Linear Continuum Mechanics for Quantum Many-Body Systems

We develop the continuum mechanics of quantum many-body systems in the linear response regime. The basic variable of the theory is the displacement field, for which we derive a closed equation of motion under the assumption that the time-dependent wave function in a locally comoving reference frame can be described as a geometric deformation of the ground-state wave function. We show that this equation of motion is exact for systems consisting of a single particle, and for all systems at sufficiently high frequency, and that it leads to an excitation spectrum that has the correct integrated strength. The theory is illustrated by simple model applications to one- and two-electron systems.

Figure 1

Unnormalized displacement fields for a few low-lying excitations of the two-electron system described in the text. The solid line is the ground-state density. Analytically we find $u_{nm}(x)\propto H_{n+m-1}[2(x-x_0/2)]$ for $x\simeq x_0/2$, with parity $(-1{)}^{n-1}$ independent of $m$. The large value of the displacement field for $x\sim 0$ does not have a physical significance since the density is exponentially small in that region.