On the Role of Irrotational Flow in Hydrodynamical Derivations of the Schrodinger Equation

 In (1), the Schrodinger equation is derived using a hydrodynamical continuity of density approach. Thus, spatial density represents x probability. There, however, is also a flow velocity field in the problem v(x) and it is assumed this is irrotational. From this it is stated in (1) that v(x) is proportional to 1/m d/dx S(x) ((1)) where S(x) is some function and m is mass.

    In this note, we argue that a particle moving in one dimension with constant p (momentum) is automatically irrotational, but it should satisfy ((1)) from (1). In other words, a constant p particle hence constant velocity is associated with a function of x, S(x) even though its density in space is not. If one considers S(x)=ln(B(x)), then ((1)) is equivalent to:   p = d/dx ln(B(x)). If one tries to associate B(x) with a probability, then ln(B(x)) is information = px. B(x), however, is a different kind of x probability from density(x) which should be constant for a free moving particle. In fact, using -id/dx instead of d/dx and exp(ipx) for B(x), one finds exp(ipx) as a second x probability. Its modulus yields the constant value of density(x). 

    Thus the irrotational flow assumption leads to p= dS/dx. Incidentally, this form is equivalent to Fermat’s least time principle for light with p=p (x component). We argue that applying p=dS/dx associates an x probability distribution even to a constant p, so the assumption p=dS/dx is essentially a quantum mechanical statement, even if one does not derive the Schrodinger equation. The time independent Schrodinger equation may be obtained by using exp(ipx) as a relative probability and using energy conservation.