Introduction

A model of a gas of particles moving in one dimension and satisfying Bose–Einstein statistics was introduced in 1963 ^[1][2] in order to study whether the available approximate theories of such gases, specifically Bogoliubov's theory, would conform to the actual properties of the model gas. The model is based on a well defined Schrödinger Hamiltonian for particles interacting with each other via a two-body potential, and all the eigenfunctions and eigenvalues of this Hamiltonian can, in principle, be calculated exactly. Sometimes it is called one dimensional Bose gas with delta interaction. It also can be considered as quantum non-linear Schrödinger equation.

The ground state as well as the low-lying excited states were computed and found to be in agreement with Bogoliubov's theory when the potential is small, except for the fact that there are actually two types of elementary excitations instead of one, as predicted by Bogoliubov's and other theories.

The model seemed to be only of academic interest until, with the sophisticated experimental techniques developed in the first decade of the 21st century, it became possible to produce this kind of gas using real atoms as particles.

Definition and solution of the model

There are ${\displaystyle N}$ boson particles with coordinates ${\displaystyle x}$ on the line ${\displaystyle [0,L]}$, with periodic boundary conditions. Thus, a state of the N-body system must be described by a wave function ${\displaystyle \psi (x_{1},x_{2},\dots ,x_{j},\dots ,x_{N})}$ that remains unchanged under permutation of any two particles (permutation symmetry), i.e., ${\displaystyle \psi (\dots ,x_{i},\dots ,x_{j},\dots )=\psi (\dots ,x_{j},\dots ,x_{i},\dots )}$ for all ${\displaystyle i\neq j}$ and ${\displaystyle \psi }$ satisfies ${\displaystyle \psi (\dots ,x_{j}=0,\dots )=\psi (\dots ,x_{j}=L,\dots )}$ for all ${\displaystyle j}$. The Hamiltonian, in appropriate units, is

${\displaystyle H=-\sum _{j=1}^{N}\partial ^{2}/\partial x_{j}^{2}+2c\sum _{1\leq i<j\leq N}\delta (x_{i}-x_{j})\ ,}$

where ${\displaystyle \delta }$ is the Dirac delta function, i.e., the interaction is a contact interaction. The constant ${\displaystyle c\geq 0}$ denotes its strength. The delta function gives rise to a boundary condition when two coordinates, say ${\displaystyle x_{1}}$ and ${\displaystyle x_{2}}$ are equal; this condition is that as ${\displaystyle x_{2}\searrow x_{1}}$, the derivative satisfies ${\displaystyle \left.\left({\frac {\partial }{\partial x_{2}}}-{\frac {\partial }{\partial x_{1}}}\right)\psi (x_{1},x_{2})\right|_{x_{2}=x_{1}+}=c\psi (x_{1}=x_{2})}$. The hard core limit ${\displaystyle c=\infty }$ is known as the Tonks–Girardeau gas.^[3]

Schrödinger's time independent equation, ${\displaystyle H\psi =E\psi }$ is solved by explicit construction of ${\displaystyle \psi }$. Since ${\displaystyle \psi }$ is symmetric it is completely determined by its values in the simplex ${\displaystyle {\mathcal {R}}}$, defined by the condition that ${\displaystyle 0\leq x_{1}\leq x_{2}\leq \dots ,\leq x_{N}\leq L}$. In this region one looks for a ${\displaystyle \psi }$ of the form considered by H.A. Bethe in 1931 in the context of magnetic spin systems—the Bethe ansatz. That is, for certain real numbers ${\displaystyle k_{1}<k_{2}<\cdots <k_{N}}$, to be determined,

${\displaystyle \psi (x_{1},\dots ,x_{N})=\sum _{P}a(P)\exp \left(i\sum _{j=1}^{N}k_{Pj}x_{j}\right)}$

where the sum is over all ${\displaystyle N!}$ permutations, ${\displaystyle P}$, of the integers ${\displaystyle 1,2,\dots ,N}$, and ${\displaystyle P}$ maps ${\displaystyle 1,2,\dots ,N}$ to ${\displaystyle P_{1},P_{2},\dots ,P_{N}}$. The coefficients ${\displaystyle a(P)}$, as well as the ${\displaystyle k}$'s are determined by the condition ${\displaystyle H\psi =E\psi }$, and this leads to

${\displaystyle E=\sum _{j=1}^{N}\,k_{j}^{2}}$

${\displaystyle a(P)=\prod _{1\leq i<j\leq N}\left(1+{\frac {ic}{k_{Pi}-k_{Pj}}}\right)\,.}$

Dorlas (1993) proved that all eigenfunctions of ${\displaystyle H}$ are of this form.^[4]

These equations determine ${\displaystyle \psi }$ in terms of the ${\displaystyle k}$'s, which, in turn, are determined by the periodic boundary conditions. These lead to ${\displaystyle N}$ equations:

${\displaystyle L\,k_{j}=2\pi I_{j}\ -2\sum _{i=1}^{N}\arctan \left({\frac {k_{j}-k_{i}}{c}}\right)\qquad \qquad {\text{for }}j=1,\,\dots ,\,N\ ,}$

where ${\displaystyle I_{1}<I_{2}<\cdots <I_{N}}$ are integers when ${\displaystyle N}$ is odd and, when ${\displaystyle N}$ is even, they take values ${\displaystyle \pm {\frac {1}{2}},\pm {\frac {3}{2}},\dots }$ . For the ground state the ${\displaystyle I}$'s satisfy

${\displaystyle I_{j+1}-I_{j}=1,\quad {\rm {for}}\ 1\leq j<N\qquad {\text{and }}I_{1}=-I_{N}.}$

The first kind of elementary excitation consists in choosing ${\displaystyle I_{1},\dots ,I_{N-1}}$ as before, but increasing ${\displaystyle I_{N}}$ by an amount ${\displaystyle n>0}$ (or decreasing ${\displaystyle I_{1}}$ by ${\displaystyle n}$). The momentum of this state is ${\displaystyle p=2\pi n/L}$ (or ${\displaystyle -2\pi n/L}$).

For the second kind, choose some ${\displaystyle 0<n\leq N/2}$ and increase ${\displaystyle I_{i}\to I_{i}+1}$ for all ${\displaystyle i\geq n}$. The momentum of this state is ${\displaystyle p=\pi -2\pi n/L}$. Similarly, there is a state with ${\displaystyle p=-\pi +2\pi n/L}$. The momentum of this type of excitation is limited to ${\displaystyle |p|\leq \pi .}$

These excitations can be combined and repeated many times. Thus, they are bosonic-like. If we denote the ground state (= lowest) energy by ${\displaystyle E_{0}}$ and the energies of the states mentioned above by ${\displaystyle E_{1,2}(p)}$ then ${\displaystyle \epsilon _{1}(p)=E_{1}(p)-E_{0}}$ and ${\displaystyle \epsilon _{2}(p)=E_{2}(p)-E_{0}}$ are the excitation energies of the two modes.

Thermodynamic limit

To discuss a gas we take a limit ${\displaystyle N}$ and ${\displaystyle L}$ to infinity with the density ${\displaystyle \rho =N/L}$ fixed. The ground state energy per particle ${\displaystyle e={\frac {E_{0}}{N\rho ^{2}}}}$, and the ${\displaystyle \epsilon _{1,2}(p)}$ all have limits as ${\displaystyle N\to \infty }$. While there are two parameters, ${\displaystyle \rho }$ and ${\displaystyle c}$, simple length scaling ${\displaystyle x\to \rho x}$ shows that there is really only one, namely ${\displaystyle \gamma =c/\rho }$.

To evaluate ${\displaystyle E_{0}}$ we assume that the N ${\displaystyle k}$'s lie between numbers ${\displaystyle K}$ and ${\displaystyle -K}$, to be determined, and with a density ${\displaystyle L\,f(k)}$. This ${\displaystyle f}$ is found to satisfy the equation (in the interval ${\displaystyle -K\leq k\leq K}$)

${\displaystyle 2c\int _{-K}^{K}{\frac {f(p)}{c^{2}+(p-k)^{2}}}dp=2\pi f(k)-1\quad {\rm {and}}\quad \int \nolimits _{-K}^{K}f(p)dp=\rho \,,}$

which has a unique positive solution. An excitation distorts this density ${\displaystyle f}$ and similar integral equations determine these distortions. The ground state energy per particle is given by

${\displaystyle e={\frac {1}{\rho ^{3}}}\int _{-K}^{K}k^{2}f(k)dk.}$

Figure 1 shows how ${\displaystyle e}$ depends on ${\displaystyle \gamma }$ and also shows Bogoliubov's approximation to ${\displaystyle e}$. The latter is asymptotically exact to second order in ${\displaystyle \gamma }$, namely, ${\displaystyle e\approx \gamma -4\gamma ^{3/2}/(3\pi )}$. At ${\displaystyle \gamma =\infty }$, ${\displaystyle e=\pi ^{2}/3}$.

Figure 2 shows the two excitation energies ${\displaystyle \epsilon _{1}(p)}$ and ${\displaystyle \epsilon _{2}(p)}$ for a small value of ${\displaystyle \gamma =0.787}$. The two curves are similar to these for all values of ${\displaystyle \gamma >0}$, but the Bogoliubov approximation (dashed) becomes worse as ${\displaystyle \gamma }$ increases.

From three to one dimension.

This one-dimensional gas can be made using real, three-dimensional atoms as particles. One can prove, mathematically, from the Schrödinger equation for three-dimensional particles in a long cylindrical container, that the low energy states are described by the one-dimensional Lieb–Liniger model. This was done for the ground state^[5] and for excited states.^[6] The cylinder does not have to be as narrow as the atomic diameter; it can be much wider if the excitation energy in the direction perpendicular to the axis is large compared to the energy per particle ${\displaystyle e}$.

References

External links

• See also Elliott H. Lieb (2008), Scholarpedia, 3(12):8712.