Legendre配置谱方法求解Bose-Einstein凝聚态的基态解

刘文杰; 王汉权

doi:10.21656/1000-0887.430257

Legendre配置谱方法求解Bose-Einstein凝聚态的基态解

doi: 10.21656/1000-0887.430257

刘文杰^1,,
王汉权^{1, 2, ,}

1.
云南财经大学统计与数学学院, 昆明 650221
2.
云南师范大学数学学院, 昆明 650504

基金项目:

国家自然科学基金项目 11871418

国家自然科学基金项目 11971120

云南基础研究计划重点项目 202101AS070044

云南省教育厅科研基金项目 2022Y547

详细信息

作者简介:
刘文杰(1996—)，男，硕士生(E-mail: 1455696828@qq.com)

通讯作者:
王汉权(1974—)，男，教授，博士(通讯作者. E-mail: hanquan.wang@gmail.com)

中图分类号: O242
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- 文章访问数: 197
- HTML全文浏览量: 86
- PDF下载量: 40
- 被引次数: 0
出版历程
- 收稿日期: 2022-08-10
- 修回日期: 2023-05-18
- 刊出日期: 2023-06-01

The Legendre Collocation Spectral Method for the Ground State Solutions of the Bose-Einstein Condensates

LIU Wenjie^1
,,
WANG Hanquan^{1, 2
, ,}

1.
School of Mathematics and Statistics, Yunnan University of Finance and Economics, Kunming 650221, P.R.China
2.
School of Mathematics, Yunnan Normal University, Kunming 650504, P.R.China

摘要

摘要: 近年来, 有关Bose-Einstein凝聚态基态解的实验研究已经取得了一系列重要的成果. 该文在相关研究成果的基础上, 首先通过降维和无量纲化方法将Bose-Einstein凝聚态基态解问题转换成能量泛函极值问题, 在离散该泛函时, 尝试使用Legendre配置谱方法离散该能量泛函的一维和二维情形. 其次, 对该能量泛函极小值问题进行了数值模拟. 最后，通过分析实验数据结果和图像得出，针对非旋转的Bose-Einstein凝聚态的基态解问题可以使用Legendre配置谱方法来求解, 且数值结果的误差较小.
- Bose-Einstein凝聚态 /
- Legendre配置谱方法 /
- 数值计算
Abstract: In recent years, a series of important achievements have been made in the experimental study of the ground state solutions of the Bose Einstein condensates. First, the ground state solution problem of the Bose Einstein condensate was converted into the extreme value problem of energy functional with the dimensionless method. In the discretization of the energy functional, the Legendre collocation spectral method was used in the 1D and 2D cases. Second, the energy functional minimum problem was numerically simulated. The analyses of the experimental data and graphs show that, the Legendre collocation spectral method is applicable to the ground state solution of the non-rotating Bose Einstein condensate, and the errors of the numerical results are very small.
- Bose-Einstein condensate /
- Legendre collocation spectral method /
- numerical calculation

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图 1 一维(左图)、二维(右图)情况下, 改变N的大小, 误差ε的变化情况

Figure 1. Changes of error ε with N in the 1D case (left) and the 2D case (right)

下载: 全尺寸图片幻灯片

图 2 β=5, 10, 100, 1 000时, 一维Bose-Einstein凝聚态的基态解ϕ_g(x)

Figure 2. Ground state solution ϕ_g(x) of the 1D Bose-Einstein condensates for β=5, 10, 100, 1 000

下载: 全尺寸图片幻灯片

图 3 β=5, 10, 100, 1 000时的ϕ_g(x, y)

Figure 3. The ϕ_g(x, y) graphs for β=5, 10, 100, 1 000

下载: 全尺寸图片幻灯片

表 1 一维情况下, 改变N的大小, 误差ε的变化情况

Table 1. In the 1D case, changes of error ε with N

N	10	20	30	40	50
error ε	6.46E-1	1.79E-1	4.67E-2	1.10E-2	1.10E-3

下载: 导出CSV

表 2 二维情况下, 改变N的大小, 误差ε的变化情况

Table 2. In the 2D case, changes of error ε with N

N	10	20	30	40	50
error ε	3.65E-1	2.23E-2	5.38E-4	9.31E-7	1.22E-9

下载: 导出CSV

表 3 一维情况下Legendre配置谱方法的能量值E_β(ϕ_g)和Fourier谱方法的能量值E_β(ϕ_g^FP)与β之间的变化情况

Table 3. Energy value E_β(ϕ_g) of the Legendre collocation spectrum method and energy value E_β(ϕ_g^FP) of the Fourier spectrum method in the 1D case, changing with β

β	0	5	10	100	1 000
E_β(ϕ_g)	0.499 9	1.316 0	1.947 2	8.508 5	39.322 4
E_β(ϕ_g^FP)	0.500 0	1.316 1	1.947 1	8.508 5	39.322 4

下载: 导出CSV

表 4 二维情况下Legendre配置谱方法的能量值E_β(ϕ_g)和Fourier谱方法的能量值E_β(ϕ_g^FP)与β之间的变化情况

Table 4. Energy value E_β(ϕ_g) of the Legendre collocation spectrum method and energy value E_β(ϕ_g^FP) of the Fourier spectrum method in the 2D case, changing with β

β	0	5	10	100	1 000
E_β(ϕ_g)	1	1.437 1	1.611 7	3.983 6	12.001
E_β(ϕ_g^FP)	0.968 7	1.501 1	1.697 1	4.004 0	12.001

下载: 导出CSV

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