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 ideal
fermions. The interaction potential corresponds to -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
Strictly speaking the wave function (2.44) does not fall into the class of Bijl-Jastrow functions (2.37) as the term 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 of the exact solution [Gir60] corresponds to lowest energy of one particle of reduced mass in a box with zero boundary conditions and the goes to one in a smooth way at the maximal allowed distance .
The drift force contribution (2.39) is given by
(2.45) |
and the local energy (2.40) equals to
(2.46) |