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Trial wavefunction

The trial wavefunction for the pure system was constructed in chapter 3.2.1. In this section the same approach will be extended to systems in the presence of quenched impurities. The wave function of the system is chosen as the product of one-body and two-body wavefunctions.
$\displaystyle \psi_T(\vec r_1,...,\vec r_N) =
\prod\limits_{i=\overline{1,N} \a...
...{1,N} \atop j=\overline{1,N}_{imp}} f_{PI}(\vert\vec r_i-\vec r^{~imp}_j\vert),$     (4.1)

where $f_{PP}$ stands for the particle-particle wavefunction, which has already been obtained in section 3.2.1 and is defined by (3.37), (3.41) and (3.42). In eq. (4.1) $f_{PI}$ describes the effect of the impurities on each particle. To construct $f_{PI}$ we use a similar procedure as for $f_{PP}$, i.e. we solve the particle-impurity Schrödinger equation
$\displaystyle \left(-\frac{\hbar^2}{2m}\triangle+V_{PI}(\vec r\,) \right)f = {\cal E} f,$     (4.2)

where the reduced particle-impurity mass is equal to the mass of a particle, because the quenched impurity is infinitely massive Let us look for the symmetric solution in spherical coordinates
$\displaystyle -\frac{\hbar^2}{2m} \left(f''+\frac{2}{r}f'\right) + V_{PI}(r)f = {\cal E}f$     (4.3)

The particle is modeled by a hard sphere of diameter $a$ and the impurity by a hard sphere of diameter $2b-a$. The particle-impurity interaction potential is
$\displaystyle V_{PI}(r) =
\left\{
{\begin{array}{ll}
+\infty,&\vert r\vert \le b\\
0,&\vert r\vert > b\\
\end{array}}
\right.$     (4.4)

where $b$ is the particle-particle $s$-wave scattering length The dimensionless Schrödinger equation has form (lengths in units of $a$ and energies in units of $\hbar^2/(2ma^2)$)
$\displaystyle \left\{
{\begin{array}{ll}
\displaystyle f(x) = 0,& \vert x\vert ...
...aystyle f''+\frac{2}{x}f' - E f = 0,&\vert x\vert > b/a\\
\end{array}}
\right.$     (4.5)

and the differential equation which has to be solved is
$\displaystyle \left\{
{\begin{array}{l}
\displaystyle f''+\frac{2}{x}f' - E f = 0\\
f\left(\frac{b}{a}\right) = 0
\end{array}}
\right.$     (4.6)

The solution is $f(x) = A\sin(\sqrt{E}(x-b/a))\,/x$, with $A$ being an arbitrary constant. Let us construct the particle-impurity wave function $f_{PI}$ in the same way as it was done for the particle-particle wavefunction. We introduce a matching point $R_{PI}$ and choose
$\displaystyle f_{PI}(x) =
\left\{
{\begin{array}{ll}
\displaystyle \frac{A}{x} ...
...p\left(-\frac{x}{\alpha}\right),& \vert x\vert > R_{PI}\\
\end{array}}
\right.$     (4.7)

The function $f_{PI}$ must be smooth at the matching point. The request of continuity for $f_{PI}$, its derivative $f_{PI}'$ and the local energy $E_{L}(R) = -(f_{PI}''(R)-2f_{PI}'(R)\,/R)\,/\,f_{PI}(R)$ is fulfilled
$\displaystyle \left\{
{\begin{array}{lll}
A&=&\displaystyle\frac{R_{PI}}{\sin(u...
...1}{u}\mathop{\rm atan}\nolimits \frac{u(\xi-2)}{u^2+\xi-2}
\end{array}}
\right.$     (4.8)


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Next: Average over disorder Up: Dilute Bose gas with Previous: Introduction   Contents