It is demonstrated that the probability density function, given by the square of a quantum mechanical wavefunction that is a real-valued eigenvector of a time-independent, one-body Schroedinger equation, satisfies a compressible-flow generalization of the Bernoulli equation, where the mass density is the probability density times the mass of the system; the pressure and velocity fields are defined by functions depending on the probability density, and the gradient and the Laplacian of the probability density, where there are two possible directions of the velocity on a streamline. The velocity given definition implies a generalization of the steady-flow continuity equation where mass is not locally conserved. The gradient of the Bernoullian equation is demonstrated to be equivalent to the steady flow Euler equation for variable mass and irrotational flow. A speed of sound quadratic equation is obtained from a spherical wave-pulse. One of the solutions indicates that the wave-pulse velocity on a streamline is equal in magnitude but opposite in direction of the fluid velocity on the streamline. The other solution is the focus of attention from that point on. It is proven that the extremums of the momentum per volume on a streamline occur at points that are Mach 1 speed. The developed formalism is applied to a particle in a one-dimensional box, the ground and first excited-states of the one-dimensional harmonic oscillator, and the hydrogen 1s and 2s states. Some behavior is repeated in all the applications examined. For example, an antinode, a point of local-maximum density, has zero velocity and zero Mach speed on the streamline, while a node, a point of minimum density, has infinite velocity and Mach 2. In between the node and antinode is an extremum of the momentum, and Mach 1. (This is a short version of the abstract.)
It is demonstrated that the probability density function, given by the square of a quantum mechanical wavefunction that is a real-valued eigenvector of a time-independent, one-body Schroedinger equation, satisfies a compressible-flow generalization of the Bernoulli equation, where the mass density is the probability density times the mass of the system; the pressure and velocity fields are defined by functions depending on the probability density, and the gradient and the Laplacian of the probability density, where there are two possible directions of the velocity on a streamline. The velocity given definition implies a generalization of the steady-flow continuity equation where mass is not locally conserved. The gradient of the Bernoullian equation is demonstrated to be equivalent to the steady flow Euler equation for variable mass and irrotational flow. A speed of sound quadratic equation is obtained from a spherical wave-pulse. One of the solutions indicates that the wave-pulse velocity on a streamline is equal in magnitude but opposite in direction of the fluid velocity on the streamline. The other solution is the focus of attention from that point on. It is proven that the extremums of the momentum per volume on a streamline occur at points that are Mach 1 speed. The developed formalism is applied to a particle in a one-dimensional box, the ground and first excited-states of the one-dimensional harmonic oscillator, and the hydrogen 1s and 2s states. Some behavior is repeated in all the applications examined. For example, an antinode, a point of local-maximum density, has zero velocity and zero Mach speed on the streamline, while a node, a point of minimum density, has infinite velocity and Mach 2. In between the node and antinode is an extremum of the momentum, and Mach 1. (This is a short version of the abstract.)
Abstract A formalism is developed, and applied, that describes a class of one-body quantum mechanical systems as fluids where each stationary state is a steady flow state. The time-independent Schrödinger equation for one-body stationary states with real-valued wavefunctions is shown to be equivalent to a compressible-flow generalization of the Bernoulli equation of fluid dynamics. The mass density, velocity and pressure are taken as functions that are determined by the probability density. The generalized Bernoulli equation describes compressible, irrotational, steady flow with variable mass and a constant specific total energy, i.e, a constant energy per mass for each fluid element. The generalized Bernoulli equation and a generalized continuity equation provide a fluid dynamic interpretation of a class of quantum mechanical stationary states that is an alternative to the unrealistic, static-fluid interpretation provided by the Madelung equations and quantum hydrodynamics. The total kinetic energy from the Bernoulli equation is shown to be equal to the expectation value of the kinetic energy, and the integrand of the expectation value of the kinetic energy is given an interpretation. It is also demonstrated that variable mass is necessary for a satisfactory fluid model of stationary states. However, over all space, the flows conserve mass, because the rate of mass creation from the sources are equal to the rate of mass annihilation from the sinks. The following flows are examined: the ground and first excited-states of a particle in a one-dimensional box, the harmonic oscillator, and the hydrogen s states.
Abstract A formalism of classical mechanics is given for time-dependent many-body states of quantum mechanics, describing both fluid flow and point mass trajectories. The familiar equations of energy, motion, and those of Lagrangian mechanics are obtained. An energy and continuity equation is demonstrated to be equivalent to the real and imaginary parts of the time dependent Schrödinger equation, respectively, where the Schrödinger equation is in density matrix form. For certain stationary states, using Lagrangian mechanics and a Hamiltonian function for quantum mechanics, equations for point-mass trajectories are obtained. For 1-body states and fluid flows, the energy equation and equations of motion are the Bernoulli and Euler equations of fluid mechanics, respectively. Generalizations of the energy and Euler equations are derived to obtain equations that are in the same form as they are in classical mechanics. The fluid flow type is compressible, inviscid, irrotational, with the nonclassical element of local variable mass. Over all space mass is conserved. The variable mass is a necessary condition for the fluid flow to agree with the zero orbital angular momentum for s states of hydrogen. Cross flows are examined, where velocity directions are changed without changing the kinetic energy. For one-electron atoms, the velocity modification gives closed orbits for trajectories, and mass conservation, vortexes, and density stratification for fluid flows. For many body states, under certain conditions, and by hypotheses, Euler equations of orbital-flows are obtained. One-body Schrödinger equations that are a generalization of the Hartree–Fock equations are also obtained. These equations contain a quantum Coulomb’s law, involving the 2-body pair function of reduced density matrix theory that replace the charge densities.
Many-body quantum-mechanical stationary states that have real valued wavefunctions are shown to satisfy a classical conservation of energy equation with a kinetic energy function. The terms in the equation depend on the probability distribution, and, in addition, pressure and velocity functions, but these functions also depend on the probability distribution. There are two possible directions of the velocity that satisfy the energy equation. A linear momentum function is defined that integrates to zero, and this property is consistent with the expectation value of the linear momentum for stationary states with real-valued wave functions. The energy equation is integrated to obtain a version of the well known energy equation involving reduced density matrices, where the kinetic energy functional of the one-particle density matrix is replaced by a function of the electron density and a velocity function. Also, the noninteracting kinetic energy functional from the Hohenberg--Kohn theorem is given as an explicit functional of the orbital densities. For the purpose of describing the behavior of particles in a stationary state, a model based on the energy equation is constructed. The model is evaluated for the two different velocity directions using the grounds state of the particle in a one-dimensional box and the hydrogen atom. For one velocity direction, equations of motions with contradictory properties are obtained, and, in the other, an unstable system is found. A discussion is given with suggestions of additional elements that might improve the model.
The Madelung equations are two equations that are equivalent to the one-body time-dependent Schroedinger equation. In this paper, the Madelung equation, whose gradient is an Euler equation, is refined by introducing interpretations of functions that are shown to depend only on the real-part of the complex-valued wavefunction. These interpretations are extensions of functions from the recently derived generalized Bernoulli equation, applicable to real-valued quantum-mechanical stationary states. In particular, the velocity and pressure definitions are extended so that they depend on the real-part of a time-dependent complex-valued wavefunction. The Bohn quantum potential is then interpreted as the sum of two terms, one involving the kinetic energy and the other involving the pressure. Substituting the interpreted quantum-potential into the Madelung equation gives a refined equation containing two kinetic energy terms, a pressure term, and the external potential. It is easily demonstrated that the refined Madelung equation, applied to the hydrogen atom states with a nonzero magnetic quantum number, gives a fluid velocity that contains both a radial component and a free vortex. Hence, the fluid particles have angular momentum and move on streamlines that terminate at infinity. It is also demonstrated that the two velocities from the refined Madelung equation are related: One is the real component and the other is the imaginary component of a complex velocity. Furthermore, an Euler equation for quantum mechanical systems is derived by taking the gradient of the refined Madelung equation.
A generalized Euler equation of fluid dynamics is derived for describing many-body states of quantum mechanics. The Eulerian Eq. can be viewed as representing the interaction of two substates, where each substate has its own velocity and pressure fields. These field quantities are given by maps of the wavefunction. For one-body systems, the Eulerian Eq. can model either a fluid or particle description of quantum states. The generalized Euler Eq. is shown to be the gradient of an equation representing the total-energy of the two substates, having two energy fields. This total-energy Eq. is a generalization of the Bernoulli Eq. of fluid dynamics. The total-energy Eq., along with a continuity-equation, is equivalent to the time-dependent Schroedinger Eq. An equation is also derived that is equivalent to the main equation of Bohmian mechanics with additional identifications: The quantum potential of Bohmian mechanics is given as a sum of a kinetic energy and pressure fields. Also, the time derivative of the wavefunction phase is replaced by an energy field. In the formalism, field quantities are identified from their placement in equations of classical mechanics and separately, by definitions that involve the wavefunction and operators of quantum mechanics. This approach yields, unintended, and unknown energy and pressure fields. These fields, however, are shown to satisfy a continuity Eq., an equation that is equivalent to the other equation of Bohmian mechanics. It is also demonstrated that energy conservation holds for both of these energy fields, if the wavefunction is a linear-combination of eigenvectors, where the eigenvectors can be nondegenerate. A detailed investigation is given on the possible behavior, or source, of an electron that has one of the velocity fields. Alternate formulae for this velocity fields are also considered.
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.