Scattering on a hard-rod potential

The hard-rod potential is a one-dimension version of the hard core potential, which in $3D$ correspond to a hard sphere (1.48). The HR potential is defined by its radius $\vert a_{1D}\vert$

$$V^{HR}(z)= \left\{ \begin{array}{cc} +\infty, & \vert z\vert<\vert a_{1D}\vert\\ 0,& \vert z\vert\ge \vert a_{1D}\vert \end{array}\right.$$ (1.79)

The scattering phase in the solution (1.62) is fixed by the condition that the function vanishes at the HR radius $\Delta = -k\vert a_{1D}\vert-\pi/2$. From (1.63) immediately follows that the radius defined as (1.79) coincides with the value of the one dimensional scattering length. Again, as in Sec. 1.3.2.2 we have a hard core potential, for which its radius, the scattering length and the range of the potential are completely the same.

The scattering solution on a hard rod potential reads as

$$f(z) = \sin(k(\vert z\vert-\vert a_{1D}\vert))$$ (1.80)