Atomic Theory of the Two-Fluid Model of Liquid Helium

Abstract

It is argued that the wave function representing an excitation in liquid helium should be nearly of the form $\Sigma i^{}f\left({\mathrm{r}}_i\right)\phi$, where $\phi$ is the ground-state wave function, $f\left(\mathrm{r}\right)$ is some function of position, and the sum is taken over each atom $i$. In the variational principle this trial function minimizes the energy if $f\left(\mathrm{r}\right)=\exp \left(i\mathrm{k}·\mathrm{r}\right)$, the energy value being $E\left(k\right)=\frac{{\hslash }^2{k}^2}{2mS\left(k\right)\mathrm{}}$, where $S\left(k\right)\mathrm{}$ is the structure factor of the liquid for neutron scattering. For small $k$, $E$ rises linearly (phonons). For larger $k$, $S\left(k\right)\mathrm{}$ has a maximum which makes a ring in the diffraction pattern and a minimum in the $E\left(k\right)\mathrm{}$ vs $k$ curve. Near the minimum, $E\left(k\right)\mathrm{}$ behaves as $\Delta +\frac{{\hslash }^2{(k-k_0)}^2}{2\mu }$, which form Landau found agrees with the data on specific heat. The theoretical value of $\Delta$ is twice too high, however, indicating need of a better trial function.