Absolute Quantum Mechanics

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Absolute Quantum Mechanics is Quantum Mechanics based on Absolute theory. Since absolute theory returns to a classical interpretation of space and time, it allows absolute quantum mechanics to be developed using a simple (and empirical) underlying physical model. This is in contrast to the relativistic quantum mechanics of the Dirac Equation, Proca equations, and Klein-Gordon Equation which are based on relativity and must therefore be covariant. Importantly, since relativity is a point-like theory of events in four-space, the relativistic quantum mechanics treatments have interpretative problems regarding quantum Wave function collapse.

In absolute theory, quantum wave function collapse can be readily understood since time is once again classical. With classical time, simultaneity is well established and the concept of instantaneous collapse holds good. This allows for an understanding of quantum collapse, since an instantaneous collapse is now instantaneous in all frames, unlike the case in a relative theory.

In addition to enabling an understanding of the process of quantum collapse, absolute theory also allows for a derivation of quantum mechanics equations in a very straightforward way. By starting with simple empirical observations, and then assuming the existence of an underlying physical wave, the equations of quantum mechanics can be derived.

It is worthy to note that the Shrodinger equation is consistent with absolute theory, as the Shrodinger equation is not covariant. However, the Shrodinger equation is only valid at low energy. To be a complete solution, it is of course important that quantum mechanics be consistent with the exact relation between energy, mass and momentum. Such a derivation is presented below. The treatment below also has the advantage of yielding exact equations for quantum mechanics and it does not need to fall back on Perturbation theory as does QED. However, due to the complexity of the final equations, mathematical approximation techniques will still be required to arrive at solutions to interesting physical problems.

The content below is from the paper: D.J. Larson (2017) "An empirical and classical approach for nonperturbative, high velocity, quantum mechanics" Physics Essays, Volume 30: Pages 264-268, 2017. That paper is copyrighted by Physics Essays Publication (PEP), http://physicsessays.org/, and it is reproduced below by permission granted from PEP.

Absolute Quantum Mechanics

Part 1 – Introduction

Schrödinger's equation[1] is quite useful for calculating the quantum mechanical wave function in low velocity situations. Application of Schrödinger's equation to two body states such as the hydrogen atom yield good results, as the spectral evidence is in excellent agreement with the theory. However, to apply high velocity corrections or to investigate the hyperfine structure caused by the spin-spin magnetic dipole interaction, it is necessary to turn to a perturbation analysis. And to account for the Lamb shift[2], the perturbative approach of QED[3], [4], [5], [6], [7] is required. While QED is spectacularly successful in its calculational ability, the perturbation approach is valid only if the perturbations are small when compared to what is being perturbed. In the case of electrodynamics, the parameterization factor is given by the fine structure constant, α, which is approximately 1/137. However, for stronger forces, such as those found in nuclear matter, the perturbation approach is ineffective. Furthermore, QED involves the handling of infinities via a process called renormalization, a process that is quite distasteful for a physical theory.

The mesons are presently believed to be two body states; yet finding a satisfactory solution for determination of their masses has proven elusive. In the ABC Preon Model [8], leptons are proposed to be two body states; yet lepton masses have not been theoretically calculable. For these reasons, it is of value to find a complete, high velocity, quantum mechanical equation for the central force, two body problem.

Here, a formulation for high velocity quantum mechanics will be developed. The approach will be to appeal to simple empirical observations, and then apply an assumption of an underlying wave in order to reach the result. With the result in hand, it will be shown that the result reduces to Schrödinger's equation in the low velocity limit.

In order to present the derivation in the most concise way, Part 2 will now contain only the minimum needed to arrive at the results. In Part 3 additional supportive remarks will be given.

Part 2 – Derivation of the Equations.

The Complete High Velocity Quantum Mechanical Equation

It is empirically observed for both light and material particles that

E = \hbar \omega

 

 

 

 

(1)

and

\mathbf{p} = \hbar \mathbf{k}

 

 

 

 

(2)

Eq. 1 is the Planck-Einstein relation and Eq. 2 is the de Broglie relation.

In Eqs. (1) and (2) \hbar is Plank's constant divided by 2\pi, \mathbf{p} is the momentum vector, E is the energy, and an underlying wave is assumed where \omega is 2\pi f, \mathbf{k} is the angular wave vector (\mathbf{k} has magnitude 2\pi/\lambda), f is the frequency, and \lambda is the wavelength. The assumption of the underlying wave has been supported by interference experiments for both light and matter.

It is also empirically observed that the total energy in the presence of certain forces can be expressed as

E = \left [ p^2c^2 + m^2c^4\right ]^{1/2} + V

 

 

 

 

(3)

In Eq. (3) m is the rest mass of the particle, p is the magnitude of the momentum, c is the speed of light and V is the potential energy associated with the forces present.

It is convenient to specify the form of our assumed underlying waves in the form

\xi = exp \left [ i \left ( \mathbf{k} \cdot \mathbf{x} - \omega t \right ) \right ]

 

 

 

 

(4)

In Eq. (4) i is the square root of minus one. Taking derivatives of Eq. (4):

\frac{\partial \xi}{\partial x} = ik_x\xi , \frac{\partial \xi}{\partial y} = ik_y\xi , \frac{\partial \xi}{\partial z} = ik_z\xi

 

 

 

 

(5)

and

\frac{\partial \xi}{\partial t} = -i\omega\xi

 

 

 

 

(6)

Differentiating Eq. (4) a second time produces

\frac{\partial^2 \xi}{\partial x^2} = -k^2_x\xi , \frac{\partial^2 \xi}{\partial y^2} = -k^2_y\xi , \frac{\partial^2 \xi}{\partial z^2} = -k^2_z\xi

which can be combined to form

\nabla^2 \xi = -k^2\xi

 

 

 

 

(7)

In Eq. (7) the usual nomenclature for the Laplacian is used, \nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}. At this point in the development it is useful to manipulate the empirical relationships E = \hbar \omega and \mathbf{p} = \hbar \mathbf{k} (Eq. (1) and Eq. (2), respectively). Taking the dot product of Eq. (2) with itself and rearranging leaves k^2 = p^2/\hbar^2 while rearranging Eq. (1) leaves \omega = E/\hbar, and substituting these values into equations (6) and (7) leaves \frac{\partial \xi}{\partial t} = -iE\xi/\hbar and \nabla^2 \xi = -p^2\xi/\hbar^2, respectively, which can be rearranged as:

E = \left ( i\hbar \frac{\partial \xi}{\partial t} \right ) /\xi

 

 

 

 

(8)

and

p^2 = - \left ( \hbar^2 \nabla^2 \xi \right ) /\xi

 

 

 

 

(9)

Substituting Eqs. (8) and (9) into Eq. (3) leaves:

\left ( i\hbar \frac{\partial \xi}{\partial t} \right ) /\xi = \left [ \left ( -\hbar^2 c^2 \nabla^2 \xi\right )  /\xi + m^2c^4 \right ]^{1/2} + V

 

 

 

 

(10)

Eq. (10) is the complete high velocity quantum mechanical equation.

(See Part 3, Remarks 1, 2, 6 and 7 for an explanation of why differential relationships derived from Eq. (4) can be used in the derivation of Eq. (10). See Part 3, Remark 3 to understand what is meant by the terminology "high velocity".)

The Equation for Stationary State Spatial Wave Functions

Next we will follow closely the typical development when dealing with stationary states where V = V(r,t) = V(r). (An excellent presentation of the typical development is given in the textbook by Anderson[9] .) For such cases, \xi can be decomposed into temporal and spatial functions:

\xi(r,t) = \Psi(r)\Phi(t)

 

 

 

 

(11)

Substituting Eq. (11) into Eq. (10) yields:

\left ( i\hbar \frac{\partial \Phi}{\partial t} \right ) /\Phi = \left [ \left ( -\hbar^2 c^2 \nabla^2 \Psi\right )  /\Psi + m^2c^4 \right ]^{1/2} + V

 

 

 

 

(12)

Since the left hand side of Eq. (12) is a function of t alone, while the right hand side is a function of r alone, each side can be set to some separation constant which we will call E_N. The resultant equation for the left hand side, \left ( i\hbar \frac{\partial \Phi}{\partial t} \right ) /\Phi = E_N can be solved by inspection, \Phi(t) = \Phi_0 exp[i\omega t], where we recall that E = h\omega. Note that this is the same solution that is found in the typical, low velocity treatment using Schrödinger’s Equation.

Turning to the spatial equation, we now have E_N = \left [ \left ( -\hbar^2 c^2 \nabla^2 \Psi\right )  /\Psi + m^2c^4 \right ]^{1/2} + V . This equation can be manipulated by bringing V over to the other side and squaring, leaving [E_N - V]^2 = \left ( -\hbar^2 c^2 \nabla^2 \Psi\right )  /\Psi + m^2c^4 , which, after we expand the square, move the mass term to the other side, and multiply through by \Psi leaves:

-\hbar^2 c^2 \nabla^2 \Psi =  [E_N^2 - 2E_NV + V^2 - m^2c^4]\Psi

 

 

 

 

(13)

Equation (13) is the complete high velocity quantum mechanical equation for stationary state spatial wave functions.

The Exact Quantum Equation for Hydrogenic S-States

As a particularly useful example, we can apply this equation to the case of the hydrogen atom, and here it is relevant to bring in two more empirical observations. First, it is empirically observed that the potential energy associated with the Coulomb force (the electric potential) is

V_E = K_CQ_1Q_2/r

 

 

 

 

(14)

In Eq. (14) Q_1 is the charge on one of the particles, Q_2 is the charge on the other particle, r is the distance between the particles and K_C is the Coulomb constant. Second, it is empirically observed that the potential energy associated with a spin 1/2 dipole-dipole interaction (the magnetic potential) is

V_M = K_M\mu_1\mu_2(1-3cos^2\theta)/r^3

 

 

 

 

(15)

In Eq. (15) \mu_1 is the dipole moment associated with one particle, \mu_2 is the dipole moment associated with the other particle, r is the distance between them, \theta is the polar angle, and K_M is a constant. For our purposes, it is desirable to combine some constants:

K_1 = K_CQ_1Q_2

 

 

 

 

(16)

and

K_2 = K_M\mu_1\mu_2

 

 

 

 

(17)

With Eqs. (14) through (17) this leaves the potential as:

V = V_E + V_M = K_1/r + K_2(1-3cos^2\theta)/r^3

 

 

 

 

(18)

Using Eq. (18) for V, and expanding out the Laplacian in spherical coordinates while suppressing the non-contributing variable leaves Eq. (13) as:

\hbar^2 c^2 \nabla^2 \Psi = [m^2c^4 + 2E_NV - E_N^2 - V^2]\Psi ->

 \frac{\partial^2 \Psi}{\partial r^2} + (2/r)\frac{\partial \Psi}{\partial r} + (1/r^2)\frac{\partial^2 \Psi}{\partial \theta^2} + (cos\theta/r^2sin\theta)\frac{\partial \Psi}{\partial \theta} =

 

 

 

 

(19)

[m^2c^4 + 2E_N\{K_1/r + K_2(1-3cos^2\theta)/r^3\} - E_N^2 - \{K_1/r + K_2(1-3cos^2\theta)/r^3\}^2]\Psi/ \hbar^2 c^2

And now expand the square:

 \frac{\partial^2 \Psi}{\partial r^2} + (2/r)\frac{\partial \Psi}{\partial r} + (1/r^2)\frac{\partial^2 \Psi}{\partial \theta^2} + (cos\theta/r^2sin\theta)\frac{\partial \Psi}{\partial \theta} =

 

 

 

 

(20)

[m^2c^4 + 2E_N\{K_1/r + K_2(1-3cos^2\theta)/r^3\} - E_N^2 - K_1^2/r^2 - 2K_1K_2(1-3cos^2\theta)/r^4 - K_2^2(1-3cos^2\theta)^2/r^6]\Psi/ \hbar^2 c^2

Equation (20) is the exact, high velocity, hyperfine-inclusive form of the quantum mechanical wave function for the Hydrogen atom for s-states. (States of non-zero angular momentum will include additional terms.)

Part 3 – Remarks.

Remark 1 – The Implicit Assumption.

Need for the Implicit Assumption.

Taking derivatives of the plane wave \xi = exp \left [ i \left ( \mathbf{k} \cdot \mathbf{x} - \omega t \right ) \right ] in the above treatment leads to relationships between derivatives of \xi and the quantities k and \omega. The empirical relationships E = \hbar\xi and \mathbf{p} = \hbar\mathbf{k} are then used in conjunction with those derivative relationships to obtain \mathbf{p}^2 = -\hbar^2\nabla^2\xi/\xi and E = \left ( i\hbar \frac{\partial \xi}{\partial t} \right ) /\xi. Finally, we use the empirical relationship E = \left [ p^2c^2 + m^2c^4\right ]^{1/2} + V, along with simple algebra to arrive at Eq. 10, \left ( i\hbar \frac{\partial \xi}{\partial t} \right ) /\xi = \left [ \left ( -\hbar^2 c^2 \nabla^2 \xi\right )  /\xi + m^2c^4 \right ]^{1/2} + V . However, \xi = exp \left [ i \left ( \mathbf{k} \cdot \mathbf{x} - \omega t \right ) \right ] is only one solution of Eq. 10, and it is not obvious that the relationship between the derivatives of \xi and the quantities p^2 and E will be the same for all solutions of Eq. 10. Nor is it obvious that E = \left [ p^2c^2 + m^2c^4\right ]^{1/2} + V will be valid within and throughout the wavefunction for all solutions of Eq. 10. This brings out the important implicit assumption that has been used in arriving at Eq. 10:

The Implicit Assumption.

It is implicitly assumed that \mathbf{p}^2 = -\hbar^2\nabla^2\xi/\xi, E = \left ( i\hbar \frac{\partial \xi}{\partial t} \right ) /\xi, and E = \left [ p^2c^2 + m^2c^4\right ]^{1/2} + V are valid relationships within and throughout all wave functions.

Discussion of the Implicit Assumption.

\xi = exp \left [ i \left ( \mathbf{k} \cdot \mathbf{x} - \omega t \right ) \right ] (the free particle solution) is simply that solution of Eq. 10 that allows us to most easily find the relations for p^2 and E in terms of derivatives of \xi based on empirical, macroscopic, observations. The derivation in Part 2 implicitly assumes that the relations found from the analysis of plane waves continue to hold good within and throughout all microscopic wave functions, even for the general case where solutions to Eq. 10 may not be plane waves.

Remark 2 – Physical Interpretations.

The relationship E = \left ( i\hbar \frac{\partial \xi}{\partial t} \right ) /\xi tells us that \left (\frac{\partial \xi}{\partial t} \right ) /\xi is proportional to the energy, and the relationship \mathbf{p}^2 = -\hbar^2\nabla^2\xi/\xi tells us that \nabla^2\xi/\xi is proportional to the square of the momentum. These facts give us interpretations for what \left (\frac{\partial \xi}{\partial t} \right ) /\xi and \nabla^2\xi/\xi are, physically, within the wavefunction.

Remark 3 – The Meaning of “High Velocity”.

It may not be obvious how Eqs. (10) and (13) are “high velocity” equations. Even more confusing might be the concept of “high velocity” in a “stationary state” as mentioned in conjunction with Eq. (13). To understand these concepts, first note that the low velocity description for the energy of a classical particle is (p^2/2m) + V E = p2/2m + V. On the other hand, the high velocity description for the energy of a classical particle is Eq. (3), E = \left [ p^2c^2 + m^2c^4\right ]^{1/2} + V. The derivation herein uses the high velocity formula for the energy to derive the quantum mechanical expressions, and that is one way that Eq. (10) is the complete high velocity quantum mechanical equation. Another aspect of “high velocity” can be understood through its use in “stationary state” wave functions by appealing to the physical interpretations introduced in remark 2 where it is explained that \left (\frac{\partial \xi}{\partial t} \right ) /\xi is proportional to the energy and \nabla^2\xi/\xi is proportional to the square of the momentum, within the wave function. The internal momentum can involve a high velocity, even though the quantum state itself may not change over long periods of time. (The state can be stationary in that it does not change, but there can still be internal motion within it.) This situation arises in single electron atoms that have highly charged nuclei, where “relativistic corrections” are needed to evaluate the energy due to high velocity effects of the wave function near the nucleus. However, as will be discussed further in remark 8 below, the treatment herein is not relativistic, and so the term “high velocity” is used instead of the term “relativistic”.

Remark 4 – Low Velocity Limit of Eq. (10).

It is useful to examine Eq. (10) in the low velocity limit. In that limit, the energy associated with the momentum is much less than the rest mass energy, \left ( -\hbar^2 c^2 \nabla^2 \xi \right ) / \xi << m^2c^4, and manipulation of the radical in this limit leaves  [ ( -\hbar^2 c^2 \nabla^2 \xi ) / \xi + m^2c^4 ]^{1/2} = m c^2  [1-(\hbar^2 c^2 \nabla^2 \xi ) /\xi m^2 c^4 ]^{1/2}  = mc^2 [1 -  (\hbar^2 c^2 \nabla^2 \xi ) /2\xi m^2c^4 ] = mc^2 -(\hbar^2 c^2 \nabla^2 \xi ) /2 \xi m c^2. Cancelling the c^2 terms and substituting this into Eq. (10) results in  ( i\hbar \frac{\partial \xi}{\partial t} ) /\xi = mc^2  -\hbar^2 \nabla^2 \xi / 2\xi m + V. At this point, since a constant in a potential does not affect the physics, we can absorb mc^2 into V, and by also multiplying through by \xi we can find the low velocity limit of Eq. (10) as  ( i\hbar \frac{\partial \xi}{\partial t} ) = -\hbar^2 \nabla^2 \xi/2 m + V\xi, which is immediately recognized as the Schrödinger equation.

Remark 5 – Low Velocity Limit of Eq. (13).

It is also of interest to investigate Eq. (13) in the low velocity limit. In this case, we can set E_N = \epsilon + mc^2 in Eq. (13), where \epsilon is an energy that is small in comparison to mc^2. This leaves -\hbar^2 c^2 \nabla^2 \Psi =  [(\epsilon + mc^2)^2 - 2(\epsilon + mc^2)V + V^2 - m^2c^4]\Psi = [\epsilon^2 + 2\epsilon mc^2 + m^2c^4 - 2\epsilon V - 2mc^2V + V^2 - m^2c^4]\Psi. V will also be small in comparison to mc^2 in the low velocity case. Noting that the terms m^2c^4 cancel, and discarding terms that are second order in small quantities, this leaves -\hbar^2 c^2 \nabla^2 \Psi = [2\epsilon mc^2 - 2mc^2V]\Psi, and dividing through by 2mc^2 leaves -\hbar^2 \nabla^2 \Psi /2m = [\epsilon - V]\Psi, which is recognized as the typical low velocity quantum mechanical equation for the spatial component of stationary states.

Remark 6 – A Low Velocity Derivation of Schrödinger’s Equation.

Note that Schrödinger’s Equation can be derived quickly if we replace Eq. 3 by the low velocity expression E = (p^2/2m) + V in the derivation done in Part 2. With our implicit assumptions of \mathbf{p}^2 = -\hbar^2\nabla^2\xi/\xi, E = \left ( i\hbar \frac{\partial \xi}{\partial t} \right ) /\xi we get E = (p^2/2m) + V = \left ( i\hbar \frac{\partial \xi}{\partial t} \right )/\xi = -\hbar^2\nabla^2\xi/\xi 2m + V, and after multiplying the latter equation by \xi we obtain  i\hbar \frac{\partial \xi}{\partial t} = -\hbar^2\nabla^2\xi/2m + V\xi, which is Schrödinger’s Equation.

Remark 7 – Low Velocity Verification of the Implicit Assumption.

The implicit assumption is presented in Remark 1: It is implicitly assumed that \mathbf{p}^2 = -\hbar^2\nabla^2\xi/\xi, E = \left ( i\hbar \frac{\partial \xi}{\partial t} \right ) /\xi, and E = \left [ p^2c^2 + m^2c^4\right ]^{1/2} + V are valid relationships within and throughout all wave functions. The implicit assumption is seen in Remark 6 to lead to Schrödinger’s Equation when the low velocity energy expression is relevant. Hence, it is reasonable to believe that the implicit assumption will also be valid when the high velocity energy expression is relevant. (The doubt about validity of the derivation arises because of the departure from the plane wave condition – not because of the form of the energy equation.)

Remark 8 – Inconsistency with Relativity.

The treatment presented here is not covariant, since Eq. (20) describes a wave function spread out over spatial coordinates, but time does not appear in the equation. This lack of covariance is the reason for the use of the term “high velocity” rather than the term “relativistic” throughout this work, since the treatment herein is not relativistic. Relativity[10] is a point-like theory of events in Four-dimensional space, and the theory presented herein is not consistent with such a foundation. Indeed, it is the constraint of relativity that has stopped others from already developing what is shown here. Instead, Dirac[11] proposed an equation that involves four by four matrices and a four component wave function for a description of electrons and positrons. Klein and Gordon[[12],[13]] and Proca[[14],[15]] developed two other covariant approaches applicable to other classes of particles. Those manifestly covariant approaches necessarily result in more complex treatments that what is described above, and they also involve a departure from the classical approach to physics.

Remark 9 – Consistency with Absolute Theories.

While the theory proposed above does violate relativity, this does not imply that it violates any known experimental results. In addition to relativity, the absolute theories of Lorentz[16] and the author[17] are also overwhelmingly consistent with all known experimental results. While not greatly appreciated in the year 2017, there is almost no predictive difference between the theory of Lorentz and that of Einstein. Indeed, the fundamental transformation equations of Einstein’s special relativity are called “the Lorentz equations” not “the Einstein equations” because those equations were first proposed by Lorentz, not Einstein. The main difference between Lorentz and Einstein lies in the interpretation of the Lorentz Equations – the equations themselves are identical. Furthermore, the absolute theories also allow for a ready understanding of Aspect, Dalibard, and Roger’s tests[18] of Bell's theorem[19], an understanding that relativity cannot easily provide. (See reference 17 (discussed here) for a more thorough discussion of the absolute versus relative theories.) The theory presented here – involving wave functions with a finite spatial spread described by a non-covariant equation – fits well within the absolute frameworks for space and time, even as it does not fit well within relativity.

Remark 10 – A Return to the Classical Approach.

The approach taken in this paper is extremely simple: it merely relies on a few empirical observations along with a single assumption of an underlying physical wave. No Hamiltonian nor Lagrangian formulations are necessary; no underlying principles (such as the relativity principle or the principle of least action) are appealed to. Instead, the approach is to return to classical physics thinking, which involves proposing a simple underlying physical model for nature, working from that physical model to develop the mathematical equations, and then analyzing the results of those equations to ensure that they accurately predict experimental reality. In this return to a classical approach, both the mathematics and the underlying physical model are readily amenable to human understanding.

Remark 11 – Handling of Infinities.

Eq. (20) has terms that involve the inverse of r to the fourth and sixth power. Those terms will go rapidly to infinity as r tends toward zero. For that reason, the author suspects that particles have some sort of finite size. (That is, Eq. (20) may only be applicable above some small limiting value, and below that limit, the wave function may be a constant.) One either accepts such an ultimate finite size, or one must deal with some unpleasantness equivalent to renormalization theory to handle the infinities. As a speculation, a small limiting size may be the result of particles being small solid balls. Such small balls may come about because of a preonic constituency of matter or perhaps from something even smaller. Alternatively, the small size limit may result from just how dense charge can become. For instance, it may be that charge density cannot exceed that of an underlying aether[20]. But no matter the source of the finite size, the important point for the present work is that a finite size can eliminate the problems that would otherwise be present for small r. Note also that the absolute theories can easily allow for finite size particles, but that relativity (a point-like theory in four-space) cannot so easily accommodate them.

Remark 12 – Missing Dirac Delta Function in the Magnetic Potential.

In contemporary treatments of the spin-spin interaction, use is made of a Dirac delta function for the return flux of the magnetic field. In this work, since it is assumed that the particles have a finite small size, the return flux will be confined within the finite size of the particle and hence the return flux does not play a role in the equations. (The equations are not to be applied within the small size of the particles.)

Remark 13 – Applicability to Stronger Forces.

While Eq. (20) is derived for the Hydrogen atom, Eqs. (10) and (13) are more general, as they allow for different potentials that may be useful in other situations. Calculations of lepton and meson masses can now in principle be treated, since the non-perturbative quantum mechanical Eqs. (10) and (13) can be applied to those situations where perturbative approaches are not feasible. Of course, it remains to specify the function V for those cases.

Remark 14 – The Lamb Shift.

The standard quantum mechanical treatment of the Hydrogen atom does not completely predict experimental results. Famously, the quantum mechanical treatment falls short as it does not predict the Lamb shift, where a very small difference is found from what is predicted by quantum mechanics. Only when a full QED treatment of radiative corrections is applied can the Lamb shift be predicted. However, as mentioned above, QED involves the process of renormalization which is itself a rather dubious technique. The equations presented herein are a new approach to quantum mechanical predictions and at this point it is not clear to the author whether additional corrections will be needed. The p state wave functions have zero value at the origin while s states have a finite value. If there is a small limiting size for the particles, this difference between p and s states near the origin may lead to a difference in energy levels due to the proposed small hard core, although this is mere speculation at this point.

Remark 15 – Difficulty of Finding Analytic Solutions.

The author was unable to find analytic solutions to Eq. (20) despite considerable effort. It may be that numerical methods will be required in order to find solutions to Eq. (20).

Remark 16 – Further Research.

Eventually, application of the treatment herein should enable advances in our understanding of forces stronger than that of electromagnetism. However, prior to those studies it would be useful to apply Eq. (20) to the hydrogen atom using numerical techniques. If Eq. (20) is truly representative of nature, both the hyperfine splitting and the high velocity corrections to the Hydrogen energy levels should be predicted, and as mentioned above, the Lamb shift may be predicted as well.  

Acknowledgements.

The author wishes to thank the anonymous reviewer for excellent comments that resulted in strengthening the clarity of this work. The author also wishes to thank the editor of Physics Essays, Dr. Emilio Panarella, for his many years of providing a home for works such as this that are presently outside of the mainstream, for it is there that true scientific breakthroughs are most likely to be found.

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