Tonks-Giradeau wave function

As was first shown by Girardeau[Gir60], the wave function of the Tonks-Girardeau gas is equal to the absolute value of the wave function of $1D$ ideal fermions. The interaction potential corresponds to $\delta$-function with infinite strength, or in other words the TG model describes impenetrable particles of a zero size. The component of an exact wave function is given by

$$f_2(z) = \vert\sin(\pi z /L)\vert$$ (2.44)

Strictly speaking the wave function (2.44) does not fall into the class of Bijl-Jastrow functions (2.37) as the term $f_2(z)$ does not go to a constant even in the large-range limit, but always experience oscillations. At the same time it does not cause problems in our calculations as it turns out that the scattering energy ${\cal E}= \pi^2\hbar^2/mL^2$ of the exact solution [Gir60] corresponds to lowest energy of one particle of reduced mass in a box with zero boundary conditions and the $f_2(z)$ goes to one in a smooth way at the maximal allowed distance $z = L/2$.

The drift force contribution (2.39) is given by

$${{\cal F}_2}(z) = \sqrt{{\cal E}} \mathop{\rm ctg}\nolimits \sqrt{{\cal E}}z$$ (2.45)

and the $1D$ local energy (2.40) equals to

$${{\cal E}_2}_2(z) = {\cal E}(1+\mathop{\rm ctg}\nolimits ^2\sqrt{{\cal E}}z)$$ (2.46)