5 And then, go ahead, we can obtain ¯ P ( t ) B = ¯ P ( t ) $^{F}$, where P = − i ℏ N ∑ i =1 ∂ ∂x$_{i}$ . (26) IV. SUMMARY AND CONCLUSIONS FIG. 4: Density-plots of the pair distribution function D ( z$_{1}$,z$_{2}$ ). Abscissa is z$_{1}$ , Ordinate is z$_{2}$ . M = 7. (a) N = 2; (b) N = 3; (c) N = 5; (d) N = 7. On solving the eigenfunction of the TG gas in the ex- ternal magnetic field, we found that the TG system with large B and ω has a bigger probability in the ground state than the TG gas raised by Girardeau in 1960 when the number of bosons N is commensurate with the num- ber of potential cycles M . It reveals that we can realize ground-stated TG system more easier. And then we have studied properties of magnetized TG gas and Fermi gas in an external magnetic field. comparing with each other when they are in the “same” state reveals that it is im- possible to distinguish them just from their average value of particle’s coordinate and potential energy, system’s otal momentum, single-particle density and pair distri- bution function. But we could distinguish them from their reduced single-particle matrices or their momentum distributions. In order to illustrate our results, we have plotted their reduced single-particle matrices and the mo- mentum distributions when they are in the ground state. Although their momentum distribution functions are dif- ferent, their total momentum are of the same. These results will be deeply helpful for understanding the TG gas. know that the Fourier transform of the original function and the transformed function is one-to-one, as the differ- ence of the reduced density matrix ρ ( x, x $^{′}$) of the hard- core Bose gas and Fermion gas, the momentum distri- bution is surely not the same. Yuan Lin and Biao Wu [10] have plotted the momentum distributions for both the TG gas and the free Fermi gas in a periodic Kronig- Penney potential, which shows the difference — Even a free Fermi gas has a broader momentum distribution than the most strongly interacting boson gas. We would like to add that, although their momentum distribution functions are different, their total momentum are of the same. With the equation This work has been supported by the National Natural Science Foundation of China under Grant No. 10975125. Acknowledgment we can prove ( ∂ ∂x$_{1}$ + ∂ ∂x$_{2}$ )sgn( x$_{2}$ − x$_{1}$ ) = 0 , (24) N ∑ i =1 ∂ ∂x$_{i}$ A ( x$_{1}$ , · · · , x$_{N}$ ) = 0 . (25) Phys. Rev. A 66 , 023606 (2002). [1] L. Tonks, Phys. Rev. 50 , 955 (1936). [2] M. Girardeau, J. Math. Phys. 1 , 516 (1960). [3] M. D, Girardeau, Phys. Rev. 139 , B500 (1965). [4] E. Lieb and W. Liniger, Phys. Rev. 130 , 1605 (1963). [5] A. Lenard, J. Math. Phys. 5 . 930 (1964). [6] L. Pitaevskii and S. Stringari, Bose-Einstein Condensa- tion (Clarendon, Oxford, 2003). [7] J. O. Andersen, Rev. Mod. Phys. 76 . 599 (2004). [8] G. J. Lapeyre, Jr., M. D. Girardeau, and E. M. Wright., [9] A. del Campo and J. G. Muga, Europhys. Lett. 74 , 965 (2006). [10] Yuan Lin and Biao Wu, Phys. Rev. A 75 , 023613 (2007). [11] G.-D. Lin, Wei Zhang, and L.-M. Duan, Phys. Rev. A 77 , 043626 (2008). [12] V. I. Yukalov and M. D. Girardeau, Laser Phys. Lett. 2, 375 (2005) [13] M. Rigol, Phys. Rev. A 72 , 063607 (2005).