The velocity field in quantum hydrodynamics

Abstract

In the usual form of quantum hydrodynamics the velocity field is defined in terms of the inverse density operator ρ⁻¹(x). It is found that this inverse does not exist. An unambiguous velocity field can, however, be constructed without the use of ρ⁻¹(x) by requiring that its Fourier components be canonical conjugates to the corresponding Fourier components of the density operator when the system is in a box with periodic boundary conditions. This velocity field has at mostN nonzero Fourier components whereN is the number of particles in the system; it satisfiesj(x)=1/2[ρ(x)v(x)+v(x)ρ(x)]; it has the proper behavior under galilean transformations; and the corresponding classical field is uniquely determined at theN particle positions, where it is equal to the particle velocity, and is a smooth interpolating field between these points. The velocity is not defined uniquely at positions where there are no particles, but this is a reflection of the atomicity of the system—a field uniquely defined at each point in space would represent a continuum with an infinity of degrees of freedom. The commutation relations of the velocity field with itself and with the current density are derived and found to be similar but not identical to those given by Landau.