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This paper will discuss the current viewpoint of the vacuum state and explore the idea of a “natural” vacuum as opposed to immutable, non-degradable vacuum. This concept will be explored for all primary quantum numbers to show consistency with observation at the level of Bohr theory. A comparison with the Casimir force per unit area will be made, and an explicit function for the spatial variation of the vacuum density around the atomic nucleus will be derived. This explicit function will be numerically modeled using the industry multi-physics tool, COMSOL, and the eigenfrequencies for the
*n* = 1 to
*n* = 7 states will be found and compared to expectation.

The current viewpoint of the quantum vacuum, or vacuum state, is that it is an immutable, non-degradable state for all observers and systems with no structure or variation. The concept of the vacuum state is typically introduced as a ground state of a harmonic oscillator, so the viewpoint that it is immutable is reasonable. How can the vacuum, being the ground state of a harmonic oscillator, be anything other than “zero” for all observers? What if, however, the vacuum could be posited to be a plenum that can be shown to be degradable, and has the capability to support particle-vacuum or particle-particle interactions that allow lower energy, ground states? It is known from experimental observation that the vacuum can exhibit characteristics that can best be associated with a degraded vacuum in the form of the Casimir force [

The idea of a vacuum “density” expectation value will be explored by first starting with the Bohr formula that relates allowed energy levels to the primary quantum numbers. The energy for the

The allowed “orbit” radius for this energy level is shown in Equation (2).

Equation (3) shows the energy relationship with the primary quantum number for the

The allowed “orbit” radius for the energy level for any atom with atomic number Z is:

The historic perspective used in the development of the above relationships was that of the electron being in “orbit” around the nucleus in a quasi-classical sense. It is appropriate to think of these energy states as a wave function [

The average density for hydrogen is calculated for the n = 1 to n = 7 states in

The question can be raised on what the calculated value for the average “density” really means―does it really indicate a perturbation (rarefication or densification) of the quantum vacuum, or is it just a number that has no physical interpretation? To help consider this question, an equation can be fitted to the radius and density data presented in

Equation (6) shows that the density value is dependent on

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

In order for there to be a universe that is accelerating as evidenced by observation, the equation of state for the vacuum,

n | ||||
---|---|---|---|---|

1 | 13.60 | |||

2 | 3.40 | |||

3 | 1.51 | |||

4 | 0.85 | |||

5 | 0.54 | |||

6 | 0.38 | |||

7 | 0.28 |

n | ||||
---|---|---|---|---|

1 | 2.96 | |||

2 | 2.96 | |||

3 | 2.96 | |||

4 | 2.96 | |||

5 | 2.96 | |||

6 | 2.96 | |||

7 | 2.96 |

vacuum is just such that

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 [

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

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

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

where the term

The term

A plot of this function for hydrogen is shown in

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

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

The way this equation is applied to a classical plasma is that the

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

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

n | Thermal vel | Orbital freq | Sound speed |
---|---|---|---|

1 | 29,476 | ||

2 | 14,738 | ||

3 | 9825 | ||

4 | 7369 | ||

5 | 5895 | ||

6 | 4912 | ||

7 | 4210 |

This analysis result shows that this eigenfrequency is

This paper has explored the idea of the quantum vacuum not being an absolute immutable and non-degradable state, and studied the ramifications of the quantum vacuum being able to support non-trivial spatial variations in “density”. These considerations showed no predictions that were contrary to observation, and in fact duplicated predictions for energy states associated with the primary quantum number. An explicit function of vacuum density spatial variation was derived such that it also predicted correct energy levels for the primary quantum numbers, and provided a simple acoustic model that could be numerically studied using the multi-physics software tool, COMSOL. This study showed that the quantum vacuum can support longitudinal wave modes with mode

n | Orbital freq | COMSOL freq | %error |
---|---|---|---|

1 | −4.98 | ||

2 | 0.05 | ||

3 | −2.48 | ||

4 | −1.59 | ||

5 | −5.36 | ||

6 | 14.28 | ||

7 | 11.16 |

shapes and frequencies commensurate with proton oscillation about the center of mass of the electron-proton “rotating” system associated with the primary quantum numbers. The spin-orbit coupling mode shapes associated with the p, d, and f orbital shapes are also viable acoustic wave mode solutions, and will be non-degenerate with slightly different frequencies, and hence, energies. It is a matter of future work to fully explore the p, d, and f orbital mode shapes using the explicit vacuum density function with a 3D model of sufficient resolution. The 2D approach was used for computational speed while maintaining fine mesh size. Some examples of the COMSOL results from a 3D model of a classical spherical resonance system with isotropic air medium are shown in

using the 2D COMSOL model, some solutions mapped to the non-spherical, but axisymmetric electron orbitals from the p, d, and f families. Three examples are provided in

There are a number of approaches detailed in the literature that seek to develop different interpretations or understandings of the origin of the wave equation, and we will only touch on few of the concepts in closing. The orthodox view is of course the Copenhagen interpretation which, in short, does not seek to assign any classical nature to the wave equation by itself, and rather only considers the wave’s statistical impact on configuration space [

The paper will close with the following thought experiment: if the vacuum around the nucleus can be considered more of a “natural” vacuum as opposed to an immutable ground state with absolutely no spatial variation, and if there are ephemeral fermion/antifermion pairs dominated by electron-positron pairs that create and annihilate with a density that increases significantly as one moves closer to the nucleus, what is so special about the orbiting electron that allows it to be a “real” electron out of this vacuum soup? Perhaps it is not a case of uniqueness, but a case of non-uniqueness. Consider the following: a room full of paired square dancers progresses through the dance moves smoothly as called by the caller, and they occasionally change partners when instructed. What if there was an additional solitary dance partner of a given gender introduced to the ranks of this evenly matched group? And the rule is established that when a trade call is issued, the free dancer will couple to the nearest available dance partner of the opposite gender, and the previously paired dancer that misses out is now the free dancer until the next trade call is issued. As the evening progresses, nearly every dance partner of the gender that had the extra dancer has had a period when they were the “unique” solitary dancer. In an analogous way, perhaps the “real” electron is also “unique”. In one instance, the “real” electron collides with a positron vacuum fluctuation elevating the now un-paired electron vacuum fluctuation to the “real” state. This real electron continues in its real state for a brief period until it too collides with a positron vacuum fluctuation, elevating the next un-paired electron vacuum fluctuation to the “real” state. This process continues ad infinitum, and the “real” electron is not unique, and rather it is non-unique in that the “real” descriptor is associated with the state, not the individual electron. So if the “real” electron is simply a unique state of the underlying natural vacuum, an unmatched dance partner in the sea of dancers, then the probability wave functions for the electron states may be a dual representation of the longitudinal acoustic wave modes that arise as a result of the dynamics of this natural vacuum.

The primary author would like to thank the Eagleworks team for their support and hearty/heated discussions about the concepts discussed and explored in this paper. The team would like to thank the National Aeronautics and Space Administration for organizational and institutional support in the exploration and analysis of the physics in this paper.

HaroldWhite,JerryVera,PaulBailey,PaulMarch,TimLawrence,AndreSylvester,DavidBrady, (2015) Dynamics of the Vacuum and Casimir Analogs to the Hydrogen Atom. Journal of Modern Physics,06,1308-1320. doi: 10.4236/jmp.2015.69136