Abstract
With Girardeau's Fermi–Bose mapping, we have constructed the eigenstates of the Tonks–Girardeau (TG) gas in an external magnetic field. When the number of bosons N is commensurate with the number of potential cycles M, the probability of this TG gas in the ground state is bigger than the TG gas raised by Girardeau in 1960. Through the comparison of properties between this TG gas and Fermi gas, we find that the following issues are always the same: their average value of particle coordinates and potential energies, system total momenta, single-particle densities and pair distribution functions. But the reduced single-particle density matrices and their momentum distributions between them are different.
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