Lieb–Liniger model: Difference between revisions
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Given <math>N</math> [[Boson|bosons]] moving in one-dimension on the <math> x </math>-axis defined from <math>[0,L]</math> with [[periodic boundary conditions]], a state of the ''N''-body system must be described by a many-body [[wave function]] <math>\psi(x_1, x_2, \dots, x_j, \dots,x_N)</math>. The [[Hamiltonian (quantum mechanics)|Hamiltonian]], of this model is introduced as |
Given <math>N</math> [[Boson|bosons]] moving in one-dimension on the <math> x </math>-axis defined from <math>[0,L]</math> with [[periodic boundary conditions]], a state of the ''N''-body system must be described by a many-body [[wave function]] <math>\psi(x_1, x_2, \dots, x_j, \dots,x_N)</math>. The [[Hamiltonian (quantum mechanics)|Hamiltonian]], of this model is introduced as |
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: <math> H = -\sum_{i=1}^N \frac{\partial^2}{\partial x_i^2} + 2c \sum_{i= |
: <math> H = -\sum_{i=1}^N \frac{\partial^2}{\partial x_i^2} + 2c \sum_{i=1}^N\sum_{j>i}^N \delta(x_i-x_j)\ , </math> |
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where <math>\delta </math> is the [[Dirac delta function]]. The constant <math>c</math> denotes the strength of the interaction, <math>c>0</math> represents a repulsive interaction and <math>c<0</math> an attractive interaction.<ref name=":1">{{Cite book |last=Eckle |first=Hans-Peter |url=https://books.google.com/books?id=g0KjDwAAQBAJ&dq=lieb+liniger+model&pg=PA549 |title=Models of Quantum Matter: A First Course on Integrability and the Bethe Ansatz |date=2019-07-29 |publisher=Oxford University Press |isbn=978-0-19-166804-3 |language=en}}</ref> The hard core limit <math>c\to\infty</math> is known as the [[Tonks–Girardeau gas]].<ref name=":1" /> |
where <math>\delta </math> is the [[Dirac delta function]]. The constant <math>c</math> denotes the strength of the interaction, <math>c>0</math> represents a repulsive interaction and <math>c<0</math> an attractive interaction.<ref name=":1">{{Cite book |last=Eckle |first=Hans-Peter |url=https://books.google.com/books?id=g0KjDwAAQBAJ&dq=lieb+liniger+model&pg=PA549 |title=Models of Quantum Matter: A First Course on Integrability and the Bethe Ansatz |date=2019-07-29 |publisher=Oxford University Press |isbn=978-0-19-166804-3 |language=en}}</ref> The hard core limit <math>c\to\infty</math> is known as the [[Tonks–Girardeau gas]].<ref name=":1" /> |
Latest revision as of 15:06, 1 July 2024
In physics, the Lieb–Liniger model describes a gas of particles moving in one dimension and satisfying Bose–Einstein statistics. More specifically, it describes a one dimensional Bose gas with Dirac delta interactions. t is named after Elliott H. Lieb and Werner Liniger who introduced the model in 1963.[1] The model was developed to compare and test Nikolay Bogolyubov's theory of a weakly interaction Bose gas.[2]
Definition
[edit]Given bosons moving in one-dimension on the -axis defined from with periodic boundary conditions, a state of the N-body system must be described by a many-body wave function . The Hamiltonian, of this model is introduced as
where is the Dirac delta function. The constant denotes the strength of the interaction, represents a repulsive interaction and an attractive interaction.[3] The hard core limit is known as the Tonks–Girardeau gas.[3]
For a collection of bosons, the wave function is unchanged under permutation of any two particles (permutation symmetry), i.e., for all and satisfies for all .
The delta function in the Hamiltonian gives rise to a boundary condition when two coordinates, say and are equal; this condition is that as , the derivative satisfies
- .
Solution
[edit]![](http://upload.wikimedia.org/wikipedia/en/f/f3/Lieb-liniger-1.gif)
The time-independent Schrödinger equation , is solved by explicit construction of . Since is symmetric it is completely determined by its values in the simplex , defined by the condition that .
The solution can be written in the form of a Bethe ansatz as[2]
- ,
with wave vectors , where the sum is over all permutations, , of the integers , and maps to . The coefficients , as well as the 's are determined by the condition , and this leads to a total energy
- ,
with the amplitudes given by
These equations determine in terms of the 's. These lead to equations:[2]
where are integers when is odd and, when is even, they take values . For the ground state the 's satisfy
Thermodynamic limit
[edit]![]() | This section needs expansion. You can help by adding to it. (June 2024) |
References
[edit]- ^ a b Elliott H. Lieb and Werner Liniger, Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State, Physical Review 130: 1605–1616, 1963
- ^ a b c Lieb, Elliott (2008). "Lieb-Liniger model of a Bose Gas". Scholarpedia. 3 (12): 8712. doi:10.4249/scholarpedia.8712. ISSN 1941-6016.
- ^ a b Eckle, Hans-Peter (29 July 2019). Models of Quantum Matter: A First Course on Integrability and the Bethe Ansatz. Oxford University Press. ISBN 978-0-19-166804-3.
- ^ Dorlas, Teunis C. (1993). "Orthogonality and Completeness of the Bethe Ansatz Eigenstates of the nonlinear Schrödinger model". Communications in Mathematical Physics. 154 (2): 347–376. Bibcode:1993CMaPh.154..347D. doi:10.1007/BF02097001. S2CID 122730941.