"Lieb-Liniger delta Bose gas"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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| 1번째 줄: | 1번째 줄: | ||
==introduction== | ==introduction== | ||
| − | * N bosons interacting on | + | * N bosons interacting on the line $[0,L]$ of length L via the delta function potential<br> |
* one-dimensional Bose gas<br> | * one-dimensional Bose gas<br> | ||
* 1963 Lieb and Liniger solved by [[Bethe ansatz]]<br> | * 1963 Lieb and Liniger solved by [[Bethe ansatz]]<br> | ||
| 12번째 줄: | 12번째 줄: | ||
:<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><br> | :<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><br> | ||
| + | |||
| + | |||
| + | ==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== | ==two-body scattering term== | ||
2013년 3월 4일 (월) 16:05 판
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
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
\[E=\sum_{j=1}^{N}k_j^2\]
computational resource
encyclopedia
articles
- http://link.springer.com/article/10.1007%2FBF02097001
- C. N. Yang and C. P. Yang Thermodynamics of a One‐Dimensional System of Bosons with Repulsive Delta‐Function Interaction, J. Math. Phys. 10, 1115 (1969)
- C.N. Yang Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett. 19 (1967), 1312-1315
- Elliott H. Lieb and Werner Liniger Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State, 1963