A trial wave function describing the ground state of a quantum system of $N$ interacting bosons is written in the Jastrow form, a product of pair functions. With the interaction potential chosen to represent liquid ${\mathrm{He}}^4$, and with the parametrized form of the pair function chosen to include a long-range term which has been found necessary to represent the zero-point motion of the long-wavelength density oscillations, a variational calculation has been performed using a new approximate integral equation for the pair distribution function. This equation, which can also be used for classical fluids, is found to be more accurate for repulsive potentials than the Percus-Yevick equation and comparable to (but much simpler than) the Percus-Yevick 2 equation. The essential results are that including the zero-point motion in the wave function tends to lower the energy, raises the equilibrium density, corrects the behavior of the structure function and the momentum distribution of the particles in the low-wave-number region, and slightly decreases the Bose-Einstein condensate fraction. The value of the lower limit on the wavelength of the density oscillations was determined variationally to be about three interparticle spacings.

Using close-coupling calculations of Burke and Moores for the scattering of electrons by ${\mathrm{Mg}}^{+}$ ions in the $3{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}$ and $3{}_{}{}^2{}_{}{}^{}P_{\frac{1}{2},\frac{3}{2}}^{}{}_{}{}^{}$ states, Baranger's expression for the impact approximation width of an isolated line is implemented for the components of the resonance doublet. These widths are extrapolated to below inelastic thresholds and averaged over elastic resonances according to theoretical threshold laws. In the experimental energy range, results compare reasonably with semiclassical approximations and with a semiempirical method involving effective Gaunt factors extrapolated to zero electron energy.