Exercise Sheet 11

Two bosons, with $X=\tfrac12(x_1+x_2)$ and $r=x_2-x_1$. (i) Show that $\partial_1^{2}+\partial_2^{2}=\tfrac12\partial_X^{2}+2\partial_r^{2}$ and $\partial_r=\tfrac12(\partial_2-\partial_1)$. (ii) Integrate $H\psi=E\psi$ over $-\epsilon<r<\epsilon$ and let $\epsilon\to0$, to show that $\partial_r\psi$ jumps by $c\,\psi$ at $r=0$. Why must $\psi$ itself be continuous? (iii) For bosons $\psi$ is even in $r$. Deduce the contact condition $(\partial_2-\partial_1-c)\psi=0$ at $x_2=x_1$, approached from $x_1<x_2$.

(i) On $x_1<x_2$ take $\psi=A_{12}e^{\ii(k_1x_1+k_2x_2)}+A_{21}e^{\ii(k_2x_1+k_1x_2)}$. Show that the contact condition holds exactly when $A_{21}/A_{12}=S(k_1-k_2)$. (ii) Show that $S=-e^{\ii\theta}$ with $\theta'(k)=2c/(k^{2}+c^{2})>0$, that $S(0)=-1$, and find the limits of $S$ as $c\to\infty$ and as $c\to0$, at fixed $k\ne0$. What do they describe? (iii) For two distinguishable particles with the same interaction, let the relative wave $e^{-\ii kr/2}$, $k=k_1-k_2>0$, come in from $r>0$: take $\psi=e^{-\ii kr/2}+\rho\,e^{\ii kr/2}$ for $r>0$ and $\psi=\tau e^{-\ii kr/2}$ for $r<0$. Find the transmitted and reflected amplitudes $\tau$ and $\rho$, and show that $\abs\tau^{2}+\abs\rho^{2}=1$ and $\tau+\rho=S(k)$.

(i) Put $k_{1,2}=\tfrac K2\pm q$ with $K=2\pi n/L$. Show that the Bethe equations reduce to $qL+\theta(2q)=\pi(n+1)\pmod{2\pi}$, and that $E=\tfrac12K^{2}+2q^{2}$. (ii) For $K=0$ the ground state has $qL+2\arctan(2q/c)=\pi$ with $q>0$. Solve it for $L=c=1$, and find its limits as $c\to\infty$ and as $c\to0$ at fixed $L$.

(i) For $N=2$ and $K=0$, look for imaginary roots $k_1=\ii\kappa$, $k_2=-\ii\kappa$ with $\kappa>0$. Show that the Bethe equations become $e^{-\kappa L}=\frac{2\kappa+c}{2\kappa-c}$. (ii) Show that for $c>0$ it has no solution, as Lecture 11’s proposition on real roots says. (iii) For $c<0$ show that it has a solution with $\kappa>\abs c/2$, and that $\kappa\to\abs c/2$ as $L\to\infty$: a bound pair, the case the lecture excludes. Solve it for $c=-1$, $L=10$.

(i) For real roots, take logarithms of the Bethe equations to get $k_jL+\sum_{l=1}^{N}\theta(k_j-k_l)=2\pi I_j$ with $I_j\in\Z+\tfrac{N-1}2$, and show $K=\frac{2\pi}L\sum_jI_j$. (ii) With $\Theta(x)=\int_0^{x}\theta$, let $B(k)=\tfrac L2\sum_jk_j^{2}-2\pi\sum_jI_jk_j+\tfrac12\sum_{j,l}\Theta(k_j-k_l)$. Show that the equations of (i) are $\partial B/\partial k_j=0$, and that the Hessian of $B$ is $L\,\Id+\sum_{j<l}\theta'(k_j-k_l)(e_j-e_l)(e_j-e_l)^{\mathsf T}$. Deduce that the solution is unique. (iii) Show that $I_j>I_l$ implies $k_j>k_l$.

Take the ground state, $I_j=-\tfrac{N-1}2,\dots,\tfrac{N-1}2$, whose roots are symmetric, $\sum_jk_j=0$. (i) Strong coupling. For $\abs k\ll c$ use $\theta(k)\approx2k/c$ to show $k_j\approx2\pi I_j/(L+2N/c)$: free fermions on a ring lengthened by $2N/c$. (ii) Weak coupling. If the roots’ differences are large compared with $c$, use $\theta(x)\approx\pi\operatorname{sgn}x-2c/x$ to show that $k_j=\sqrt{2c/L}\,h_j$ with $h_j=\sum_{l\ne j}1/(h_j-h_l)$. Check that the zeros of the Hermite polynomial $H_3$, $0$ and $\pm\sqrt{3/2}$, solve this for $N=3$. (iii) (Computational.) For $N=5$, $L=5$, solve the equations of Exercise 5 at $c=0.1$, $1$ and $10$, and compare with (i) at $c=10$ and with (ii) at $c=0.1$, using the zeros of $H_5$.

The displacement. In this exercise the two bosons move on the infinite line, not on a ring, so that they meet once. Superpose two-boson solutions of Exercise 2 into packets: $\psi=\iint f_1(k_1)f_2(k_2)\big[e^{\ii(k_1x_1+k_2x_2)}+S(k_1-k_2)e^{\ii(k_2x_1+k_1x_2)}\big]e^{-\ii(k_1^{2}+k_2^{2})t}\,\dd k_1\dd k_2$ on $x_1<x_2$, with $f_1$, $f_2$ real, positive and narrow around $\bar k_1>\bar k_2$ (a real envelope puts the packet’s centre at the origin at $t=0$; a phase $e^{-\ii ka}$ would shift it by $a$). Work to leading order in the width, following the central rays. (i) A packet sits where its phase is stationary in the momentum it carries. Show that before the collision ($t\to-\infty$) the particles are at $x_1=2\bar k_1t$, $x_2=2\bar k_2t$, and after it ($t\to+\infty$) at $x_2=2\bar k_1t-\theta'(\bar k_1-\bar k_2)$ and $x_1=2\bar k_2t+\theta'(\bar k_1-\bar k_2)$. (ii) In the centre-of-mass frame, $\bar k_1=-\bar k_2=\bar k>0$, show that both particles come out delayed, and find the delay. What happens to it as $c\to\infty$ and as $c\to0$?