General approach

In this section we will formulate the generic scattering problem in a three-dimensional space. At low density the interaction between particles of a gas is well described by binary collisions.

Consider a collision of two particles having coordinates $r_1$ and $r_2$, masses $m_1$ and $m_2$. The aim is to find the stationary solution $f_{12}({\vec r_1},{\vec r_2})$ of the Schrödinger equation:

$$\left(-\frac{\hbar^2}{2m_1}\Delta_{{\vec r_1}}-\frac{\hbar^2}{2m_... ...ight)f_{12}({\vec r_1},{\vec r_2}) = {\cal E}_{12}f_{12}({\vec r_1},{\vec r_2})$$ (1.34)

In absence of an external confinement the problem is translationary invariant and the center of mass moves with a constant velocity. The solution of problem gets separated in the center of the mass frame. The Schrödinger equation for the movement of the center of mass ${\vec{\cal R}}=(m_1{\vec r_1}+m_2{\vec r_2})/M$ is trivial:

$$-\frac{\hbar^2}{2M}\Delta_{\vec{\cal R}}f_{\vec{\cal R}}({\vec{\cal R}})= {\cal E}_{\vec{\cal R}}f_{\vec{\cal R}}({\vec{\cal R}}),$$ (1.35)

here $M=m_1+m_2$ is the total mass. The solution of Eq. 1.35 is a free wave^1.3 $f_{\vec{\cal R}}({\vec{\cal R}}) = \exp\{i\vec k_{\vec{\cal R}}{\vec{\cal R}}\}$ with $\vec k_{\vec{\cal R}}$ being the initial wavenumber of the system and ${\cal E}_{\vec{\cal R}}=\hbar^2k_{\vec{\cal R}}^2/2M$.

The equation for the relative coordinate $\r={\vec r_1}-{\vec r_2}$ involves the interaction potential:

$$\left(-\frac{\hbar^2}{2\mu}\Delta_{\r}+V_{int}(\vert\r\vert)\right) f(\r)= {\cal E}f(\r),$$ (1.36)

where

$$\mu = \frac{m_1m_2}{m_1+m_2}$$ (1.37)

is the reduced mass. Once solutions of Eqs. 1.35-1.36 are known the solution of the scattering problem (1.34) is given by

$$\left\{ \begin{array}{cll} f_{12}({\vec r_1},{\vec r_2}) &=&f_{\v... ...}) f(\r)\\ {\cal E}_{12}&=&{\cal E}_{\vec{\cal R}}+{\cal E} \end{array}\right.$$ (1.38)

In order to proceed further we will assume that the energy of the incident particle $E$ is small and the solution has a spherical symmetry $f(\r) = f(\vert\r\vert)\equiv f(r)$. In this case the Laplacian gets simplified $\Delta = \frac{\partial^2}{\partial r^2}+\frac{2}{r}\frac{\partial}{\partial r}$ and the Eq. 1.36 is conveniently rewritten by introducing function $g(r)$

$$u(r) = \frac{f(r)}{r}$$ (1.39)

$$u(0) = 0$$ (1.40)

in such a way that its form reminds a one-dimensional Schrödinger equation:

$$-\frac{\hbar^2}{2\mu}u''(r)+V_{int}(r)u(r) = {\cal E}u(r)$$ (1.41)

The solution of this equation in a general form can be written as

$$u(r) = \sin(kr+\delta(k)),$$ (1.42)

where

$$\hbar k=\sqrt{2mE}$$ (1.43)

is the momentum of the incident particle and $\delta(k)$ is the scattering phase.

The scattering at low energy (which describes well a binary collisions in a dilute gas) has a special interest as it becomes universal and can be described in terms of one parameter, the $s$-wave scattering length $a_{3D}$:

$$a_{3D} = -\lim\limits_{k\to 0}\frac{\delta(k)}{k}$$ (1.44)

In the asymptotic limit of slow particles $k\to 0$ the scattering solution (1.42) can be expanded

$$f(r)\to const \left(1-\frac{a_{3D}}{r}\right)$$ (1.45)

and has the node at a distance equal to $a_{3D}$. It gives an equivalent definition of the three-dimensional scattering length as a position of the first node of the positive energy scattered solution in the low-momentum limit.

In the next several sections we will solve the problem of the scattering on a hard-sphere potential (1.3.2.2) and a soft-sphere potential (1.3.2.3). We will find explicit expressions for the scattered functions, which are of a great importance, as in many cases can provide a physical insight into properties of a many body problem. Indeed, at a certain conditions the correlation functions can be related to the scattered function $f(r)$. Another point is that the two-body Bijl-Jastrow term $f_2(r)$ (2.37) in the construction of the trial wave function is very often taken in a form of $f(r)$. Thus such calculations are very important for the implementation of the Quantum Monte Carlo methods.

We will also find expressions for the scattering length $a_{3D}$ in terms of the height (or depth) of the potential $V_0$:

$$V_0 = \max\limits_{r}\vert V_{int}(r)\vert$$ (1.46)

and the range of the potential $R$, which in this Dissertation will be understood as a characteristic distance on which the potential acts. In other words the potential can be neglected for distances much larger than $R$:

$$R = \min\limits_{r} \{V(\vert r\vert) \approx 0\}$$ (1.47)