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The Fluid Dynamics of the One-Body Stationary States of Quantum Mechanics with Real Valued Wavefunctions

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.)

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. The antinode is also an unstable equilibrium points, where, for downhill flow, the fluid particles are moving and accelerating away from the unstable equilibrium point where they end of at a node, if one exist, with infinite speed and zero mass. Another type of behavior is identified, called a fall, where the electron density is strictly decreasing away from the nucleus, as in the hydrogen 1s state. So that different flows can be compared with ease, and flows can be characterized, a reference flow, called ordinary Q1 flow, is defined. (All the figures appear after the citations.) * Electronic address: james.finley@enmu.edu

I. INTRODUCTION
When one acquires knowledge of the fundamentals of quantum mechanics, it becomes clear that the meaning of the word 'understand', as in the phrase 'to understand quantum mechanics' is difficult to "define," and a definition is needed for the comprehension of the statement "nobody understands quantum mechanics" by Feynman [1]. However, for any reasonable definition of the word "understand," it is certainly true that, for all things being equal, it is much more difficult to understand the information contained in a state of quantum mechanics than one from classical mechanics. Therefore, it is not surprising that entire monographs are available that go beyond the mathematical structures used in quantum mechanics, including algebras and analysis [2,3], and into topics such as quantum logic [4,5].
For many applications of quantum mechanics, the values of physical properties, called observables, are given by a finite or infinite series, in particular, the electronic energy, and in these mathematical objects, very few terms can be assigned a "physical meaning," as indicated by Mullikan: ... the more accurate the calculations became, the more the concepts tended to vanish into thin air. [6] However, for one-electron stationary-states, whose wavefunctions satisfy the timeindependent, electronic Schrödinger equation, there is one function, −e|ψ| 2 , where ψ is the complex-valued wave-function and −e is the electron charge, that is often given an interpretation, as described by Raimes: In many problems it is useful, if only as an intuitive aid, to picture the electron charge as spread out in a cloud having charge density −e|ψ(r)| 2 at position r. [7] This gives a simple meaning from classical electrostatics of the potential-energy functional vρ dr, where ρ = |ψ| 2 is the probability density of the quantum state, and v is the potentialenergy of the corresponding classical system. Also, for many electron systems with wavefunctions that are approximated as a single Slater determinant, there is a simple classical interpretation given for the Coulomb integral: It is the classical Coulomb-repulsion between two charge clouds [7,8]. For chemistry, many of the simple, but very useful, models used to describe chemical stability and reactivity have components of charge density, including Lewis dot structures. Crystal field theory also uses a simple electrostatic repulsion model, based on the charge densities determined by occupied orbitals, to describe the energy splitting of otherwise degenerate orbitals of metal complex ions.
In this paper, this idea of a state of a bound electron-and other one-body systems in a stationary state-being described as a continuum, with charge density −eρ, is specialized and generalized to a steady-flow state of a fluid, such that mρ(r) is the mass density of the state with (total) mass m. Also, the momentum per volume and the pressure are taken as functions of the density ρ and a subset of the partial derivatives of ρ: The momentum per volume vector field is proportional to the gradient of ρ, as in Fick's law of diffusion [9], and the pressure, a scalar field, is proportional to the Laplacian of ρ, where in both cases, the proportionality constant depends on Plank's constant . However, only one-body quantum states with real-valued wavefunctions are considered.
The precise mathematical statements giving momentum, pressure, and other conditions given below can be viewed as postulates. However, at this point, it is more convenient to categorize these statements as definitions of a type of fluid. This is similar to the approach used by Currie [10] when he describes a Newtonian fluid, except that he calls the four statements given for a Newtonian fluid, which can be viewed as definitions, postulates.
This paper is not about challenging the fundamental axioms that give probabilities of outcomes of the measurement of observables of a quantum state, or any other essential axiom of quantum mechanics. Instead, the results presented below are about building upon and contributing to what has been done.
Below (in Sec. II A) it is proved that the probability density function ρ, such that ρ = φ 2 , where φ is a real-valued eigenvector of a time-independent, one-body Schrödinger equation for a system with mass m, satisfies a compressible-flow generalization of the Bernoulli equation, where the mass density ρ m = mρ, pressure p, and vector velocity u fields are defined by (composite) functions depending on ρ, ∇ρ and ∇ 2 ρ. Also, there are two possible velocity directions u ± , where u − = −u + , called uphill u + and downhill u − flow. Furthermore, the "Bernoullian" equation, mentioned above, is a statement of the local spatial and temporal conservation of (total) specific energyĒ, an intensive variable, e.g., the energy per mass, that is a constant of motion for each fluid element. It is not a statement of the local conservaton of the (total) energy, an extensive variable.
Relationships between the mathematical objects from the one-body quantum states con-sidered, called Q1 (or Q1.0) probability states, and the corresponding objects from the steady flow states, called Q1 flow states, are developed (II B). For example, the integrand of the expectation value of the kinetic energy for a Q1 probability state is the sum of the kinetic-energy per volume and the pressure of the corresponding Q1 flow state. Also, if the pressure of the Q1 flow state is not the zero function, then it cannot be a nonnegative function. Terminology are also developed (II B). For example, the kinetic energy per mass u 2 /2 for a classical fluid, "corresponds" to mu 2 /2, for the quantum fluid, where mu 2 /2 is called the kinetic energy per amount, m is the (total) mass of the system, and mu 2 /2 has units of energy. (The scalar fields u 2 /2 and mu 2 /2 are also classified as specific open and closed quantities, respectively, with respect to the dimension of energy.) The velocity, pressure and all terms terms within the Bernoullian equation, mentioned above, are examined (II C), for the 1s state of the hydrogenic atom with atomic number Z.
Also, the average pressure in the subspace where the pressure is positive is calculated.
The velocity definition for the Q1 flow states is shown to imply a generalization of the steady-flow continuity equation (III), where mass is not locally conserved, and where the pressure is proportional to the mass creation rate per volume. For downhill flow u − , a positive and negative pressure implies a positive and negative mass creation-rate, respectively.
The total and positive mass creation rate, involving summing over the region of space where the pressure is positive, is calculated for hydrogen 1s state with downhill flow. In general, Q1 flow state do not conserve mass and energy locally. However, over all space, they, of course, do conserve mass and energy.
There is nothing novel presented in Sec. IV. It is the derivation of a form of the Euler equation for variable mass and irrotational flow, obtained by using a standard derivation [11] (and a vector calculus approach like one the one used by Kelly [12]), where the continuity equation is not used. This equation (41) is derived because it is not so common and it is needed in Sec. V, where it is proven that the gradient of the Bernoullian equation from Sec. II A is equivalent to the steady flow version of (41). This implies that Q1 flow is a flow with variable mass, that is steady, irrotational, inviscid, and compressible. The properties of Q1 flows are compared to classical flows that are mass conserving, irrotational, steady, inviscid, compressible, and with no body force. For convenience, these classical flows are called the corresponding classical potential-free flows. The speed of sound quadratic equation from (VI D) is easily modified to yield the speed of sound equation for the corresponding classical potential-free flows, and it is the same formulas as the well known one that was derived for a static ambient state with one-dimensional flow [11,13]: ± dp/dρ m .
It is proven that the extremums of the momentum density ρ m u ± on a streamline occur at points r ∈ L of space where ρ(r) = 0 and Ms(r) = Ma(r) = 1 (VII).
Applications (VIII) of the formalism developed is applied to a fluid (or particle) in a onedimensional box (VIII A), the ground and first excited-state of the one-dimensional harmonic oscillator (VIII B), and the hydrogen 2s state (VIII C). For the ground state of a fluid in a one-dimensional box of length [0, 1] in atomic units, where the density is maximum at 0.5 and zero at 0 and 1, the point r = 0.5 of maximum ρ is an unstable equilibrium point.
For r > 0.5, and a flow direction that moves away from the maximum ρ at 0.5, the fluid particles are moving and accelerating to the right; for r < 0.5 they move and accelerate to the left. A similar type of behavior is seen by all states examined for similar regions where the density has a local maximum. Another behavior repeated is a Mach speed of 0 and 2 at the maximum and minimum densities, and in between these point, Mach 1 is obtained, where, as predicted, the momentum per volume is an extremum. Another type of behavior is identified, called a fall, where the electron density is strictly decreasing away from the nucleus. Also, in the limit of a fluid particle reaching a node, the particle have infinity speed and zero mass. So that different flows can be compared with ease, and flows can be characterized, a reference flow, called ordinary Q1 flow, is defined.

II. A BERNOULLIAN EQUATION FOR QUANTUM SYSTEMS.
A. The theoretic foundation Definition. A quantum 1 (Q1) wavefunction φ is a real-valued eigenfunction of a onebody time-independent Schrödinger equation. A Q1 probability state is a state represented by a quantum 1 wavefunction φ, where all information about the state is determined by the wavefunction φ (and the axioms of quantum mechanics).
Let φ be a Q1 wavefunction that is a solution of a one-body Schrödinger equation with external potential V and mass m: We next show that ρ = φ 2 is a solution of the following: and u 2 = |u ± | 2 . Equation (2) is a compressible-flow generalization of the Bernoulli equation [10,11] with a mass density ρ m = mρ, and ρ is the probability density of the Q1 state represented by φ. Henceforth, we call u ± and p the velocity and pressure, respectively, regardless of their "actual" physical meaning. Note that there are two possible velocities u + and u − , giving two possible directions along each streamlines. (An N-body generalization of (2), from the N-body generalization of (1), has been obtained elsewhere [14].) For the proof below, the external potential V is required to be a function such that the subspace {r ∈ R 3 |ρ(r) = 0} has measure zero and the eigenfunctions of (1) are three times continuously differentiable. Hence, the condition ρ = 0 for (2) implies that ρ satisfies the equation almost everywhere a.e, since {r ∈ R 3 |ρ(r) = 0} has measure zero.
The proof involves first obtaining the following: To show that this is an identity, we obtain two other auxiliary identities. The first one, given by is obtained from the result of For the second one, we take the vector dot product of with itself, giving an equation that is equivalent to (7): ∇ρ · ∇ρ = 4φ 2 ∇φ · ∇φ = 4ρ∇φ · ∇φ Substituting (7) into (6) gives (5). Multiply (1) by φ and using (5) we have (3) and (4) gives (2). Hence, if φ is a solution of (1), then ρ = φ 2 satisfies (2). It is easily seen that the derivation is reversible giving the following: If ρ is a solution of (2), then a three times continuously differentiable function φ such that ρ = φ 2 , satisfies (1). We indicate this state of affairs by stating that (1) and (2) are equivalent.
For later use for systems with a (total) charge q = 0, we write (2) as where Φ is defined by V = qΦ, and the charge density is qρ.
B. Terminolgy, interpretations, and relationships of quantum particle states of (1) and the corresponding classical fluid states of (2) The terminology and conventions used for states of a classical fluid and states of a quantum (bound) particle, are different and similar in ways that can lead to confusion when investigating relationships and the assignment of meaning to mathematical objects. Therefore, it is necessary to introduce some terminology so that statements are clear and unambiguous.
Alias. The kinetic energy per volume ρmu 2 /2 has the alias the 'kinetic-energy density,' and this is also done for other per volume quantities, especially ones involving energy.
Definitions. A Q1 (or Q1.0) density ρ is a real valued solution of the Bernoullian (2), where the velocity and pressure are given by (3) and (4) Definitions. Flow state with velocity fields u + and u − are said to have uphill and downhill flows, respectively.
Note that it follows from definition (3) that the direction of u + and u − on a streamline are towards increasing and decreasing density ρ, respectively. The velocity field definition (3) also indicates that Q1 flow states are irrotational. To obtain the velocity potential ω from (3), and avoid taking the natural logarithm of a dimensioned quantity, both densities are replaced by the dimensionless densityρ = a 3 0 ρ, where a 0 is the Bohr radius, giving ω = ± 2m lnρ (9) If follows from (3) that the potential of the momentum density ρ m u ± is ±( /2m)ρ. As in the Bernoulli equation for irrotational flow [10,11], (2) holds for the entire flow field-there is only one constantĒ, instead one for each streamline.
Alias. The pressure p is also called the 'compression-energy density.' Consider the following two equations: 1) The equation obtained by multipling the Schrödinger' (1) by φ and then followed by substituting ρ = φ 2 ; and, 2) the Bernoullian' (2) multiplied by ρ. Subtracting of these two equations gives Hence, the integrand of the expectation value of the kinetic energy for a Q1 probability state is the sum of the kinetic-energy and compression-energy densities of the corresponding Q1 flow state.
Definitions. Recall that m and q are the total change and mass of a system.
These are also called the classical and quantum density forms, respectively. Other energy quantities are similar.
The various forms of (8) can be written 1 2 mu 2 ρ + p + qΦρ = E m ρ, quantum density form (15) where E m = mE =Ē. If there is no chance of confusion, the words 'open' and 'closed' in the above definitions can be suppressed.
For each cartesian coordiate α ∈ {x, y.z} we require that the wavefunction satisfy This result combined with (4) gives This equation implies that if p(r) ≥ 0 a.e, then p(r) = 0 a.e. Hence, if p(r) = 0 a.e, then we cannot have p(r) ≥ 0 a.e. Hence, the pressure from quantum flows differ in this regard from classical flows where the pressures are required to be thermodynamic pressures, and such pressures satisfy p(r) ≥ 0.
If V is a constant function, and there are no boundary condition imposed in R 3 , then there is no real-valued eigenfunction of (1). Since (1) and (2) where mu 2 ± /2 is the specific, closed kinetic-energy of the Q1 flow state. Analogous meanings also hold for the summed compression-, and (electrostatic) potential-energy. Since the scalar fieldĒ is uniform and ρ is normalized, the summed total energy satisfies Hence, the summed total energy of the Q1 flow state is equal to the specific, closed total-energyĒ, whereĒ is also the energy of the corresponding Q1 probability state that satisfies (1). (Note that the summed kinetic-energy, defined above, can also be called the total kinetic energy, but we cannot do this when speaking aboutĒ, sinceĒ is a kind of total energy.) Let S ∈ R 3 . The kinetic energy over the subspace S (of the state represented by ρ) is given by (17)  By setting S = R 3 for the integral on the left hand-side (lhs), and using (10) and (16), we find that Hence, the expectation value of the kinetic energy for a Q1 probability state is equal to the summed kinetic energy of the corresponding Q1 flow state.
C. The ground state of hydrogenic atoms.
Henceforth, a large semicolon, ending a displayed equations with a dedicated line, as in (19) below, is read "where." Next we examine the velocity, pressure and the energy per amount terms within (8)  respectively. Also, the Bohr radius a 0 is 0.5292 × 10 −10 meter to four figures.
In spherical coordinates the open density, the probability density for hydrogenic 1s states [15], is a 0 is the Bohr radius and m is the electron mass. The velocity field is In other words, Hence, the speed |u ± | is constant. For Z = 1, |u ± | = /ma 0 , and this constant is both the derived Hartree unit of velocity and the speed of the electron in the first Bohr orbit of hydrogen [15]. The second equation from (20), for use below, is the specific closed kinetic energy, and u can be chosen to be |u ± |, but there are other choices, e.g., u = u ± ·r.
(For higher accuracy, the electron mass m can be replaced by the reduced mass µ; a 0 then becomes the modified Bohr radius a µ .) Letz = Z/a 0 . Next we calculate the compression-energy density p and the compression energy per amount pρ −1 , starting with the Laplacian of ρ: The pressure p, defined by (4), is Usingz = Z/a 0 , this equation can also be written Substitute (19) for ρ gives Finally we express the pressure in a manner so that it is clearly dependent on r/a 0 : where 2 /ma 5 0 is the derived Hartee unit of pressure. For the hydrogen atom, the pressure is zero at r = a 0 ; it is positive and negative for r < a 0 and r > a 0 , respectively. The pressure is plotted for the hydrogen atom in Fig. 1 for a calculation with a 0 in the Angstrom unit and 2 /ma 5 0 in the Pascal unit. (All the figures appear after the citations.) The positive P + and negative P − pressure subspaces of R 3 for a Q1 flow are P + = {r ∈ R 3 |p(r) > 0} and P + = {r ∈ R 3 |p(r) < 0}, respectively. The compression energy X over the positive pressure subspace P + is Since the summed compression energy (over all space) is zero, as pointed out in the previous subsection, and since the set {r ∈ R 3 |p(r) = 0} does not contribute to X, the compression energy over the negative pressure subspace P − is −X.
For the hydrogenic 1s state, the pressure p is a function of the radial sperical-coordinate only and P + = {r ∈ R 3 |r < a 0 }. Hence and we substitutied (4). Expressing the Laplacian in spherical coordinates, we discover that Substituting this equation into the one above it and then using (19) we get and 2 /(ma 2 0 ) is the derived energy unit from the Hartree atomic units. For Z = 1, X ≈ 0.27 a.u.
The average valuep + of the pressure in P + is where we used (23), (24) and Vol(P + ) is the volume of P + , a ball of radius a 0 . For Z = 1, we havep about nineteen hundred gigapascals. The average valuep − of the prssure in P − is zero, because the volume Vol(P − ) is infiite.
Returning to the Eq. (21), the specific compression energy pρ −1 is Using the definition [15] 2 a 0 m = e 2 4πε 0 ; e is the proton charge, we obtain the desired form: Adding the second equation from (20) to equations (25) and (11), we get In other words, Comparing this result with (8) for q = −e, we obtain The second form of the energy is obtained from 1/(a 0 m) = e 2 /(4πε 0 2 ). As expected,Ē is the eigenvalue of the Schrödinger equation (1) for the 1s ground state with atomic number For later use we compute (pρ −1 + (−e)Φ), a constant function, using (25) and (11):

III. THE CONTINUITY EQUATION FOR QUANTUM SYSTEMS
The differential and integral forms of the time-dependent continuity equations are [10,11] where ∂ t = ∂/∂t and ∇ · ρ m u is the mass creation-rate density. For the quantum flows under consideration, a generalizations of the differential form of the continuity equation (27) for steady flows is obtained by multiplying the velocity definition (3) by the mass density ρ m and then taking the divergence of the result: Hence, for this steady-flow generalization of the continuity equation (27), mass is not locally conserved. Using ∇ 2 ρ = ∇ · ∇ρ with the divergence theorem, we obtain a generalization of (28) for steady flows: Generalizations of both (27) and (29), and both (28) and (30) are, respectively, Reversing the order of (29), followed by multiplying the equations by ± /(2m), and using Comparing this result with the pressure definition (4) gives Hence, the pressure is proportional to the mass creation rate per volume ∇ · ρ m u. Note that both velocity vectors u ± give the same pressure scalar field, but their formulas differ. The previous equation can also be written Hence, for a point r ∈ R 3 on a streamline with downhill flow u − , a positive pressure (r ∈ P + ) implies a positive creation rate [∇ · ρ m u ± ](r) > 0; a negative pressure (r ∈ P − ) implies a negative creation rate [∇ · ρ m u ± ](r) < 0. The analogous "switched around" conclusion for uphill flow is easily determined by noting the following: [∇ · ρ + u + ](r) > 0 if and only if [∇ · ρ − u − ](r) < 0 and vice versa. Using (34), the total (and positive) mass-creation rateṀ for a Q1 downhill flow state iṡ X is the compression energy, defined by (23). To computeṀ for a hydrogenic 1s state we use (24), givingṀ The constant /a 2 0 , which looks like a derived Hartee unit, is a "natural" generalization. A mass increase caused by a process of an object moving to a higher speed, may, somehow, slow the mass creation and annihilation rates, which could be a fundamental for the relative speed that time passes.

IV. THE VARIABLE MASS EULER EQUATION FOR IRROTATION FLOWS
In the vast majority of derivation of the Euler equation, the differential form of the Here we derive the variable-mass Euler equation for irrotation flows.
We start with the momentum balance equation for a fluid subject to a Coulombic body-force with force per charge (−∇Φ): where σ σ σ is the stress tensor and n is the normal unit vector to the surface S, the border of the subspace V . First we work on the lhs. Using the Reynolds' transport theorem, and the definition, where g is an arbitrary function, we obtain and the continuity equation for systems that conserve mass is M = 0. Note that the definition above permits the use of a product rule of differentiation on the factors of the integrand: ρ m × u × dV , where dV is "considered" a factor, even though the symbol is excluded in some notations for integrals.
For the surface integral of Eq. (35), we apply the divergence theorem: Substituting Eq. (36) and (38) into (35) gives the differential momentum-balance equation with variable mass: where we removed the integrations and obtained a true statement, since the equation with the integrations holds for all subspaces V .
Next we consider only inviscid flows. By definition, these satisfy σ σ σ = −pI, where p is the pressure. Using this equality and a vector identity, the first term on the right-hand side (39) for inviscid fluids becomes Next we require u to be irrotational, i.e., ∇ × u = 0. This permits the use of the following equality: Substituting the above two equations into (39) we obtained the desired equation: and we also used (37). This equation can also be written can be written This equation is the variable-mass Euler equation for the special case of irrotational flow.
In other words, Eq. (41) is applicable to flows that are irrotational, compressible, invsicid, and variable-mass. We also require the body force to be Coulombic, but, obviously, qρ∇Φ can be replaced by ρ m F , where F is a force per mass. In the special case where the flow is steady, incompressible, and mass is conserved, the division of (41) by ρ m , followed by integration, yields the Bernoulli equation. Consider the gradient of the Beroullian (8) In this section we show that the time dependent generalization of this equation, given by is equivalent to the variable-mass Euler' (41) with the condition that the velocity and pressure are given by (3) and (4).
The gradient of the Beroullian (43) implies variable-mass Euler' (41). Substituting the function p, defined by Eq. (4), into (43) and multiplying the result by ρ we obtain For later use, note that Next we expand out a factor from the second term from (44) and then multiply it by a constant: ρ∇ Using (45) and the continuity Eq. (29) for the u − velocity field, we obtain Substituting this into (44), we obtain the Euler equation (41) for a steady flow. Hence, (43) implies (41). Also, (42) implies the steady flow form of (41).
SinceĒ is determined by m, q, ρ, and Φ, (42) determinesĒ and its integral, the Bernoullian (8). Therefore, the Bernoullian (8) and its integral (42) are equivalent. Hence, the Beroullian (8) is equivalent to the variable-mass Euler' (41) for steady flows with the condition that the velocity and pressure are given by (3) and (4), respectively. A Q1 flow is a flow with variable mass, that is steady, irrotational, inviscid, and compressible.
A Newtonian fluid is defined by four equations: 1) continuity, 2) Navier Stokes, 3) energy and 4) an equation of state for the pressure [10]. For Q1 flows, there is a generalized continuity equation and a fluid that satisfies the Euler equation is a special case of one that satisfies the Navier Stokes [10]. However, there is no equation of state of the pressure, as in the pressure as a function of the density that is used for an isentropic flows an isentropic mediums [11,13]. Instead, Q1 flow has the pressure as a functioan of the Laplacian of the density ρ (4), where the actual form of the function p depends on the "formula" for the density ρ. An energy equation has been derived for Q1 flow [16], but it does not provide any useful information. Henceforth, we ignore any thermodynamic properties a Q1 state may have, and see if this approach leads to any short comings. For example, we do not define an isentropic process for Q1 fluid particles.
For later use, we write (8) and (43) in a more condensed form where the effective pressure P is defined P = p + qρΦ, which for all r ∈ R 3 such that ρ(r) = 0, can be written and P ρ −1 is the specific effective compression-energy or the (total) specific potential-energy.
For the hydrogen 1s flow, P ρ −1 can be obtained by combining (50) and (26) with q = −e: Hence, the hydrogen ground-state has both a constant specific potential-energy P ρ −1 and, from (20), a constant specific kinetic-energy mu 2 /2. 3) The wave-pulse frame. A coordinate frame that moves with the wave pulse. The fluid velocity in the wave-pulse frame is s = sẑ, right in front of the wave pulse, and s = (s+dū)ẑ, right behind the wave pulse. All three coordinate frames have a Cartesian coordinate axis that contains theẑ unit vector. The difference between the velocity right behind the wave pulse, and the velocity right in front of the wave pulse, denoted dū, is the same from all three frames. It not necessary to know or to determine dū, because it is eliminated below by combining the momentum-balance and continuity equations.
In order to derive a solvable equation, we only consider the special case where the speed of sound |s| is constant. With this additional constraint, the wave-pulse coordinate frame, like the the other two frames, is inertial. For the static frame, let the parameter z be the position along the u-velocity streamline; hence, each vector-field and scalar-field restriction to the u-velocity streamline is a function of z. At time t, let z 1 and z 2 be the positions right in front and right behind the wave pulse, respectively. Except for the velocity field u, the values of the pertinent fields for the velocity-and wave-pulse frames are equal to, and can be obtained from, the static frame, including velocity differences between two point on the u-streamline. The later convenience, unit vectorŵ is defined byŵ = −ẑ. Next we derive and combine both the continuity and momentum-balance equations for an arbitrary inertial coordinate frame with a fluid velocityū ± =ū ±ẑ , where final equations also holds in the special where the wave pulse is absent. The combined general equation is applied to the wave pulse frame by settingū ± = s ± , in other words,ū ± | L = s ± | L , yielding equations containing s ± , where |s ± | is the speed sound. Note the the steady flow case is covered by settingū ± = u ± and by removing the time dependent terms.
In the derivations that follow, for a wave pulse propagating on an ambient state, we assume that the time-dependent linear-momentum balance equation (49), a generalization of (42), holds; we also assume that the time-dependent continuity equation (32), a generalization of (30), holds.

B. The Momentum Balance equation
In order to improve efficiency, imperative sentences are used for some derivations that start and end with a large vertical line.
|Use the definition ∂ t = ∂/∂t. After dividing the momentum balance equation (49) by m, take the dot product of the resulting equation with dz =ẑ dz, and then rearrange: and solve (52) for −dū(z): Setū =ū ± and multiply the result by the factor −ρ mū± /dρ m to put it into a more convenient form: For later use (53) is rearranged: |Consider the orthonormal control volume represented in Figure 2, where, for spherical flow, each streamline is a signed ray, and all rays begin at the same starting point. Let z = 0 be the center of the spherical flow, the starting point. The area A(z 1 ) of the frontdoor S(z 1 ) is az 2 1 , where a ≤ 4π. Since, later the limit z 2 → z 1 is taken, make a first-order approximation for the area A(z 2 ) of backdoor S(z 2 ): The volume V (z) of the fluid between the origin and front door S(z 1 ) is (a/3)z 3 1 . Make a first-order approximation for the volume of the system between the origin and the backdoor S(x 2 ): is the approximate volume of the control volume.
|Next compute the individual terms from (32), which, for later convenience, is written Let u = u ± . For the integral from the second term on the lhs, separate it into two parts: Treat the front door term of the rhs of this equation in two steps: In a two-step evaluation of the backdoor term on the rhs of (60), neglect the second-order terms and the product of differentials: Substite (61) and (62) into (60), set z 1 = z, divide by A(z), and use dA/A(z) = 2 dz/z from (56): For the last term on the lhs of (58), first set ρ(r) = 1 in (63): To obtain the desired result, substitute the values into the previous equation and multiply the equation by −α ± : For the first term on the lhs of (58), treat the integral using the mean value theorem, approximating the average value [∂ t ρ m ](z * ) of the integrand over the volume dV by [∂ t ρ m ](z 1 ), also use (57) for dV and set z 1 = z: Obtain the satisfaction of (58) by adding the rhs of (63), (64), and (65), and setting the sum to zero: Obtain the desired result by solving for −dū: Setū ± =ū and multiply it by the factor ρ mū± /dρ m to put it into a more convenient form: For uniform flow, (67) still holds with This follows because dA = 0 for uniform flow, and since dA/A(z) = 2 z −1 dz, the terms with z −1 factor do not appear for uniform flow.
D. The speed of sound quadratic equation |Next eliminate dū from (67) and (54) by adding the two equations in two steps: Consider the equation obtain by settingū ± = s ± in (69). This equation represents the case of a wave pulse contained in, and moving with, the control volume. The function ρ m is an independent variable at our disposal with the constraint that ρ m (z 1 ) is taken from the ambient state. Let ρ m be the limit of a sequence φ 1 , φ 2 , · · · of continuous function that converge point wise such that ρ m has a jump discontinuity at z 1 and ρ m (z 2 ) = lim z→z + 1 φ(z). Hence, lim n→∞ dφ n (z 1 )/dz = ∞, giving lim n→∞ dz/dφ n (z 1 ) = 0, and (69) becomes where the following notation is used for the directional derivative ∇f ·ẑ in the direction of the unit vectorẑ: probably holds for a variety of flow geometries, with a similar limit taken that removed the γ term from (69).

E. The hydrogenic 1s state
Since, for hydrogenic atoms in the 1s state, the specific effective-compression-energy P ρ −1 and (71) reduces to A solution is s ± = 0 for both velocity directions. This is Mach ∞. Hence, if P ρ −1 as a function of ρ is a constant, then the fluid has Mach speed ∞. Such a flow is "silent" since it cannot propagate a wave pulse. A disturbance is only carried downstream with the fluid velocity.
|For the second solution, divide (73) by s: To evaluate the expression use ρ from (19) and notation (72) with the choiceẑ =r, wherê r is the radial spherical-coordinate: Interpretations of this result are given at the end of the next subsection and in (IX B).
F. Another speed of sound quadratic equation.
In order to apply (71), it is assumed that all the functions from this equation can be taken from the ambient medium without the wave pulse. Let X be a scalar field that can be taken from the ambient medium. For many of the derivation and calculation that follow, it is useful to be aware that, we have and notation (72) is used. Note that the brackets (· · · ) matter: (∂X)/∂ρ is the ratio of directional derivatives and the not the same as the partial derivative ∂X/∂ρ.
In order to assign another meaning to dX/dρ, besides being a ratio of differentials of the variable r ∈ R, consider a subspace S ⊂ R 3 such that both ρ and X are strictly monotone.
In that case, the map ρ → X exists with domain W = Range (ρ| S ). Hence, the derivative dX/dρ with domain Range (ρ| S ) exists, and it satisfies (74). This also holds for other scalar fields besides ρ, in particular the streamline velocity u ± .
In the special case of X = ∂ρ, from (74), we obtain where the last statement defines (∂ 2 ρ). Examples of the use of this notation is obtained from speed of sound equation (71) and the definition (3) of u ± : For use below, we obtain an equality by taking the open-density derivative of this equation where u = u + : The use of (77) and (75) gives the result In order to find another speed of sound equation, we derive a relationship between the two density derivatives in (76). Solving (48) for P ρ −1 , followed by taking the density derivative and the use of (78), with u = u + , gives Substituting u + = −u − gives another equation. The combination of the two equations can be written For later use, it can also be written Substituting (79) into (76) we obtain the desired equality: For later use, we write down the quadratic formula for this equation: By inspection, it is immediate obvious that one solution of (81) is s ± = u ± .
For the case of a corresponding classical potential-free flow, (52) and (32) hold with d(P ρ −1 m ) replaced by ρ −1 m dp and replaced by zero, giving, from (76), the well known result that was derived for a static ambient state with one-dimensional flow [11,13]: Hence, (76) is a generalization of (84). However, (81) does not have a classical corresponding equation with gravity neglected, because it is derived from (76) using the velocity definition By inspection, this result also covers the case where u ± satisfies the quadratic formula with the negative sign (−) in front of the square root. Substituting (83) into the previous equation gives the desired result: Also, using −u + = u − we discover that (88) gives us another formula for the speed of sound |s|: Recall that the fluid velocity on the L streamline relative in the static frame is u ±ẑ ; recall that the velocity of the wave pulse in the fluid velocity frame is −s ±ẑ . In other words, the one-dimensional velocity of the wave pulse on the L streamline is −s ±ẑ relative to the velocity of the fluid on the L streamline. With this in mind, and utilizing (89), let the signed Mach speed Ms, a generalization of the the Mach speed Ma, be defined by

VII. EFFECT OF VARIATION IN FLOW CROSS-SECTIONAL AREA
The following equation is applicable on a streamline of the corresponding classical, onedimensional potential-free flows [11]: A is the variable cross sectional area. This equation indicates that an area increase along the streamline results in a decrease in fluid speed, for subsonic flow (Ma < 1), and an increase in fluid speed, for supersonic flow (Ma > 1). A rearrangement of this equation for du = 0, indicates that Mach 1 is an extremum for the cross sectional area A in any region on the streamline where A is a function of the the velocity u. Both Eqs. (92) and (93) are important for characterizing duct flow [11]. For example, if a duct is sufficiently long, a convergingdiverging duct will obtain sonic speed (Ma = 1) at the minimum cross sectional area, called the throat, for either subsonic or supersonic flow.
The above two equations are, of course, applicable for spherical flow.
In this section we derive equations that are analogous to (92) and (93) for Q1 flows: We follow the procedure explained by Munson, Young, and Okiishi [11] in their derivation of (92) and (93), deviating as little as possible, but using, of course, the analogous equations for Q1 flows. At key points of the derivation, the derived equations for Q1 flow are compared to the corresponding classical, one-dimensional potential-free flow equations.
Let u 2= u 2 − = u 2 + . Let the restriction of the velocity field u to the streamline L be uẑ, whereẑ is the unit tangent vector of the streamline in the direction of the velocity field.
Taking the dot product of the momentum balance equation (49) with dz =ẑ dz, for steady flow, yields mu ± du ± + d(P ρ −1 ) = 0, or u ± du ± + d(P ρ −1 m ) = 0, which, for u = 0, can be written and the second one is the corresponding equation for a potential-free classical flow: A special case of the Q1 with d(P ρ −1 m ) = ρ −1 m dp. The continuity equation (32) for an othonormal control region with doors S f and S b is α ± is defined by (59). Sincen(r 1 ) = −ẑ(r 1 ) andn(r 2 ) =ẑ(r 2 ), for r 1 ∈ S f and r 2 ∈ S b , we ∂ρ is the directional derivative in the direction ofẑ. We require the flow to be onedimensional, implying that [ρ m u ± ](r) = [ρ m u ± ](r ′ ), ∂ρ(r) = ∂ρ(r ′ ); r, r ′ ∈ S i , and these equations are satisfied in two separate cases: Since we are interested in the limit of r 2 → r 1 , we approximate this equation to first order The second one, again, is for the corresponding classical flow: A special case with α ± = 0.
|Add equations (94) and (95), and then continue with some algebraic manipulations so that the signed Mach Ms (90) can be substituted: The second one, again, is for the corresponding classical flow. The significant difference between the two arises from the difference in the formulas for Ma and Ms. |Substitute (94) into (96): The corresponding classical flow equation is (92), and (92) implies (93).
It is our objective to obtain equations that are analogous to (92) and (93) for Q1 flows.
In order to accomplish this objective, below we expand out and then simplify the two terms involving the cross sectional A in (97). However, before we do that, we first we obtain a useful equality by taking the dot product of dz =ẑ dz with the velocity equation (3) in the form u ± ρ m = ±α∇ρ, giving, for the streamline, the momentum per volume equation: and u ± ρ m is also called the 'streamline momentum-density.' Substitute this equality into the expansion of the factor from (97) involving the cross sectional area A, we obtain Hence, the area terms cancel in (97), giving Hence, for the Q1 flows, we do not obtain the same type of relationship involving the cross sectional area, briefly mentioned in the introduction to this section, and described in detail elsewhere [11]. Instead, next we obtain a different relationship involving the streamline momentum-density ρ m u ± .
Substituting (98) into (99), we get If du ± = 0, this equation can also be written Hence, the extremums of the momentum density ρ m u ± on a streamline occur at points r ∈ L of space where ρ(r) = 0 and Ms(r) = 1. Also, since Ms ≥ 0 implies Ma = Ms, Ms(r) = 1 implies Mach 1.
Consider a point r ′ ∈ R such that ρ(r ′ ) = 0. In that case, for the flows considered in the next section, we find that the limit u ± → 0 as r → r ′ does not exists and that |u ± | → ∞ as r → r ′ . Hence, the derivative d(ρ m u ± )/du ± does not exists at points where ρ(r ′ ) = 0.
Hence, extremums of the momentum density ρ m u ± do not necessarily occur at such points.

Definition.
A connected surface such that the momentum density is minimum and maximum on the streamlines are called nodes and antinodes, respectively. With the assumption mentioned above, the nodes and antinodes are defined by ρ(r) = 0 and Ma(r) = 1, respectively.
In the special case where u ± = 0 and u ± is constant, (100) reduces to

VIII. APPLICATIONS
The formula for the speed of sound for classical compressible flows (84), s = + dp/dρ m , is derived with the restriction that the one-dimensional velocity u ± and wave pulse s ± are constant along the streamline (and the flow is either spherical or uniform). However, this same formula is, apparently, also very successful in describing compressible flows without these restrictions [11,17]. With this in mind, it is assumed that (88), and other derived speed of sound formulas, including (89), hold, or are useful approximations, for Q1 flows.
In this section a set of Q1 flows are examined using the formulas derived in the previous sections for many physical properties, including the speed of sound. Flow that are neither uniform nor spherical are not considered, but I suspect that (88) could be derived with a variety of flow geometry with the restriction of constant u ± and s ± . (The speed of sound equations derived actually also hold if the sum u ± + s ± is constant, but u ± and s ± are not constant, since the wave pulse frame is still inertial.) All the calculations in this section use atomic units.
A. A fluid (or particle) in a one-dimensional box and ordinary Q1 flow For a particle, or fluid, in a one-dimensional box of length ℓ, let ℓ be the Bohr radius a 0 . In one-dimensional cases, the directional derivatives reduce to ordinary derivatives, but the general notation is still used below, with the definition ∂ρ= dρ/dr, and r ∈ R is the position variable for this subsection. |Substitute (cos 2 (nπr) = 1 − sin 2 (nπr)) into (105) and use (4), i.e., (p = −∂ 2 ρ/4), to calculate the pressure: Use ρ −1 (r) = csc 2 (nπr)/2 and csc 2 x = 1 + cot 2 x to calculate the specific compression energy pρ −1 : Calculate the specific kinetic-energy mu 2 /2 using (103) and the well known total energyĒ n using the formula for the specific total-energy (48), given byĒ n = mu 2 /2 + pρ −1 :  accelerating to the right, in other words, they move downhill with respect to the "density hill," so to speak; for r < 0.5 they move and accelerate to the left. In the limit of a fluid particle reaching a node, i.e., r → 0 or r → 1, the particle will have infinity speed  Definition. An equation of state for a Q1 flow, when it exists, for the pressure p and the effective pressure P , are equations that define the maps ρ(r) → p(r) and ρ(r) → P (r), respectively, where the maps hold for Range(ρ).
Note that the above definition gives a stronger condition for an equation of state to exist than one where the pressure p is determined by ρ and a given nonempty set of its partial derivative, or the pressure is a functional of the density ρ. Also, the definition is similar to how a thermodynamic equations of state from static fluids are used for substances undergoing laminar flow, or when a homogeneous substances becomes only a continuous one with independent variables represented by scalar fields [19].
By the definition above, (51), which can be written is an effective pressure equation of state for the ground-states of hydrogenic atoms, and P is a linear function of ρ(r) ∈ Range(ρ). Combining (109) with ρ = 2 sin 2 (nπr) gives the pressure equation of state for all flow states in a one-dimensional box: and p is an affine function of η = ρ(r).
A function f : R −→ R is said to be symmetric with respect to the point c ∈ R within   So that different flows can be compared with ease, and flows can be characterized, a reference flow, called ordinary Q1 flow, is defined below. The properties defining ordinary Q1 flow are the ones viewed to be important that are exhibited repeatedly in the applications; they might hold for all Q1 flows. These properties are considered to be fluid (no pun intended), so they can be modified without hesitation.
Definitions. A node r node ∈ R 3 is a point on the streamline such that ρ(r node ) = 0.
An antinode r antinode ∈ R 3 is a point on the streamline such that ∇ρ(r antinode ) ·ẑ = 0. Let 2) There exist a map ρ(r) → Ma(r), r ∈ S, such that the restriction of the Mach speed Ma to Range(ρ| S ) is a strictly decreasing function of ρ(r) ∈ Range(ρ| S ).
3) For r restricted to L and s ∈ {s + , s − }, u ∈ {u + , u − }, the following conditions are satisfied for all nodes r anti-node , antinotes r node , and points at infinity r ∞ that are members of the closure of L: i) As r → r anti-node , u → 0, Ma → 0.
An ordinary Q1 flow is a Q1 flow that is an ordinary Q1 streamlime flow for each streamline.
Definition. Let P be the set of all primitive flow segments of streamline L. The primitive flow segments I and I ′ are related if they contain the same antinode end-point r anti-node , and this state of affairs is denoted by (r 0 , r anti-node , r ′ 0 ), where I = [r 0 , r anti-node ] and I ′ = [r anti-node , r ′ 0 ], and (r 0 , r anti-node , r ′ 0 ) is called a flow pair or a flow hill. For a particle in a one dimensional box, the number of flow hills is (n − 1), where n is the number of nodes including the end points for the fluid in a box.
By examining the pertinent functions, ρ, u, ρu, s, and Ms, it is easily verified, as it is illustrated in figures 3, 5 and 6, that the ground state fluid in a box is an ordinary Q1 flow containing one flow hill (0, 1/2, 1).
It is also easily verified that the first excited state is an ordinary Q1 flow containing the two flow hills (0, .25, .5), (.5, .75, 1). Figure (7) and (8) are plots of the same variables as in (5) and (6), respectively, but for the first excited state. Fig. (8) is for the [3/4, 1] primitive flow segment, and the behavior of the functions in r ∈ [0, 3/4] can be obtained by noting that the ±-functions and ρ are antisymmetric and symmetrical, respectively, with respect to the points 1/4, 2/4 and 3/4, all within 1/4, as displayed in Fig. (7) for ρu − and ρ.

B. The harmonic oscillator
In this subsection we treat the ground state and first excited state of the harmonic oscillator, with quantum number n = 0, 1. The Q1 wavefunctions for the ground-and first excited-states of the harmonic oscillator are [7,18] where α = mω/(2 ), ω = k/m, and the specific potential-energy is V (x) = mω 2 x 2 /2. We choose m, k = 1, giving ω = 1 and α = 1/2 in atomic units.
It goes without saying that the behavior of a ground-state fluid-particle from the harmonicoscilator flow with the specific potential-energy P ρ −1 , and with an unstable equilibrium point at x = 0, is totally different, and in many ways opposite, of a point-mass By examining the pertinent functions, it is readily verified, and it is also illustrated in figures 9, 10 and 11, that the ground-state harmonic-oscillator flow is an ordinary Q1 flow with speed limit ∞ for both primitive flow segments, i.e., |u| → ∞ as r → ±∞.
One significant difference with this flow compared to the fluid in a box flows is that ground-state harmonic-oscillator flow contains points at infinity r ∞ = ±∞, while the flows for the fluid in a box contain node points r node . The two different types of points, r node and r ∞ , yield the following differences for the functions Ma and |s|: As r → r node , Ma → 2 and |s| → ∞, and as r → r ∞ , Ma → ∞ and |s| → 0.

Figure (13) and
By examining the pertinent functions, it is readily verified, and it is also illustrated in the three figures, that the first excited-state harmonic-oscillator flow is an ordinary Q1 flow.
Also, as in the ground state, the speed limit is ∞ for both primitive flow segments.

C. The hydrogen 2s state
In this subsection we treat the 2s state of the hydrogen atom. In spherical coordinates, each streamline depends on the radial coordinate r only. So the results hold for any streamline given by a ray r ∈ [0, ∞) for and any polar and azimuth angles.
Calulate the second term on the rhs of (A1) using (136).
Obtain the pressure p, by adding this result to the rhs (138) as indicated in (A1).
The 2s state has the primitive flow segment [0, 2] and the flow hill [2, 4, ∞). The segment [0, 2] contains the nucleus r nuc = 0 and the node r node = 2, but it does not contain an antinode. As discussed below, there two Mach points at 2.59 and 5.41, and these are represented by vertical lines in the following plots.  Fig. (12), but the two differ significantly on the rhs of the unstable equilibrium point. Also, unlike the harmonic oscillator flows, the speed limit of 1/2 is finite. Fig. 16 presents the same scalar fields as in Fig. 15, but for the primitive flow segment |Use φ(r) = η(2 − r)e −hr and (136) to calculate ∂ρ(r) and ∂ 2 ρ(r).
By examining a plot of the function Ms, as in Figures (17) and (19), it is almost certain that Ms is non-negative. Also, there are two Mach 1 points, 2.59 and 5.41, and these are on either side of the only antinode at 4.0. Fig. 17 presents the same type of Mach-pressure plot as (10) and (13), but for the hydrogen 2s state with the flow-hill subset [2,6] ⊂ [2, 4, ∞).
The overall behavior exhibited by the functions are similar to the ones displayed in Fig. (13).
However, as expected, the two points r ∈ R 3 where p(r) = 1, do not satisfy Ma(r) = 1. Fig. 18 presents the same type of wave-pulse-velocity plot as (11) and (14), but for the hydrogen 2s state with the flow-hill subset [2,6] ⊂ [2, 4, ∞). The overall behavior exhibited by the functions are similar to the ones displayed in Fig. (14). Fig. 19 presents the same functions as in both (17) and (18) but for the electron fall [0, 2].
All the functions, except for the pressure p, have finite limits at the point-charge location r nuc = 0. In particular, note that Ma(r nuc ) = 8. Also, the pressure has a similar shape as the one for the 1s flow, Fig. 1 is a strictly increasing, not decreasing, function of ρ(r) ∈ Range(ρ| [0,2] ).
By examining the pertinent functions, it is not difficult to demonstrate that |ρu ± | → 0, |s ± | → 0 and Ma → ∞, as r → ∞. Furthermore, it is readily verified, and it is also illustrated in the three figures, that the hydrogen 2s flow is ordinary Q1 flow.
The behavior of the hydrogen 1s flow, with the single electron fall [r nuc , r ∞ ) of a given streamline, can be compared to the 2s flow at the limiting points r nuc and r ∞ . The r ∞ limits of |ρu ± |, |s ± |, and |Ma| for the hydrogen 1s state are the same as the 2s state, and the limit of |u ± | differ, but they are both finite. Also all the functions, except for the pressure p, have finite limits at the point-charge location r nuc = 0. However, the hydrogenic flow is different from the others considered, because it his a single fall with at point at infinity r ∞ as an end point.

IX. DISCUSSION
A. The compatability with quantum mechanics and the continuum assumption For later use, we note that we have used the time-dependent momentum balance (49) and continuity (32) equations for the derivation of the speed of sound equation (71 Let φ be a normalized Q1 wavefunction. Let χ be a normalized, nondegenerate eigenvector with eigenvalue λ of a Hermitian operatorÔ with a discrete spectrum representing an observable, such that the inner product (φ, χ) of the L 2 Hilbert space satisfies |(φ, χ)| 2 ∈ (0, 1). Hence, φ and χ are linderly independent, but not orthogonal. According to an axiom of quantum mechanics, if a measurement is made of the observable with operator O for the probability state represented by φ, then the probability of the measured value being equal to λ is |(χ, φ)| 2 . Also, if the measured value is λ, then the quantum mechanical system is in the probability state χ immediately after the measurement is made. In other words, the measurement has transformed the state φ into χ. Such a transformation for the corresponding Q1 flow state involves a time-dependent process and non-steady flow. Hence, it is outside the range of applicability of Q1 flow. The same conclusion is easily shown to hold for observers represented by bounded self adjoint operators such that λ corresponds to a subspace, and to cases where the operatorÔ is unbounded, but still self-adjoint, with a continuous spectrum. Therefore the assumed existence of a Q1 flow state does not lead to any contradictions with the corresponding existence of the Q1 probability state. Both states can coexist, within the very limit range of applicability of Q1 flows.
In this work, certain one-body quantum systems have been endowed with a continuum with vector and scalar fields. I do not have sufficient knowledge of experimental quantummechanics to know if the properties represented by these fields can be measured, to determine if they "exist" by some reasonable definition. Therefore, in order to proceed, and make some progress on this question, suppose these properties cannot be measured, nor can an experiment be performed that shows that their existence contradicts experimental facts.
Under these circumstances, how is it to be determined at some point in the future if this endowment is successful? To approach an answer to this question we turn to Marion and Thorton, where they discuss the appending of conservation postulates, including energy, on systems that do not have conservative potentials: We therefore extend the usual concept of energy to include "electromagnetic energy" to satisfy our preconceived notion that energy must be conserved. This may seem an arbitrary and drastic step to take, but nothing, it is said, succeeds as does success, and these conservation "laws" have been the most successful set of principles in physics.
Hence, the endowment will succeed if useful applications can be found.
B. Q1 flow with the solution s ± = u ± Here two interpretations are given for a Q1 flow with the solution s ± = u ± from (81).
1) Consider a a corresponding classical potential-free flow with Mach speed 1 and uniform velocity uẑ. If a wave pulse travels downstream and upstream, the speed of the wave pulse in the static frame is 2u and 0u, respectively. (This same behavior is also observed for a streamline L of spherical flow with the wave pulse moving up or down the streamline L.) Assuming this type of behavior also holds for Q1 flow, the solution s ± = u ± for Q1 flow is consistent with Mach 1, giving u ±ẑ − s ±ẑ = 0 for the wave-pulse velocity in the static frame, even though, for some reason, the corresponding solution s ± = −u ± , giving u ±ẑ − s ±ẑ = 2u ±ẑ for wave-pulse velocity in the static frame, is absent.

2a)
Since, for s ± = u ± , the velocity of the wave pulse in the static coordinate frame is u ±ẑ − s ±ẑ = 0, the solution s ± = u ± is for the case where the wave pulse frame and the control volume are at rest relative to the static coordinate frame. Hence, the solution is simply the one for a steady flow with velocity u ± , and the "speed-of-fluid" equation is (69) withū = u and γ = 0, and there is no wave pulse.
2b) The possibility of time dependent flow cannot be ruled out because (69), withū = u and γ = 0, also yields (71) with s ± = u ± , where ρ m is taken as the limit of a sequence of functions. (There is no reason to continue with the derivation to (81), since s ± = u ± is set.) Such a flow would be silent with respect to the static frame, since there is no moving wave-pulse relative to the static frame. Also, for such a flow, a disturbance is not carried downstream with the fluid velocity, instead, just like steady flow, fluid particles are perturbed as they pass through a disturbed region, but, since γ = 0 is permitted, the disturbance could be time dependent. Hence, the existence of a Q1 system that cannot carry a wave pulse is conceivable. The situation is different for classical systems, since the wave pulse velocity of (−s ± ) = −u ± is not always one of the solutions of the quadratic wave-pulse equation. |Use ∇ρ = 2φ∇φ and (3) to obtain an orbital expression for the velocity u ± , the velocity component u ± = u ± ·ẑ, and the kinetic energy density mu 2 /2.         (17) and (18)