Equation (6) shows that the density value is dependent on, which is a similar dependency to the Casimir force. At this point, the equation should not be viewed as a real continuous function; rather, it should be considered as a function that can only be used with the allowed, discrete values for the radius as input values. Recall that the Casimir force per unit area is (added for unit consistency):

One can use Equation (7) to calculate a Casimir “density” value for the hydrogen primary quantum numbers 1 - 7 by equating the distance, d, to twice the allowed orbit radius, 2r. In a sense, the electron establishing a “boundary” at this radius could be envisioned as setting up some sort of boundary condition that mimics a Casimir cavity of sorts. These values are calculated and compared to the average density with a ratio provided in Table 2 (again Casimir force per unit area is multiplied by for units consistency).

Table 2 shows that the values will be equal if there is a factor of 1/3 added to the Casimir force per unit area ´. As a short side discussion to help possibly provide an explanation for the 1/3 factor, consider the Friedmann equation:

In order for there to be a universe that is accelerating as evidenced by observation, the equation of state for the vacuum, must be such that, or [12] . If the equation of state of the quantum

vacuum is just such that, this might explain the need for the addition of a factor of 1/3 in Table 2. It could be posited that the predicted altered ground state of the vacuum around the hydrogen nucleus resulting from the presence of the “orbiting” electron may be responsible for helping establish the stable energy state and preclude the “orbiting” electron from radiating down to the nucleus. An approach similar to this was also successfully explored in [13] . Equipped with this insight, one can combine Equation (5) and Equation (7) to yield Equation (8)

The significance of this equation is that it indicates that the calculated “density” expectation value using the Bohr relationships for the energy and radius may have physical meaning as opposed to just a calculated number.

Consider the Casimir force (and Casimir equation) as it has been explored in the lab to date by numerous experimentalists with the work done by Steven Lamoreaux in 1996 establishing convincing experimental evidence of the phenomenon [2] . The Casimir force equation is used in practice to assess the force developed for a given geometry associated with a real test article that will be studied in the laboratory. When the force per unit area equation is applied to the real test article being studied in the lab, the equation provides a number that really does have physical meaning as evidenced by the real force that is measured during the study of the test article. The state of the quantum vacuum between two plates that establish a Casimir cavity is perturbed (rarified) as a result of the plates precluding certain vacuum fluctuation modes from being manifested in the cavity. The pressure inside the cavity is less than the pressure outside of the cavity.

Since the “density” using the Bohr relationships has been shown to make predictions of the energy density around the hydrogen nucleus that are identical to the modified Casimir force per unit area equation, this may indicate that these numerical values do have physical meaning and are not just a numerical calculation with no basis in nature. To be explicit, these values may indicate that the quantum vacuum around the hydrogen nucleus is not an immutable and non-degradable medium with no variation or structure, rather the vacuum appears to have a perturbation around the hydrogen nucleus that exhibits a strong dependency on.

What if one considers the scenario when the atomic number Z is varied? The course is similar to the above treatment for hydrogen, except the Bohr relationships used are the equations with the Z dependency included. For this discussion, the primary quantum number n will be spanned from 1 to 7, and the atomic number Z will be spanned from 1 to 7, which corresponds to hydrogen, helium, lithium, beryllium, boron, carbon, and nitrogen respectively. The expectation value for the “density” is shown in Figure 1 for each atomic number. The observed “density” is determined using Equation (5), and the Casimir “density” is determined using Equation (9):

A trend line has been added to each series to visually link each set of primary quantum numbers together for a given atomic number Z, and help illustrate the dependency. A mark is plotted for each combination of Z and n for the “density” expectation value. The modified Casimir force per unit area value is also calculated for each point with the inclusion of Z in the numerator of the equation as shown in Equation (9), and is plotted as an open circle. The figure shows that as the atomic number Z increases, the curves trends upward, and the state progresses to the left as it should. For higher atomic numbers, the perturbation of the quantum vacuum is slightly higher than for lower atomic numbers.

To this point, the discussion has been about the expectation value for the “density” of the quantum vacuum for a given primary quantum number with no consideration of substructure or variation within the given spherical region. The interpretation is that the predicted “density” is an isotropic state throughout the orbital defined by the corresponding allowed radius. Since it was just shown that the expectation value for the density at each allowed orbit radius is dependent on, it is reasonable to expect that there is a more continuous representation of the density of the quantum vacuum that varies continuously over the radial distance from the nucleus. This continuous function can be found by establishing the requirement that the volumetric integral of the density (multiplied by) is equal to the energy level allowed for each primary quantum number. The differential element to integrate is a thin spherical shell element that starts at the “radius” of the proton and continues to the allowed radius associated with the target quantum number:

where the term is a proportionality constant that is to be determined to ensure that the continuous function will yield the energy level for each quantum number. The integral to solve is simple enough (neglecting the constant of integration):

The term is the effective “radius” of the proton, and is the allowed radius of the quantum number n. There are three options to use for the proton radius in this calculation. The first option is to use the charge radius of the proton which has a measured value of 0.88 fm [14] , the second is to use the Compton wavelength of the proton which is 1.32 fm, and the third option is to use the historical Rutherford empirical equation, where, and Z again is the atomic number yielding a radius of 1.2 fm for the hydrogen nucleus. The third option of 1.2 fm will be used for the remainder of this treatment. Using this yields a value for of. The continuous density function for the quantum vacuum is shown in Equation (12) and includes the necessary dependency on the atomic number Z:

A plot of this function for hydrogen is shown in Figure 2. Rather than being a series of step functions stepping down from a large value for n = 1 to a small value when n = 7 as was previously the scenario in the isotropic view, this curve is now a continuous function with slight shifts at the allowed radii which could be construed as soft boundaries between the energy levels. So rather than the density of the quantum vacuum being a uniform over the entire spherical region defined by the Bohr radius for n = 1, it starts out considerably higher just outside the nucleus and tapers down to at the Bohr radius. On average, it is, but it varies continuously over radial distance r.

If the vacuum is indeed not an immutable and non-degradable medium, but rather a medium that can vary, as first evidenced by direct observation of the Casimir force, what can be said about what has been developed in

this discussion? A thing to note is that the integral of the perturbation of the quantum vacuum around the nucleus for a given atomic number Z and quantum number n is exactly equal to the energy level of the electron in that state. The energy level of the electron is a function of its potential energy and kinetic energy. Does this mean that the energy of the quantum vacuum integral needs to be added to the treatment of the captured electron as another potential function, or is the energy of the quantum vacuum somehow responsible for establishing the energy level of the “orbiting” electron? The only view to take that adheres to the observations would be the latter perspective, as the former perspective would make predictions that do not agree with observation. It was shown earlier that the perturbation of the vacuum around the nucleus appears to have characteristics very similar to that of the Casimir force per unit area, and since the Casimir force per unit area is negative; the integral of the vacuum perturbation would also be negative. So for n = 1, Z = 1, the energy for the captured electron is −13.6 eV, and likewise, the integral of the vacuum perturbation is −13.6 eV.

If the quantum vacuum is indeed not a static immutable medium, can be locally perturbed as the above assessment indicates may be the case, and that this perturbed medium can be shown to be related to the binding energy of trapped electrons, what other characteristics might the medium have that should be considered? If the quantum vacuum is a sea of vacuum fluctuations consisting of virtual photons and virtual fermions (e.g. electron-positron pairs and others), then it may be useful to study the types of wave modes that are possible for this medium that has a density dependency. The discussion will be specific to acoustic wave modes, as solutions to acoustic wave modes in a spherical region with soft boundaries are determined using the Helmholtz equation, and will yield wave modes with mathematical forms that are identical to the electron probability wave functions. A couple of points need to be made before proceeding. First, acoustic modes in air are longitudinal waves that propagate as a result of collisional process between the neutrals that make up the air. Acoustic wave modes in a plasma medium are not communicated by means of collisional process, rather the longitudinal wave propagates as a result of the charged particle motion being relayed to other charged particles by means of the intermediary electric field. Second, consider an acoustic mode analog that looks like one of the three 2p orbitals as shown in Figure 3 [15] .

In the acoustic case, the mode has two areas of maximum and minimum pressure separated by a region of neutral pressure that is defined by the nodal surface. In an acoustic mode, particles oscillate from the high pressure region to the low pressure region where they will reflect back again as the wave cycle oscillates. The particles are at their slowest, minimal displacements from reflection, and spend the most time in these 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 pressure lobes and lower that the particle will be found at the nodal surface. This is a classical analogy to the probability function that determines the likelihood of observing an electron (in a particular state) at some point around the nucleus.

The vacuum density function for hydrogen defined by Equation (12) and plotted in Figure 2 can be modelled and studied using the high fidelity industry tool, COMSOL Multiphysics. The vacuum medium can be modelled as a virtual plasma consisting of predominantly electron-positron ephemeral pairs. Although there will be a spectrum of other fermion-antifermion particle pairs, the dominant members of the medium at any given time will be the pairs. The speed of sound in a plasma can be calculated using Equation (13) [16] .

The way this equation is applied to a classical plasma is that the is the temperature of the electrons, and M is the mass of the ions. In the initial consideration of the hydrogen atom, the mass of the proton will be used for M. It should be noted that it is likely that M will be a lower mass value that is representative of either relativistic pairs, or heavier charged fermion-antifermion pairs. The “average” temperature of the quantum vacuum modelled as a virtual plasma within a given energy state around the hydrogen nucleus will be set equal to the equivalent kinetic temperature of the orbiting electron at that state. In this view, the real electron at the n = 1 state has a thermal speed of. The temperature corresponding to this thermal speed is found from the equipartition theorem using the relationship, or for the n = 1 case.

A possible source of longitudinal waves is the hydrogen nucleus. Continuing with the quasi-classical viewpoint, the electron in the n = 1 state is “orbiting” around the proton at an average distance of the Bohr radius with an orbital speed of. Similarly, the proton is oscillating around the electron-proton system center of mass at the same frequency and it acts as a longitudinal wave source for the surrounding quantum vacuum medium. The frequency of this oscillation can be considered the eigenfrequency of the longitudinal wave associated with this quantum state. When running a COMSOL acoustic analysis of this system, the objective is to find the COMSOL-determined acoustic resonance frequency for each of the seven primary quantum number states (spherical acoustic modes defined by their corresponding Bohr radii) and compare it to the target frequency determined using the electrons orbital frequency. The target frequencies for the primary quantum numbers are given in Table 3.

Since this phase of the analysis is centered on finding the spherical acoustic modes for all of the primary quantum numbers, a 2D axisymmetric model was used. This allowed for very fine mesh size when studying the mode. Figure 4 shows a close-up of the innermost region roughly defined by the Bohr radius,. The mesh size for this region of the model was set to one picometer within the spherical region defined by the Bohr radius.

Figure 5 shows the eigenfrequency associated with the spherical acoustic mode that falls within the n = 1 spherical region of the model. Note that the quantum vacuum density used in the COMSOL analysis is the continuous curve defined by Equation (12), and there are no “hard” boundaries in the model. The picture on the left of Figure 5 is the view from the outer extents of the model, and the picture on the right is a close-up of the mode with a cutaway depicting the intensity distribution of the mode.

This analysis result shows that this eigenfrequency is, whereas the hope was to find that this frequency was at. In order to ensure that the eigenfrequency falls at the target value, the sound speed model will need to be scaled up by a factor of 11. This means that all of the sound speeds listed in Table 3 will be an order of magnitude higher. This also requires that the mass in the denominator of Equation (13) is not the proton mass, but an effective mass of 15.2´ that of the electron. It should be noted that in [17] [18] , the authors determined that in order to accurately predict the permittivity, , and permeability, , of the vacuum of space arising as a result of quantum vacuum fermion pair fluctuation interaction with propagating photons, the quantum vacuum would need to routinely produce fermion pairs in a relativistic condition with a mass factor of 31.9. A scaling factor of 11 was used for all subsequent COMSOL analysis to find eigenfrequencies representative of the seven primary quantum numbers for the hydrogen model.

Figure 6 shows the expanded view of the COMSOL 2D axisymmetric model used for finding eigenfrequencies. The model mesh sizing is provided to the right of the model picture, and the dense mesh sizing is apparent as the elements become unresolvable at the resolution of the figure below the primary quantum number, n = 6.

Table 4 presents the longitudinal mode eigenfrequencies for the n = 1 to n = 7 orbitals. For each row, the expected value is listed in the left column, and the COMSOL eigenfrequency value is provided to the right along with a % error. For a primary quantum number, the expected value is the frequency that the proton would oscillate about the system center of mass that consists of the proton and the orbiting electron. Figure 7 shows the eigenfrequency solution plots that correspond to those recorded in Table 4.