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Recent theoretical and experimental results demonstrate a close connectionbetween the super Tonks-Girardeau (sTG) gas and a 1D hard sphere Bose (HSB) gaswith hard sphere diameter nearly equal to the 1D scattering length $a_{1D}$ ofthe sTG gas, a highly excited gas-like state with nodes only at interparticleseparations $|x_{j\ell}|=x_{node}\approx a_{1D}$. It is shown herein that whenthe coupling constant $g_B$ in the Lieb-Liniger interaction$g_B\delta(x_{j\ell})$ is negative and $|x_{12}|\ge x_{node}$, the sTG and HSBwave functions for $N=2$ particles are not merely similar, but identical; theonly difference between the sTG and HSB wave functions is that the sTG wavefunction allows a small penetration into the region $|x_{12}|<x_{node}$,whereas for a HSB gas with hard sphere diameter $a_{h.s.}=x_{node}$, the HSBwave function vanishes when all $|x_{12}|<a_{h.s.}$. Arguments are givensuggesting that the same theorem holds also for $N>2$. The sTG and HSB wavefunctions for N=2 are given exactly in terms of a parabolic cylinder function,and for $N\ge 2$, $x_{node}$ is given accurately by a simple parabola. Themetastability of the sTG phase generated by a sudden change of the couplingconstant from large positive to large negative values is explained in terms ofthe very small overlap between the ground state of the Tonks-Girardeau gas andcollapsed cluster states.

Wave functions of the super Tonks-Girardeau gas and the trapped 1D hard sphere Bose gas