Zhao Liang Wang ∗ and An Min Wang † Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui, China. With Girardeau’s Fermi-Bose mapping, we have constructed the eigenstates of a 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 of the same: their average value of particle’s coordinate and potential energy, system’s total momentum, single-particle density and the pair distribution function. But the reduced single-particle matrices and their momentum distributions between them are different. PACS numbers: 05.30.Jp, 03.65.Ge momentum, single-particle density and the pair distri- bution function. But both their reduced single-particle matrices and momentum distributions are different. The problem of a one-dimensional (1D) hard-core gas was first studied classically by Tonks [1]. Girardeau [2, 3] continued the research in quantum, and put forward the Fermi-Bose mapping to solve energy spectrum and the corresponding wave function of a one dimensional Tonks- Girardeau (TG) gas. After that, Lieb and Liniger [4] considered N Bose particles interacting via a repulsive δ -function potential with a coupling constant γ . In fact, TG gas can be considered as the Lieb-Liniger gas when γ → ∞ . During the past decade, the study on TG gas has un- dergone a rapid development in theoretical and experi- mental [5–17]. In particular, A. Lenard [5] indicated that the off-diagonal parts of the one-body density matrix and the momentum distribution show distinct differences in both the TG gas and the Fermi gas. G. J. Lapeyre et al. [8] studied the momentum distribution of a harmon- ically trapped gas. The most important progress is the realization of the strongly interacting bosons [16] in a one-dimensional optical-lattice trap and the TG gas [17] by trapping $^{87}$Rb atoms by trapping them with a combi- nation of two light traps. Such an interparticle interaction could be represented by the following subsidiary condition on the wave func- tion ψ : In order to let the TG gas wave function, which must be symmetric with respect to the permutations of any x$_{i}$ and x$_{j}$ , satisfy this condition (2), we need the ideas of Fermi-Bose mapping proposed by Girardeau [2, 3]. That is, using the fact that the Fermi wave function, denoted by ψ $^{F}$, satisfies condition (2) naturally and is antisym- metric, a bosonic wave function ψ B of a TG gas can be constructed by We have known some properties of one dimensional TG gas, and we are interested in its properties and fea- tures in external magnetic field. Consequently, we first reveal some properties of both gases in the same exter- nal magnetic field and compared them. With the Fermi- Bose mapping [2, 3], we obtained all the eigenenergies and eigenstates of this TG gas. When the number of bosons N is commensurate with the number of potential cycles M , the TG system in an external magnetic field will be in the ground state with a bigger probability than the TG gas raised by Girardeau in 1960. Many properties of the TG gas and the Fermi gas are always of the same even if it’s time-dependent, such as their average value of particle’s coordinate and potential energy, system’s total $^{∗}$Electronic address: wzlcxl@mail.ustc.edu.cn $^{†}$Electronic address: anmwang@ustc.edu.cn in which and ψ $^{B}$( x$_{1}$, · · · , x$_{N}$, t ) = A ( x$_{1}$, · · · , x$_{N}$ ) ψ $^{F}$( x$_{1}$, · · · , x$_{N}$, t ) (3) A ( x$_{1}$, · · · , x$_{N}$ ) ≡ N ∏ i>j sgn( x$_{i}$ − x$_{j}$ ) , (4) sgn( x ) ≡ x | x | = { 1 , x > 0 , − 1 , x < 0 . ψ ( x$_{1}$, · · · , x$_{N}$ , t ) = 0 if x$_{i}$ = x$_{j}$, 1 ⩽ i < j ⩽ N. (2) U ( x$_{i}$, x$_{j}$ ) = { 0 , x$_{i}$ = x$_{j}$, ∞ , x$_{i}$ = x$_{j}$. (1) II. TG GAS IN AN EXTERNAL MAGNETIC FIELD I. INTRODUCTION In a TG gas, the boson is assumed to have an “impen- etrable” hard core characterized by a radius of a . From Girardeau’s work, with the hard core radius a → 0, the interparticle interaction is given by One Dimensional Magnetized TG Gas Properties in an External Magnetic Field arXiv:1001.0116v1 [quant-ph] 31 Dec 2009