Quantum Mechanical Description of Fluid Dynamics

In this paper, we deal with fluid motion in terms of quantum mechanics. Mechanism accounting for the appearance of quantum behavior is discussed. Consider a ideal fluid which is composed of discrete identical particles, its mass and charge are m and q respectively, it is convenient to consider the fluid to be a flow characterized by a 4-vector velocity field u(x1, x2, x3, x4 = ict) in a Cartesian coordinate system (in a laboratory frame of reference). The particle will be affected by the 4-vector force due to electromagnetic interaction. According to the relativistic Newton’s second law, the motion of the particle satisfies the following governing equations m duμ dτ = qFμνuν (1) uμuμ = −c 2 (2) where Fμν is the curl of electromagnetic 4-vector potential A. Since the Cartesian coordinate system is a frame of reference whose axes are orthogonal to one another, there is no distinction between covariant and contravariant components, only subscripts need be used. Here and below, summation over twice repeated indices is implied in all case, Greek indices will take on the values 1,2,3,4, and regarding the mass m as a constant. Eq.(1) and (2) is valid at every point for every particle. As mentioned above, the 4-vector velocity u can be regarded as a 4vector velocity field, then duμ dτ = ∂uμ ∂xν dxν dτ = uν∂νuμ (3) qFμνuν = quν(∂μAν − ∂νAμ) (4) E-mail: [email protected] Substituting them back into Eq.(1), and re-arranging their terms, we obtain uν∂ν(muμ + qAμ) = uν∂μ(qAν) = uν∂μ(muν + qAν)− uν∂μ(muν) = uν∂μ(muν + qAν)− 1 2 ∂μ(muνuν) = uν∂μ(muν + qAν)− 1 2 ∂μ(−mc ) = uν∂μ(muν + qAν) (5)