"Lieb-Liniger delta Bose gas"의 두 판 사이의 차이

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==introduction==
  
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* N bosons interacting on the line <math>[0,L]</math> of length L via the delta function potential
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* one-dimensional Bose gas
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* 1963 Lieb and Liniger solved by [[Bethe ansatz]]
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* In 1963, Lieb and Liniger solved exactly a one dimensional model of bosons interacting by a repulsive \delta-potential and calculated the ground state in the thermodynamic limit
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==Hamiltonian==
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*  quantum mechanical Hamiltonian
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:<math>H=-\sum_{j=1}^{N}\frac{\partial^2}{\partial x_j^2}+2c\sum_{1\leq i<j\leq N}^{N}\delta(x_i-x_j)</math>
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==wave function==
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* <math>\psi(x_1, x_2, \dots, x_j, \dots,x_N)</math>
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* <math>\psi(x_1, \dots, x_N) =  \sum_P a(P)\exp \left( i \sum_{j=1}^N k_{Pj} x_j\right)</math>
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:<math>
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a(P) = \prod\nolimits_{1\leq i<j \leq N}\left(1+\frac{ic}{k_{Pi}  -k_{Pj}}\right) \ .
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</math>
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==two-body scattering term==
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* <math>s_{ab}=k_a-k_b+ic</math>
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==Bethe-ansatz equation==
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:<math>\exp(ik_jL)=\prod_{l=1}^{N}\frac{k_j-k_l+ic}{k_j-k_l-ic}</math>
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==energy spectrum==
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* energy of a Bethe state
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:<math>E=\sum_{j=1}^{N}k_j^2</math>
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==related items==
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==computational resource==
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* [http://msstp.org/?q=node/275 Day 5 - Yang-Baxter, Delta Bosons, Contact Terms]
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** [http://msstp.org/sites/default/files/Problems4.pdf Bose-Einstein Condensation and BAE exercise .pdf]
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** [http://msstp.org/sites/default/files/problem4.nb Bose-Einstein Condensation and BAE solution .nb]
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==encyclopedia==
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* http://en.wikipedia.org/wiki/Lieb-Liniger_model
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==articles==
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* Tracy, Craig A., and Harold Widom. “On the Ground State Energy of the Delta-Function Bose Gas.” arXiv:1601.04677 [math-Ph], January 18, 2016. http://arxiv.org/abs/1601.04677.
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* Zill, J. C., T. M. Wright, K. V. Kheruntsyan, T. Gasenzer, and M. J. Davis. “A Coordinate Bethe Ansatz Approach to the Calculation of Equilibrium and Nonequilibrium Correlations of the One-Dimensional Bose Gas.” arXiv:1601.00434 [cond-Mat, Physics:hep-Th], January 4, 2016. http://arxiv.org/abs/1601.00434.
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* Veksler, Hagar, and Shmuel Fishman. “A Generalized Lieb-Liniger Model.” arXiv:1508.02011 [cond-Mat, Physics:math-Ph], August 9, 2015. http://arxiv.org/abs/1508.02011.
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* Flassig, Daniel, Andre Franca, and Alexander Pritzel. “Large-N Ground State of the Lieb-Liniger Model and Yang-Mills Theory on a Two-Sphere.” arXiv:1508.01515 [cond-Mat, Physics:hep-Th], August 6, 2015. http://arxiv.org/abs/1508.01515.
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* Dorlas, T. C. “Orthogonality and Completeness of the Bethe Ansatz Eigenstates of the Nonlinear Schroedinger Model.” Communications in Mathematical Physics 154, no. 2 (June 1, 1993): 347–76. doi:10.1007/BF02097001.
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* Yang, C. N., and C. P. Yang. “Thermodynamics of a One‐Dimensional System of Bosons with Repulsive Delta‐Function Interaction.” Journal of Mathematical Physics 10, no. 7 (July 1, 1969): 1115–22. doi:[10.1063/1.1664947 http://dx.doi.org/10.1063/1.1664947].
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* C.N. Yang [http://dx.doi.org/10.1103/PhysRevLett.19.1312 Some exact results for the many-body problem in one dimension with repulsive delta-function interaction], Phys. Rev. Lett. 19 (1967), 1312-1315
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* Elliott H. Lieb and Werner Liniger [http://link.aps.org/doi/10.1103/PhysRev.130.1605 Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State], 1963
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[[분류:integrable systems]]
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[[분류:math and physics]]
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[[분류:migrate]]
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==메타데이터==
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===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q6543926 Q6543926]
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===Spacy 패턴 목록===
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* [{'LOWER': 'lieb'}, {'OP': '*'}, {'LOWER': 'liniger'}, {'LEMMA': 'model'}]

2021년 2월 17일 (수) 03:16 기준 최신판

introduction

  • N bosons interacting on the line \([0,L]\) of length L via the delta function potential
  • one-dimensional Bose gas
  • 1963 Lieb and Liniger solved by Bethe ansatz
  • In 1963, Lieb and Liniger solved exactly a one dimensional model of bosons interacting by a repulsive \delta-potential and calculated the ground state in the thermodynamic limit


Hamiltonian

  • quantum mechanical Hamiltonian

\[H=-\sum_{j=1}^{N}\frac{\partial^2}{\partial x_j^2}+2c\sum_{1\leq i<j\leq N}^{N}\delta(x_i-x_j)\]


wave function

  • \(\psi(x_1, x_2, \dots, x_j, \dots,x_N)\)
  • \(\psi(x_1, \dots, x_N) = \sum_P a(P)\exp \left( i \sum_{j=1}^N k_{Pj} x_j\right)\)

\[ a(P) = \prod\nolimits_{1\leq i<j \leq N}\left(1+\frac{ic}{k_{Pi} -k_{Pj}}\right) \ . \]


two-body scattering term

  • \(s_{ab}=k_a-k_b+ic\)


Bethe-ansatz equation

\[\exp(ik_jL)=\prod_{l=1}^{N}\frac{k_j-k_l+ic}{k_j-k_l-ic}\]


energy spectrum

  • energy of a Bethe state

\[E=\sum_{j=1}^{N}k_j^2\]


related items

computational resource

encyclopedia


articles

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'lieb'}, {'OP': '*'}, {'LOWER': 'liniger'}, {'LEMMA': 'model'}]
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