Equivalent wave-function approach to theHe4structure factor and excitation spectrum

Abstract

A new method for the calculation of the excitation spectrum of liquid $^{4}\mathrm{He}$ is developed based on the wave function ${\ensuremath{\chi}}_{\ensuremath{\alpha}}({x}_{1},{x}_{2})=〈0|T{\ensuremath{\Psi}}^{\ifmmode\dagger\else\textdagger\fi{}}({x}_{1})\ensuremath{\Psi}({x}_{2})|\ensuremath{\alpha}〉$ that is directly related to and completely determines the dynamic structure factor $S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},\ensuremath{\omega})$ of liquid $^{4}\mathrm{He}$, and then employing this method a calculation of the sound spectrum of liquid $^{4}\mathrm{He}$ is performed taking explicitly into account the depletion effect although it is based on the McMillan's data for the ground-state momentum distribution of liquid $^{4}\mathrm{He}$. First, by employing the exponential-decay and effective-mass approximation for the single-particle Green's function, the two-time Bethe-Salpeter integral equation for ${\ensuremath{\chi}}_{\ensuremath{\alpha}}({x}_{1},{x}_{2})$ is transformed into the one-time Schr\odinger equation. This is thus an approximate Schr\odinger equation the solutions of which give $S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},\ensuremath{\omega})$ of liquid $^{4}\mathrm{He}$. In the exponential-decay approximation, the numerators of the single-particle Green's function are the true momentum distribution ${N}_{ \mathrm{q}}^{\ensuremath{\rightarrow}}$ of the ground state of liquid $^{4}\mathrm{He}$, which then enters into the Schr\odinger equation in an essential way. Through this ${N}_{ \mathrm{q}}^{\ensuremath{\rightarrow}}$, both the presence of a condensate and the large depletion of particles from the zero-momentum state in $^{4}\mathrm{He}$ can be properly taken into account. And the effective mass of a helium atom is determined using the McMillan's data and the Lennard-Jones potential. The Schr\odinger equation is then solved in the limit $k\ensuremath{\rightarrow}0$ using the McMillan's data, and an excitation spectrum of $^{4}\mathrm{He}$ of the form ${[{\ensuremath{\epsilon}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}^{\ensuremath{'}2}+\frac{2\ensuremath{\alpha}nv(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}){\ensuremath{\epsilon}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}^{\ensuremath{'}}}{(1+D)]}}^{\frac{1}{2}}$ in the limit $k\ensuremath{\rightarrow}0$ is obtained, which differs from the Bogoliubov form by the factor ${(1+D)}^{\ensuremath{-}1}$ and the replacement of the bare mass $m$ by the effective mass ${m}^{*}$. $D$, being proportional to $(1\ensuremath{-}\ensuremath{\alpha})n$, represents the depletion effect. The expansion of the above expression gives for the sound velocity of liquid $^{4}\mathrm{He}$ $c=190$ m/sec and verifies the positive phonon dispersion.