with a straight current carrying wire. PACS numbers: 11.30.Pb, 03.65.Fd, 11.10.Ef ∗This research was supported in part by the Brazilian Research Agencies CNPq and Finep. 1 Consider an electrically neutral spin 1/2 particle of mass M and a magnetic moment μ~σ (a Neutron) in interaction with an infinite straight wire carrying a current I and located along the z-axis. The magnetic field generated by the wire is given by (we use units with c = 1 = h̄) ~ B = 2I (−y, x, 0) (x2 + y2)3/2 (1) where x, y are cartesian coordinates in the plane perpendicular to the wire. The Hamiltonian of the particle is given by H = ~ p 2 2m + μ~σ. ~ B. (2) The motion along the z-axis is free and will be ignored in the following. Thus we get a two-dimensional problem with H = ~ p 2 2m + 2Iμ (−yσ1 + xσ2) (x2 + y2)3/2 . (3) The problem formulated above has a dynamical symmetry [1]. From the form of the Hamiltonian there is one obvious constant of motion J3 = xpy − ypx + σ3 2 . (4) Also, the ‘Runge-Lenz vector’ defined by Ai = εijxj σ1x2 − σ2x1 + 1 4IμM [pi, J3]+, (i = 1, 2) (5) satisfies the following commutation rules [J3, Ai] = i εij Aj, [Ai, H ] = 0, [Ai, Aj] = −i H J3 2M (2IμM)2 . (6) Thus for negative energy (−E) eigenstates of H we may put Ji = Ai { (2IμM) −2M E }1/2 , (i = 1, 2) (7)