On the Theory of Quantum Liquids. I. Surface Tension and Stress

Abstract

Pressure and stress of monatomic liquid is calculated by a modification of perturbation method in which the boundary of the system is subject to change. If we use the phase space distribution function of Wigner the expression becomes identical to classical stress tensor. The same method applies to the calculation of surface tension. The result is
γ=\frac1A\left[\frac2Tr\left{−\cfrach²2m∑\limits_j=1^N\left(\cfrac∂²∂z_j²−\cfrac∂²∂x_j²\ ight)\ ight}ρ(x,x′)Tr ρ(x,x′)+∫∫\fracx₁₂²−z₁₂²2r₁₂\fracdφ(r₁₂)dr₁₂n(z₁)n(z₂)g(x₁, r₁₂)dx₁ dx₂\ ight]
in which ρ(x, x′) is the density matrix for the states with a definite liquid film with area A perpendicular to the z-axis. The first term represents quantum effect. And the second term is identical in form to the expression obtained by Harasima for classical liquids. It is shown that the relation
γ=∫(P−P_T)dz
holds also for quantum liquid. Here P and P_T are respectively the pressure in the liquid and the pressure component tangential to the surface z=const. Surface effect on ideal gases and comparison with experiments are also discussed.