General approach

We already have explored some aspects of the scattering problem in three-dimensions in Sec. 1.3.2. Here we will consider the problem of a one-dimensional scattering.

The scattering solution in a uniform system separates in center of the mass frame, as the property (1.38) is valid also in a 1D case. Thus in the following we will skip the trivial solution for the movement of the center of the mass and we will address the most interesting part due to solution for the relative coordinate $z = z_1-z_2$. The one dimensional Schrödinger equation for the relative motion is written as

$$-\frac{\hbar^2}{2\mu}f''(z) + V_{int}(z) f(z) = {\cal E}f(z),$$ (1.60)

where the reduced mass $\mu$ is given by (1.37). We will always consider scattering with a positive energy, even if the interaction potential itself might be attractive. Then the scattering energy can be written as ${\cal E} = \hbar^2 k^2/2\mu$, where $\k$ is real. The equation (1.60) becomes

$$f''(z) + \left(k^2-\frac{2\mu V_{int}(z)}{\hbar^2}\right) f(z) = 0$$ (1.61)

Its general solution can be written as^1.5

$$f(z) = \cos(k\vert z\vert+\Delta(k))$$ (1.62)

The one-dimensional scattering length is defined as the derivative of the phase $\Delta(k)$ in the limit of low-energy scattering^1.6

$$a_{1D} = -\lim\limits_{k\to 0}\frac{\partial \Delta(k)}{\partial k}$$ (1.63)