A discussion on a dynamic vacuum model: Derivation of Helmholtz equation from Schrödinger equation

Physics Open 1 (2019) 100009

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A discussion on a dynamic vacuum model: Derivation of Helmholtz equation €dinger equation from Schro H. White *, P. Bailey, J. Lawrence, J. George, J. Vera NASA Johnson Space Center, Houston, TX, 77058, USA

A R T I C L E I N F O

A B S T R A C T

Keywords: PACS: 67.10.Jn 43.20.-f 47.37.þq

In this paper, Madelung’s quantum Euler equations will be re-derived, which will be shown to lead to the familiar form of the acoustic wave equation for a medium. A model of the density variation in the vacuum field around the nucleus of the hydrogen atom is then introduced and used to create a simplified representation of the atom in COMSOL. The time-independent form of the acoustic wave equation (Helmholtz equation) is used to find the eigenfrequency acoustic modes of the simulated atom. The analysis technique for the single atom case is extended to build a model of molecular hydrogen, and the numerical analysis results are presented and discussed. The paper concludes with some speculative discussion on the nature and make up of the dynamic vacuum medium being modeled around the hydrogen nucleus. This polar form of the Schr€ odinger equation can be separated into its real and imaginary parts respectively:

1. Background The Copenhagen interpretation is the dominant point of view taken by the physics community today, and has been so since early in the last century. Stated simply, the Copenhagen interpretation of the physical world at a quantum level is that the nature of reality is probabilistic and not deterministic. This is made manifest in a mathematical sense by means of the Born rule, which states that the probability density of finding a particle at a particular point in space can be obtained by taking the square of its wave function amplitude. While the Copenhagen interpretation enjoys the distinction of being the favored line of thinking, it is not without alternatives. One of the alternative lines of thinking in the ontological debate about the nature of reality at the quantum level is pilot wave theory. The pilot wave theory viewpoint proposes that the statistical nature of quantum mechanics arises as a result of an unknown, underlying real dynamics - sometimes referred to as (nonlocal) hidden variables. Stated another way, at the quantum level, reality is not probabalistic, rather it is in fact deterministic; and the stochasticity that is observed arises from the manifestation and influence of the quantum potential. The concept was first proposed by de Broglie in 1923 [1], and was in fact a considerable focus of discussion during the 1927 Solvay conference in physics [2]. Colloquially stated, de Broglie intimated that a quantum particle generates and interacts with some type of real wave at the quantum level, and the wave interacts with the particle directing it on

a real path, or “pilots” the particle. During the Solvay conference however, Pauli raised some objections that the concept would potentially violate relativity (now known as quantum entanglement), so the idea did not gain wide acceptance at the time. This did not deter some from continuing to develop and explore the concept further. In 1927, Madelung took an approach that converted the Schr€ odinger equation into its polar form, and after some manipulation yielded two continuity equations that resemble the Euler equations from hydrodynamics. These equations are today commonly referred to as quantum Euler equations, or equations of quantum hydrodynamics. David Bohm published two papers in 1952 [3,4] detailing his approach to the pilot wave concept, and his work significantly increased awareness of the idea in the community. An approach known as Stochastic Electrodynamics (SED) [5] explores the idea that the underlying real dynamics arise as a result of quantum vacuum fluctuations, rather the zero-point (electromagnetic) field, serving as the source of stochasticity observed in the quantum realm. This paper will focus on the Madelung equations that can be categorized as quantum Euler equations. The form of these equations bear close resemblance to the Euler equations of classical hydrodynamics, one of which is representative of the continuity equation, the other of the force equation in a fluid medium. These equations will be shown to lead to the traditional form of the acoustic wave equation, which can be used to solve for the acoustic eigenfrequencies of a system by switching to the time independent form of the equation known as the Helmholtz equation.

* Corresponding author. E-mail address: [email protected] (H. White). https://doi.org/10.1016/j.physo.2019.100009 Received 11 October 2019; Received in revised form 12 November 2019; Accepted 21 November 2019 Available online 29 November 2019 2666-0326/Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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2. Madelung

Hamiltonian or some external source, rather it appears to be representative of some underlying internal dynamics, possibly in the form of a field, for the system. It has been observed in the literature [7] that the probability density may be associated with a mass density ρm ¼ mρ that is identified with a continuous medium or fluid that extends throughout space, consisting of corpuscles of matter with mass m, and with v being representative of the velocity field of the fluid medium. If these equations are indeed indicative of the presence of an inviscid and continuous “vacuum-fluid” medium that can manifest characteristics similar to its classical counterparts, then it should be able to express spatial and temporal variations in that vacuum fluid. Consequently, it should be possible to manifest rarification and densification of the medium, and hence sustain the generation and transmission of longitudinal, that is, acoustic waves. As such, it will be helpful to explore the derivation of the acoustic wave equation for a continuous medium as described herein, which will be done in the next section adapting the traditional approach of deriving an acoustic wave equation for a generic fluid medium detailed in the literature [9].

Madelung provided a hydrodynamic-analog physical interpretation of the Schr€ odinger equation that yields a set of quantum Euler-like equations by rewriting the Schr€ odinger equation iℏð∂ψ =∂tÞ ¼ ððℏ2 =2mÞr2 þVÞψ in polar form utilizing ψ ¼ ReiS=ℏ , where R is the amplitude of the wave function ψ and S=ℏ is the phase. To make explicit the assumptions involved in producing the Madelung equations, and to illuminate the difference between the Madelung approach and the standard Copenhagen interpretation, it will be instructive to go through the detailed derivation process here. Readers already familiar with this derivation process detailed in the literature [6,7] may want to skip to the next section. Utilizing ψ ¼ ReiS=ℏ in the Schr€ odinger equation and multiplying through by eiS=ℏ yields the following expanded, polar form of the equation:    
∂R iR ∂S ℏ2 R þ ¼ r2 R  2 ðrSÞ2 iℏ ℏ ∂t ∂t 2m ℏ  
   2 R rR  rS þ r2 S þ VR: þi ℏ ℏ

(1)

∂S ðrSÞ2 ℏ2 r2 R þ V ¼ 0; þ  ∂t 2m 2m R

(2)

∂R 1  2rR  rS þ Rr2 S ¼ 0: þ ∂t 2m

(3)

3. Acoustic wave equation The probability density ρ can be identified with the mass density ρm by multiplying by the mass such that ρm ¼ mρ, with m being representative of a fundamental unit of matter (notionally, ephemeral vacuum pairs) that travels in and through the quantum “fluid” with a velocity v. Multiplying Eq. (4) by ρm and recasting Eq. (5) into a function of ρm by multiplying through by m yields the following form:

Equation (2) has a term,  ðℏ2 =2mÞðr2 R =RÞ, that represents the curvature of the amplitude of the wave function that has come to be known as the quantum potential Q. This will be discussed further in a moment. Taking the gradient of Eq. (2) and mutiplying through by 1= m yields the modified form:   ∂ rS rSr2 S 1 þ þ rðQ þ VÞ ¼ 0: m2 m ∂t m

ρm

∂ρm þ r  ðρm vÞ ¼ 0: ∂t



(6)

(7)

As part of deriving the acoustic wave equation for the inviscid “vacuum” fluid medium, the following nomenclature will be used:

(2a)

Making use of the relationship ∂R2 =∂t ¼ 2Rð∂R =∂tÞ for the first term of Eq. (3); and that rR2 ¼ 2RrR and r ðR2 rSÞ ¼ rR2 rS þ R2 r2 S successively for the parenthetical term, the equation can be brought into the following form:

ρm ¼instantaneous fluid density at a point; ρm0 ¼equilibrium density at a point;

s ¼condensation at a point s ¼ ðρm  ρm0 Þ = ρm0



∂ R2 R2 rS ¼ 0: þr m ∂t

∂v ρ þ ρm ðv  rÞv ¼  m rðV þ QÞ; ∂t m

(3a)

ρm  ρm0 ¼ ρm0 s ¼acoustic density at a point

The following field relationships are now introduced: ρ ¼ R2 and v ¼ rS=m where ρ can be recognized as the probability density and v is the flow velocity of a fundamental unit of mass traveling along with the continuous medium. Using these relationships with Eq. (2’) and Eq. (3’) allows them to be rewritten into the following forms:

∂v 1 þ ðv  rÞv ¼  rðV þ QÞ; m ∂t

(4)

∂ρ þ r  ðρvÞ ¼ 0: ∂t

(5)

P ¼instantaneous pressure at a point; P 0 ¼equilibrium pressure at a point; p ¼acoustic pressure at a point p¼P  P 0

cs ¼propagation speed of longitudinal wave through the medium Considering the continuity equation (7), one can write that ρm ¼ ρm0 ð1 þ sÞ. If it is assumed that s is very small, and that ρm0 is a weak function of time, we get the following linear continuity equation:

These are known as Madelung’s equations, his core result. The Madelung equations have forms similar to those of Euler’s equations of hydrodynamics with Eq. (4) being the force equation for fluid flows and Eq. (5) being the continuity equation. It is interesting to note that the quantum potential term emerges naturally from the process of reexpressing the Schr€ odinger equation in polar form (from the real part), which suggests that its nature is inherent in the framework of the model. That is, the implications of the quantum potential’s presence are always felt, but it requires that a suitable choice of coordinate system be made for its role in the theory of quantum mechanical motion to become mathematically explicit. Complementary to this thought, Hiley made the observation [8] that the quantum potential term does not come from the

ρm0

∂s þ r ðρm0 vÞ ¼ 0 ∂t

(8)

Looking at the force equation (6), in the case of no acoustic excitation, it can be stated that the classic potential term is representative of the equilibrium pressure gradient,  ρm0 rV=m ¼ rP 0 , and that the quantum potential term represents the instantaneous pressure gradient encoding both the stochastic and equilibrium nature of the medium yielding ρm rQ=m ¼ rP ¼ rp þ rP 0 . Incorporating these statements changes the force equation to the following intermediate form:

2

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  1 rV ∂v s ¼ ð1 þ sÞ þ ðv  rÞv :  rp  m ρm0 ∂t

and phase speed being functions of position. For harmonic waves of the form eiωt , the pressure perturbation will take the form of pðr;tÞ ¼ pðrÞeiωt . Putthing this into 17 leads to the time independent form of the acoustic wave equation, the Helmholtz equation, ðr2 þk2 ÞP ¼ 0 (where k ¼ ω=c), which can be used to solve for the acoustic eigenfrequencies of a physical system.

(9)

If the assumptions are made that jrVs =mj ≪ jrpj=ρm0 , that jsj ≪ 1, and that jðv  rÞv ≪ j∂v=∂tj, one arrives at the linear Euler’s equation, valid for low amplitude longitudinal waves.

ρm0

∂v ¼ rp ∂t

(10)

4. Dynamic vacuum model of hydrogen atom

The final piece necessary to complete this process is to consider the isentropic equation of state for the continuous and inviscid medium. The relationship between total pressure P and the mass density ρm can be represented as a Taylor series expansion: 

∂P ∂ρm

P ¼P 0 þ

þ

The previous sections have completed a painstaking process showing a deep and explicit connection between the Schr€ odinger equation and the Quantum potential which led to the detailed derivation of the acoustic wave equation for the quantum medium. Armed with the insight that this quantum medium might be viewed as a spatially and temporally varying quantum fluid, one could ask how this viewpoint may be utilized to develop deeper insights into the quantum regime. With this objective in mind, the mechanics of the preceding sections will now be brought to bear on the hydrogen atom, where the Helmholtz formulation will be used to find the acoustic eigenfrequencies (or resonances) for the system. These frequencies will then be compared to the observed characteristics of the electron orbital states. In previous work [10], the authors considered a heuristic model of the hydrogen atom. The heuristic model consisted of the assumptions of a quantum fluid surrounding the proton, a density distribution of that fluid, and of a velocity field describing the spatially varying speed of propagation of a disturbance in the density field. In this dynamic system, acoustic-like resonances could be established describing the electron orbitals as determined by the Helmholtz wave equation. The authors found that the acoustic eigenfrequencies of the system matched observation with a reasonable degree of accuracy (ψ 100 error was e5%). Equipped with the rigorous derivation of the Helmholtz equation from the Schr€ odinger equation, we now proceed by developing a more direct derivation of the density distribution function and velocity field for the system. Having derived the Helmholtz equation from the Schr€ odinger equation, we need two input parameters to develop and analyze a model utilizing the Helmholtz equation: the density and velocity field of the quantum fluid around the proton, as well as some boundary conditions. Note that this is the explicit difference in our approach versus the standard approach of using the Schr€ odinger equation with the Copenhagen interpretation. This is a deterministic approach modeling the vacuum as a fluid in which longitudinal waves can manifest. This is compared to the Copenhagen interpretation in which nothing is explicitly waving, rather the Schr€ odinger wave equation describes probabilities only. Our view may then be considered complementary to the Copenhagen interpretation in that the density and velocity fields play the role of hidden variables with the pilot wave interpretation. In order to derive an equation for the mass density of the presumed quantum fluid surrounding the proton, consider the electrostatic field E around the proton: the electrostatic pressure field is PE ¼ ε0 E 2 =2 where

 ρm0

ðρm  ρm0 Þ

(11)

  1 ∂2 P ðρ  ρm0 Þ2 þ … 2 2 ∂ρm ρm0 m

The partial derivatives are calculated with respect to the compression and expansion of the medium about the equilibrium density. If the assumption is made that the magnitude of the longitudinal waves are small, then only the lowest order term need be retained which yields the linear relationship between the pressure and density fluctuations: P  P 0  B ðρm  ρm0 Þ = ρm0 :

(12)

Equation (12) makes use of the adiabatic bulk modulus term B ¼

ρm0 ð∂P =∂ρm Þρm0 , and can be rewritten in terms of acoustic pressure p and condensation s, with the restriction that condensation is small. p  Bs

(13)

The linearized Eq. (8), Eq. (10), and Eq. (13) can now be utilized to yield a partial differential equation that is a function of one variable, specifically the acoustic pressure p. The first step is to take the time derivative of Eq. (8) maintaining the assumption that ρm0 is a weak function of time. 

ρm0



∂2 s ∂v ¼0 þ r  ρm0 ∂t 2 ∂t

(14)

The next step is to take the divergence of Eq. (10) to yield: 

 r  ρm0

∂v ¼  r2 p: ∂t

(15)

Equation (14) and Eq. (15) can be combined to eliminate the divergence term to yield the following form: r2 p ¼ ρm0

∂2 s : ∂t 2

(16)

E ¼ kq=r 2 and k ¼ ð4πε0 Þ1 . The energy density in the field can be determined by making use of the equation of state relationship between pressure and energy density w ¼ P=ε with w ¼ 1=3 for radiation. Using the Einstein mass-energy equivalence, the predicted mass density in the field around the nucleus is quite close to the heuristic derivation relationship just discussed (Fig. 1a shows the semi-log plot):

Now Eq. (13) can be rewritten to express the condensation as a function of acoustic pressure and bulk modulus s ¼ p= B , and this can be used in Eq. (16) to arrive at the desired acoustic wave equation where c represents the propagation speed of longitudinal waves in the medium, c2s ¼ B =ρm0 : r2 p þ

1 ∂2 p ¼0 c2s ∂t2

(17)

ρm ¼

Some observations about Eq. (17) should be made at this point. This is a linear and lossless wave equation that describes the propagation of longitudinal waves in a continuous medium where the phase speed is defined as cs . Similar to the traditional derivation of this equation for the classical scenario, there were no restrictions put on B and ρm0 with respect to position, so Eq. (17) is valid for a medium with both density

3q2 1 3:06  1046 ¼ : 2 2 32c π ε0 r4 r4

(18)

The derivation of the spatial dependency of the longitudinal wave speed in the medium takes a similar approach to the heuristic method applied to the density from the cited literature. The azimuthal “effective” speed of the electron is calculated for n ¼ 1 to n ¼ 7 states using the Bohr model, and a curve is fitted to the tabulated radius-velocity data which pffiffi shows that v∝1= r . The final equation for wave propagation speed used 3

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Fig. 1. Plot of (log) density and velocity fields for the molecular hydrogen model.

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in the simulation is cs ðrÞ ¼ ðαcÞ=ð2π r=rBohr Þ, and Fig. 1b shows a plot of the function. As will be shown, incorporating this velocity ansatz into the model yields eigenfrequencies that match observational data very well. A two-dimensional axis-symmetric finite element model was built such that the extent of the modeling space was sufficient to contain the first 7 primary quantum number states, n ¼ 1 to n ¼ 7. The density distribution and velocity relationships discussed previously were used with the numerical model to represent the salient characteristics of the continuous medium. External sources were provided to stimulate the model across a target frequency range to facilitate the determination of the eigenfrequencies (acoustic resonances of the system by solving the Helmholtz equation). The results for the first seven primary quantum numbers are presented in Table 1. The corresponding eigenfrequency pressure distributions from the numerical analysis tool for the solutions presented in Table 1 are depicted in Fig. 2. The resonance shapes depict large pressure changes as bright regions (lobes) and unchanging pressure as dark areas (nodal surfaces). Additionally, the numerical analysis found eigenfrequencies representative of the 2p, 3p, and 3d states, which are shown in Fig. 3.

ρtotal ¼ ρðr1 Þ þ ρðr2 Þ:

This sum assumes that energy and mass density is additive, and this formulation is applied to the areas filling the entire computational space between and around the atoms. The longitudinal wave speed determination required a different technique, however, since a straightforward linear addition could not as easily be justified. Instead, a new formution is required. First, Eq. (18) is rearranged and solved for r as a function of density:  rðρÞ ¼

6:579  8:224  2:437  1:028  5:263  3:046  1:918 

Model 1015 1014 1014 1014 1013 1013 1013

6:553  8:324  2:479  1:019  5:358  3:239  2:248 

(20)



3q2

32π 2 ε

0

18 :

(21)

c2 ρ

Using Eqs. (18), (19) and (21), the density and speed of sound for the entire computational space around and between the atoms is mapped. Fig. 4 shows the log of the composite density field and velocity field for the molecular hydrogen model. The resonance response spectrum for the H2 model as measured by the Quality Factor (QFðf Þ ¼ Pmax ðf Þ=Pstimulation ) is presented in Fig. 5. The scan range of the spectral response search is 5% the ionization energy frequency value of 1:0928  1015 Hz. The linear plot is presented in Fig. 5a, and the semi-log plot is presented in Fig. 5b. The two greatest peaks found in this scan, and several subsequent high-resolution scans around each individual peak, are at f1 ¼ 1:1109230  1015 Hz and f2 ¼ 1:1120691  1015 Hz. Fig. 6 shows the normalized pressure distributions for the resonant system responses at f1 and f2 . The plots show a high pressure standing wave with the rough shape of an ellipse that surrounds the two atoms with a major axis length of 0.44 nm and minor axis length of 0.36 nm. These plots show a pressure that is greatest in the space opposite the two nuclei and a pressure that is slightly smaller in the space between them. The plots for the two frequencies also show a slightly different shape between the two with a slight hourglass shape becoming more prominent in the response of f2 (Fig. 6b) and a slightly lower pressure in the internuclear region relative to the max pressure opposite the nuclei. The numerical analysis of the model predicts one eigenfrequency for the twoatom system of 1:110923  1015 Hz which is within 1.66% of observation, and a second eigenfrequency of 1:112069  1015 Hz which is within 1.76% of observation. These analysis results indicate that the modeling approach used for a single atom case can be extended to a multi-atom scenario with no notable modifications to the overall “mechanics” of

Error 1015 1014 1014 1014 1013 1013 1013

;

pffiffiffiffiffiffiffiffiffi

αc rBohr 2π

Table 1 This table shows the theoretical and model eigenfrequencies (Hz) for the first seven primary quantum numbers. Theory

32π 2 ε0 c2 ρ

cs ðρÞ ¼

With the completion of the analysis of a single hydrogen atom, the following question was posed: would it be possible to use the single atom model to create a two atom model of molecular hydrogen? A new two atom model was constructed that included two instantiations of the single atom model placed at an inter-atomic spacing of 74 pm. At this separation, the binding/ionization energy, that is, the energy required to completely separate the two H atoms from their bound state is at its greatest level at 436 kJ/mol, or 4.5188 eV. Using the equation f ¼ E= h, this corresponds to a frequency of 1:0928  1015 Hz. Modeling H2 adopts a very similar approach to the method used to model the single hydrogen atom. The density distribution between the two atoms is performed using a simple linear summation of the radially-dependent density function of Eq. (18):

1 2 3 4 5 6 7

14

3q2

then rðρÞ is substituted into the speed of sound equation derived earlier for the hydrogen atom to yield the following form:

5. Dynamic vacuum model of molecular hydrogen

n

(19)

 0:395% 1:217% 1:740%  0:885% 1:803% 6:323% 17:171%

4

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Fig. 2. Eigenfrequency pressure distributions from the numerical analysis tool for the n ¼ 1 to n ¼ 7 states. Note that the panels are zoomed at different levels and the horizontal axis provides the scale.

the solution finding process. Further, even though the assessment here deals with molecular hydrogen as a simple analysis problem of modest complexity, the results are encouraging enough to reason that it will be possible to successfully extend this new modeling approach to more complicated molecules.

6. Discussion In the analysis presented, beginning with the postulate of a medium in which the Schr€ odinger wave equation is modeling the propagation of a real wave, we have derived both a density and a wave propagation

Fig. 3. Eigenfrequency pressure distributions from the numerical analysis tool representative of the 2p, 3p, and 3d states. Note that the panels are zoomed at different levels and the horizontal axis provides the scale. 5

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properties of an acoustic wave. In an acoustic wave expressed in air, water, or some other familiar medium, the unperturbed state of the medium is such that individual particles are arranged with some nominal spacing yielding an average number density of particles. In the presence of an acoustic wave disturbance, the particles are alternately compressed and rarefied about the unperturbed number density as the wave transits through the medium. Said another way, the specks of mass of the medium are moved closer together and farther apart by the transit of the wave, as compared to their rest state. Applying the concept to the case modeled in this paper, it is conjectured that the individual components of the medium are to first order, ephemeral fermion-antifermion pairs appearing and disappearing in their individual bubbles of Heisenberg uncertainty, and that there is an average unperturbed separation of these bubbles. In addition to temporal variations imposed on the medium due to the acoustic wave, the number density is also spatially affected by the energy density associated with any fields expressed in the medium as evidenced by the predicted 1=r 4 spatial variation of the density due to the proton’s electrostatic potential. A question then to address in this discussion is what exactly is

Fig. 4. Plot of (log) density and (log) velocity fields for the molecular hydrogen model.

velocity for that medium, presuming the propagation of a longitudinal acoustic wave. As part of the process of attempting to understand the implications of the analysis findings and the nature of the conjectured medium, it will be useful to briefly consider the generally understood

Fig. 5. Wide band spectral response near H2 bond energyl.

Fig. 6. Normalized pressure distributions for molecular hydrogen model representing the system resonance response at a frequency closely corresponding to the bond energy of H2 . 6

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result of hydrodynamic terms. The first sentence of Section 4 notes this saying, “The previous sections have completed a painstaking process showing a deep and explicit connection between the Schr€ odinger equation and the Quantum potential which led to the detailed derivation of the acoustic wave equation for the quantum medium.” That is, in terms of a hydrodynamic flow velocity, a flow field mass density, a quantum potential, and a related quantum pressure tensor. These all in pursuit of an answer to the question, “What is waving?” at the heart of all quantum theories involving any form of wave equation. Consequently it was then possible to describe the quantum vacuum as a medium subject to acoustic, that is, longitudinal, pressure/density waves. Using the potential to derive a density field and then a velocity field for the quantum vacuum medium, it was shown that one can describe the simplest case, the electron orbitals of the hydrogen atom, with good accuracy. Thus we can propose a reasonable answer to what is waving, and derive useful insights into the physical reality not obtained by other theories. This logic does not follow if one only considers the time-independent form of the Schr€ odinger equation. There is still an interesting thought process to consider when comparing the operator, specifically the variables representative of the wave number, in equation (23) to the variables representative of the wave number in our acoustic version. The standard form of the Helmholtz equation acting on a variable is ðr2 þk2 ÞA ¼ 0 which suggests from

occurring in the continuous medium that is manifesting an acoustic resonance associated with an energy state of the hydrogen atom? Consider for a moment the purely classical scenario of a basketball struck by a blunt rod - this case is detailed in the literature [11] as a purely educational exercise on acoustic resonance. This blunt impact will induce broadband resonance in the air trapped inside of the ball and the medium will resonate at most of the natural frequencies of the system. One of the resonance modes for this spherical resonance chamber will be the acoustic analog for the 2p orbital around the hydrogen atom, which is depicted in Fig. 7 [12]. In the acoustic case, the mode has two areas of maximum and minimum pressure separated by a region of unchanging pressure that is defined by the nodal surface depicted as a mesh disc. In an acoustic mode, particles oscillate across nodal surface from one region to another. As they oscillate about the nodal surface, they densify in one region manifesting as a high pressure lobe with an adjoining low pressure lobe, then reflect back across the nodal surface to the adjoining region reversing the polarity of the high/low pressure regions. The particles are at their slowest, minimal displacements from reflection, and spend the most time in the extreme pressure regions, whereas they are at their fastest and largest displacements when they cross the nodal surface (or surfaces depending on mode). If one were to “mark” a particle that is a member of the acoustic continuum medium and try and “find” that particle or observe that particle at a particular moment in time, the odds are higher that the particle will be found within the high(low) pressure lobes and lower that the particle will be found at the nodal surface. This is a classical analogy to the probability density that determines the likelihood of observing an electron (in a particular state) at some point around the nucleus. Said simply, the high/low pressure lobes correspond directly to high probability lobes in the Copenhagen view. It is claimed here that this is the scenario that is being manifest in the continuous dynamic medium.

the time independent Schr€ odinger equation (23) that kðrÞ2 ¼ a  b=r where the arbitrary constant variables a and b just condense the physical constants for brevity. Neglecting the constant term a for a moment, this wave number relationship can be restated as kðrÞ2 ∝b=r. Reconsidering the Helmholtz equation that was solved numerically for the acoustic resonances of the hydrogen atom, ðr2 þk2 ÞP ¼ 0 (where k ¼ ω=c), and plugging in the derived equation for longitudinal wave speed will yield the following form:   2    2π r r2 þ ωðrÞ2 P ¼ 0: rBohr αc

€ dinger? 7. Simpler path to obtain Helmholtz from Schro

Note that omega, just like wave speed, will not, in general be a constant due to the presence of the potential. If these two wave numbers of equations (23) and (24) are to have the same spatial dependency, then ωðrÞ∝1=r. This will also result in a dispersion relationship for these longitudinal waves propagating in this vacuum fluid medium filling the atomic system of ωðkÞ∝k2 . This insight may prove useful in some of the future work efforts briefly detailed in the following section to explore the bulk modulus of the quantum fluid.

One might take a shorter path to arrive at a Helmholtz type equation from the Schr€ odinger equation, and it is worth discussing the difference in this abridged approach to the method explored in the preceeding sections. Consider the time-independent form of the Schr€ odinger equation. It is already in the form of a Helmholtz equation:  r2 þ

 2m 2 ðE  VðrÞÞ ψ ¼ 0: ℏ

(23)

(22)

While this is a Helmholtz equation, it represents a different view on the physical characteristics of the system. This differential equation is acting on the wave function ψ, and not the pressure field of system P. The

8. Future work In the traditional approach to quantum mechanics, the primary quantum number n determines the main energy level of the shell, the angular momentum quantum number l determines the orbital shape, the magnetic quantum number ml determines the orientation, and the spin quantum number ms determines the spin of the electron. The angular momentum quantum number and magnetic quantum number are related to the quantization of angular momentum, and these variables are integral in establishing the familiar s, p, d, and f orbital shapes and their canonical orientations. While our dynamic vacuum model numerical analysis yielded the familiar s, p, d, and f orbitals, it did so without the explicit incorporation of the quantization of angular momentum. It is anticipated that future refinement of the model to develop a correspondence of the quantization of angular momentum in the context of the Helmholtz equation approach will improve the precision of the eigenfrequencies found by the numerical analysis. In the derivation presented, as seen in Sec. 4, it was necessary to provide an equation for the spatial variation of the propagation speed c of a disturbance in the presumptive medium as an input to the numerical model. The heuristic velocity field for the simple atomic model developed was first expressed as a function of radius, and in Sec. 5 it was then

units of the wave function are by observation m2 compared to the units of the pressure field, N=m2 . The Madelung process was deliberately followed in order to justify the characterization of various quantities in the 3

Fig. 7. Plot of the 2p orbital from Orbital Viewer Software. The nodal surface is the mesh disc separating the two lobes. 7

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the work reported in this paper.

re-expressed in a form to demonstrate the dependency on density in Eq. (22). In air or water, this propagation speed corresponds to the speed of sound, and in general for acoustic media, the speed of sound depends on the density of the medium, along with other properties. Immediate future focus will seek to develop a first principles understanding of the physical mechanism(s) for the velocity field around the atom as a function of the density and bulk modulus of the quantum fluid, or dynamic vacuum. Along with the implications of density variation in the quantum vacuum induced by the presence of a charged particle - the proton in the case studied here - the corresponding variation in wave propagation velocity, c, is considered to provide a fundamental insight into the structure of reality unavailable to investigations employing the standard probabilistic interpretation of quantum mechanics. That interpretation, we believe, has demonstrated astonishing explanatory and predictive power, but we believe it is incomplete.

Acknowledgments This work was supported by the DARPA Defense Science Office Nascent Light Matter Interaction Program (HR001118S0014 NLM). Special thanks goes to NASA for organizational and institutional support. References [1] L. De Broglie, Interpretation of quantum mechanics by the double solution theory, Ann. Fond. Louis Broglie 12. [2] G. Bacciagaluppi, A. Valentini, Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference, Cambridge University Press, 2009, https://doi.org/ 10.1017/CBO9781139194983. [3] D. Bohm, A suggested interpretation of the quantum theory in terms of "hidden" variables. i, Phys. Rev. 85 (1952) 166–179, https://doi.org/10.1103/ PhysRev.85.166. https://link.aps.org/doi/10.1103/PhysRev.85.166. [4] D. Bohm, A suggested interpretation of the quantum theory in terms of "hidden" variables. ii, Phys. Rev. 85 (1952) 180–193, https://doi.org/10.1103/ PhysRev.85.180. https://link.aps.org/doi/10.1103/PhysRev.85.180. [5] T.H. Boyer, Any classical description of nature requires classical electromagnetic zero-point radiation, Am. J. Phys. 79 (11) (2011) 1163–1167, https://doi.org/ 10.1119/1.3630939. [6] E. Madelung, Quantentheorie in hydrodynamischer form, Z. Phys. 40 (3) (1927) 322–326, https://doi.org/10.1007/BF01400372. [7] P.R. Holland, The Quantum Theory of Motion: An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics, Cambridge University Press, 1993, https://doi.org/10.1017/CBO9780511622687. [8] D. Bohm, B. Hiley, The Undivided Universe: an Ontological Interpretation of Quantum Theory, Taylor & Francis, 1995. https://books.google.com/books? id¼ZBXVnLtbphEC. [9] L.E. Kinsler, A.R. Frey, A.B. Coppens, J.V. Sanders, in: Lawrence E. Kinsler, Austin R. Frey, Alan B. Coppens, James V. Sanders (Eds.), Fundamentals of Acoustics, 4th Edition, Fundamentals of Acoustics, fourth ed., Wiley-VCH, December 1999, ISBN 0-471-84789-5, p. 560, 1. [10] H. White, J. Vera, P. Bailey, P. March, T. Lawrence, A. Sylvester, D. Brady, Dynamics of the vacuum and casimir analogs to the hydrogen atom, J. Mod. Phys. 06 (2015) 1308–1320, https://doi.org/10.4236/jmp.2015.69136. [11] D. Russell, Basketballs as spherical acoustic cavities, American Journal of Physics AMER J PHYS 78 (2010), https://doi.org/10.1119/1.3290176. [12] D. Manthley, Orbital viewer. http://www.orbitals.com/orb/ov.htm, 2001. (Accessed 8 April 2019).

9. Conclusion In this paper, some of the key governing equations (Madelung quantum Euler equations) of pilot wave theory were re-derived. A meticulous and detailed derivation of the acoustic wave equation was then presented utilizing the noted Madelung equations. The implications of the applicability of the acoustic wave equation to model the behavior of a continuous vacuum medium was discussed, and a model of the hydrogen atom was both cited from the literature and derived herein. The details of a numerical modeling analysis of the hydrogen atom were presented along with the results. The analysis technique for the single atom case was extended to build a model of molecular hydrogen, and the numerical analysis results were presented and discussed. The paper concluded with some speculative discussion on the nature and make up of the dynamic vacuum medium, and identified some areas for future focus. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence

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