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The ideas of noncommutative space originate from the research on time-space coordinate on an extremely small scale. Subsequently, the noncommutative space has gradually attracted some attention. The researchers started to explore noncommutative effect in some other fields. With the establishment of noncommutative quantum mechanics, it becomes significant to explore the noncommutative effect of exactly solvable models. The kinds of harmonic oscillators are very important and fundamental models in physics. But in noncommutative phase space, coordinate and coordinate are noncommutative, and momentum and momentum are also noncommutative. These results in the difficulty in obtaining the energy spectra of oscillators systems. In this paper the quantum properties of a two-dimensional coupling harmonic oscillator in noncommutative phase space are studied. Firstly, the Hamiltonian of the system is constructed, which includes all possible coupling types, namely, coordinate-coordinate coupling, momentum-momentum coupling, and coordinate-momentum cross-coupling. Secondly, the explicit expression of the noncommutative energy spectrum for the Hamiltonian is obtained by using the invariant eigen-operator method. In this work it is shown explicitly that the changes in the energy levels are related to the noncommutative parameters and coupling parameters. Thirdly, the effects of coupling parameters and non-commutative parameters on the energy spectra are analyzed in detail in the form of graphs. The results show that the energy levels under the influence of non-commutative parameters are non-degenerated. As the values of non-commutation parameters
$ \theta $ and$ \phi $ increase, some energy levels increase and tend to change linearly, and other energy levels first decrease and then increase. If the limit values of the non-commutative parameters are taken as follows:$ \theta \to 0 $ and$ \phi \to 0 $ , then the noncommutative energy spectra will be consistent with the energy spectra of the two-dimensional harmonic oscillator in the commutative space in general. On the other hand, the energy levels will split under the influence of coupling parameters. Moreover, the degree to which the energy levels split can increase as the kinds of couplings in the system increase. It is found that the coordinate coupling parameter$ \eta $ and the momentum coupling parameter$ \sigma $ have the same influence on the energy levels, but the coordinate momentum cross-coupling parameter$ \kappa $ has less influence on the energy levels than$ \eta $ and$ \sigma $ . Overall, the above results are completely different from those of two-dimensional oscillator in the usual commutative space, which is degenerated except for the ground state.-
Keywords:
- coupling harmonic oscillator /
- non-commutative phase space /
- invariant eigen-operator method
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图 1
$ E_{{n_1}, {n_2}}$ 随θ的变化曲线(n1 + n2 = 2, ϕ = 1, η = 0.5, σ = 0.5, κ = 0.5)Figure 1.
$ E_{{n_1}, {n_2}}$ versus θ (n1 + n2 = 2, ϕ = 1, η = 0.5, σ = 0.5, κ = 0.5).表 1 能级
$ E_{{n_1}, {n_2}}^{(1)} $ 的值(n1+n2 = 2, η= σ = κ = 0)Table 1. Values of
$ E_{{n_1}, {n_2}}^{(1)} $ (n1+n2 = 2, η = σ = κ = 0)$ E_{2, 0}^{\left( 1 \right)} $ $ E_{1, 1}^{\left( 1 \right)} $ $ E_{0, 2}^{\left( 1 \right)} $ θ = ϕ = 0 3 3 3 θ = 0.1, ϕ = 0 3.1025 3.0025 2.9025 θ = 0, ϕ = 0.1 3.1025 3.0025 2.9025 θ = 0.1, ϕ = 0.1 3.2 3.0 2.8 
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表 2 能级
$ E_{{n_1}, {n_2}}^{(2)} $ 的值(n1+n2 = 2, θ = ϕ = 0)Table 2. Values of
$ E_{{n_1}, {n_2}}^{(2)} $ (n1+n2 = 2, θ = ϕ = 0)$ E_{2, 0}^{(2)} $ $ E_{1, 1}^{\left( 2 \right)} $ $ E_{0, 2}^{\left( 2 \right)} $ η = 0.1, σ = 0, κ = 0 3.09762 2.99749 2.89737 η = 0, σ = 0.1, κ = 0 3.09762 2.99749 2.89737 η = 0, σ = 0.1, κ = 0.1 3.0000 2.9798 2.95959 η = 0.1, σ = 0.1, κ = 0 3.2 3.0 2.8 η = 0.1, σ = 0, κ = 0.1 3.05913 2.97825 2.82038 η = 0, σ = 0.1, κ = 0.1 3.05913 2.97825 2.82038 η = 0.1, σ = 0.1, κ = 0.1 3.16333 2.98167 2.80000
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[1] Snyder H S 1947 Phys. Rev. 72 68
Google Scholar
[2] Seiberg N, Witten E 1999 J. High Energy Phys. 3 32
Google Scholar
[3] Doplicher S, Fredenhagen K, Roberts J E 1994 Phys. Lett. B. 331 39
Google Scholar
[4] Zupnik B M 2007 Class. Quantum Grav. 24 15
Google Scholar
[5] Polychronakos A P 2001 J. High Energy Phys. 6 70
Google Scholar
[6] Bolonek K, Kosinski P 2002 Phys. Lett. B. 547 51
Google Scholar
[7] Gamboa J, Loewe M, Rojas J C 2001 Phys. Rev. D. 64 067901
Google Scholar
[8] Nair V P, Polychronakos A P 2001 Phys. Lett. B. 505 267
Google Scholar
[9] Falomir H, Pisani P A G, Vegaf, Cárcamo D, Méndez F, Loewe M 2016 J. Phys. A: Math. Theor. 49 055202
Google Scholar
[10] Kupriyanov V G 2013 J. Phys. A: Math. Theor. 46 245303
Google Scholar
[11] Sinhad D, Giri P R 2011 Mod. Phys. Lett. A. 26 2213
Google Scholar
[12] Wu H, Fan H Y 2008 Commun. Theor. Phys. 50 1348
Google Scholar
[13] Fan H Y, Li C 2004 Phys. Lett. A. 321 75
Google Scholar
[14] Fan H Y, Xu X F, Li C 2004 Commun. Theor. Phys. 42 824
Google Scholar
[15] 任益充, 范洪义 2013 62 156301
Google Scholar
Ren Y C, Fan H Y 2013 Acta Phys. Sin. 62 156301
Google Scholar
[16] 张科, 范承玉, 范洪义 2018 67 170301
Google Scholar
Zhang K, Fan C Y, Fan H Y 2018 Acta Phys. Sin. 67 170301
Google Scholar
[17] Fan H Y, Wu Z 2018 Chin. Phys. B. 27 080301
Google Scholar
[18] 吴泽, 范洪义 2019 68 220301
Google Scholar
Fan H Y, Wu Z 2019 Acta Phys. Sin. 68 220301
Google Scholar
[19] Lin Q, Fan H Y 2018 Journal of University of Science and Technology of China 48 643
[20] Lin B S, Heng T Y 2011 Chin. Phys. Lett. 28 070303
Google Scholar
[21] 琚泽志, 李文波 2013 大学物理 32 35
Ju Z Z, Li W B 2013 Coll. Phys. 32 35
[22] 张秀兰, 刘恒, 余海军 2011 60 040303
Google Scholar
Zhang X L, Liu H, Yu H J 2011 Acta Phys. Sin. 60 040303
Google Scholar
[23] Han D, Kim Y, Noz M 1995 J. Math. Phys. 36 3940
Google Scholar
[24] Han D, Kim Y, Noz M 1999 Am. J. Phys. 67 61
Google Scholar
[25] Kim Y, Noz M 2007 J. Opt. B Quant. Semiclass. Opt. 7 S458
Google Scholar
[26] Jellal A, Kinani E, Schreiber M 2005 Int. J. Mod. Phys. A 20 7
Google Scholar
[27] Lin B S, Jing S C, Heng T H 2008 Mod. Phys. Lett. A. 23 445
Google Scholar
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