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1. The Isolated Electron at Rest

Viewing an isolated electron (or any particle) at rest as an orbiting quantum:
The total relativistic energy E is the electron rest mass energy, and the equivalent quantum energy is hf where h is Plank’s constant and f is the de Broglie frequency of the electron. If c is the velocity of light, then from Special Relativity and elementary quantum mechanics:

      (1.1)

the corresponding wavelength

      [2.43e-10 cm]       (1.2)

is the Compton wavelength of the electron.

1.1 Spin

If we specify the orbit, which we identify as the electron’s spin orbit, as circular with a circumference of one wavelength, λ_e, then the spin radius is
       [3.86e-11 cm]       (1.3)

and the magnitude of the spin angular momentum is

      (1.4)

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1.2 The Fine Structure Constant

figure 1

The Bohr radius for the Hydrogen ground state is

      [5.3e-9 cm]       (1.5)

where e is the electron charge. The ratio

       [1/137]       (1.6)

is the Fine Structure Constant. An interesting physical correspondence I have not seen elsewhere.

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2. The Free Electron

Consider an electron moving in the z direction at constant velocity, v_z. For simplicity we assume it to be oriented with its angular momentum in the z direction as well so the orbiting quantum follows a helical path as shown in fig. 2. If it can indeed be in any random orientation, then its exact position and momentum will be uncertain :-)

figure 2

For constant              (2.1)

and                     (2.2)
                            (2.3)

Since a quantum’s velocity is always c, we have

      (2.4)

from which:

and             (2.5)

The de Broglie postulate E = hf specifies E as the total relativistic energy of the particle, which in the observer frame is

       (2.6)

so             (2.7)

and from (2.5)

       (2.8)

We see that r = |r| decreases with v_z, which is to be expected since E = hf means higher energies correspond to shorter wavelengths and thus to smaller r. Interestingly, as v_z approaches c, r approaches 0 and the helix approaches the straight line path of a quantum with infinite energy.

3. Schrödinger’s Equation Revisited

Einstein postulated the quantization of em radiation in the photoelectric effect by assuming the energy content E, of a quantum of energy of frequency f is

       (3.1)

where h is Plank’s constant.

The wavelength of the radiation is obtained by the usual relation between its wavelength λ, f, and v, the propagation velocity of the wave. This is

      (3.2)

For em radiation v = c, the velocity of light, so

      (3.3)

The quantum is characterized as a particle of zero rest mass, and total relativistic energy, E which is entirely kinetic. The general relation between total relativistic energy E, momentum, p, and rest mass, m_o, is

      (3.4)

Since the rest mass of a quantum is zero we have for its momentum

      (3.5)

3.1 De Broglie’s Postulates

De Broglie figured if radiation waves could have a particle nature then maybe particles could have a wave-like nature, and thus was born quantum mechanics. He postulated that the wavelength, λ, and the frequency, f of the “pilot waves” associated with a particle of momentum p and total relativistic energy E are given by

      (3.6)

and that the motion of the particle is governed by the wave propagation properties of the “pilot waves.”

3.2 Schrödinger’s Requirements

Schrödinger picked up from there searching for the corresponding propagation equation. Changing de Broglie’s “pilot waves” terminology to “wave function” to describe both the waves and the mathamatical function Y(x,t), he adopted de Broglie’s postulates, but not de Broglies definition of E as the total relativistic energy,

      (3.7)

choosing instead the classical definition of total energy,

      (3.8)

He then specified the following three requirements which the wave equation, Y(x,t), must satisfy:

1. It must be consistent with the de Broglie postulates (3.6)

2. It must be linear in , i.e., if and are solutions, then any linear combination of these must also be a solution for any values of the constants and . This accomodates superposition of waves to produce interference effects which we know occur.

3. The potential energy V is, in general, a function of x and t. In the special case , a constant, the momentum p and the energy E will also be constant. This is the situation of a free particle moving at constant velocity for which it is required that the wave equation have oscillatory traveling wave solutions of constant wave number k = p/h and frequency f = E/h.



From these conditions Schrödinger derived his famous equation

      (3.9)

Solutions to Schrödinger’s equation for a free (constant velocity) particle moving on the z axis are:

       (3.10)

where k = 1/l = p/h = the wave number, and K = 2πk, and w = 2πf as given by the de Broglie relations: f = E/h, and λ= h/p, where p is the particle momentum.

Y (z,t) is visualized as a traveling wave on the x axis with a real part and an imaginary part which have previously been given only statistical significance in calculating the probability of finding the particle at a coordinate x, i.e., not considered as pertaining to any physical reality on an individual particle basis.

In fact, 3.10 also describes a traveling helical wave propagating in the x direction normal to the complex plane.

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4. The Helix Equation

From the treatment of section 2. of the postulated helical path of a quantum which constitutes an electron, if we switch coordinate labels so Schrödinger’s x is our z and our x and y become the complex plane, we write:

       (4.1)

Which describes a quantum traveling at constant velocity v_z on a helical path around the z axis in terms of the complex plane coordinates. This is not a solution to Schrödinger’s equation. Without the Kz term there is no traveling wave, but what does it mean? Lets take a few derivatives and see where it takes us.

The solution (4.1) to the helix equation can be written



or in primed notation

       (4.2)

Taking time derivatives assuming v_z and thus w and r constant

      (4.3)

and        (4.5)

Defining substituting from (2.7) and from (2.8) gives

      (4.6)

       (4.7)

       (4.8)


Multiplying both sides of (4.8) by the total relativistic energy

      (4.9)

gives      

i.e.      

            

from (4.7)      

so      

i.e.      


which in full notation gives

       (4.10)

Which is the full relativistic equation for the helix model free particle (constant v_z). Of course in a non-zero potential V(z,t) v_z is not likely to be constant. The relativistic version of the helix equation for the general case is derived in appendix A. Meanwhile, we continue with the non-relativistic equation to see what happens in any potential V(z,t).

Unlike Schrödinger’s equation the second derivitive is a time derivative instead of a space derivative. This is because there is no traveling wave.

Equation (4.10) is easily shown to satisfy requirements 1 and 2 of section 3.2, but not 3 which requires a traveling helical wave when what we have is a traveling particle on a helical wave trajectory.

Requirement 3 was based on the assumption that a particle is the more or less localized wave packet resulting from the superposition of traveling waves. It probably seemed the only way to account for the dual particle-wave nature of reality. This seems no longer necessary.

In the general case with the potential energy V(z,t), v_{z =} v_z(z,t) will not be constant.

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4.1 The Non-Relativistic Helix Equation

For non-relativistic v_z << c (4.10) becomes

      (4.11)

The free particle solutions, Y(z,t) = re^iwt can also be written (substituting from (2.6) and (2.8))

      (4.12)

Where      

Note that unlike the Schrödinger equation, there is no space dependent part, Y(z,t) is just Y(t) in the non-relativistic free particle case.

4.1.1 Probability Not

Lets investigate the function



for the free particle case.

This is clearly not Born's probability function, but who needs it. We know where the damn particle is. It's just not always in the "real" plane, in fact it hardly ever is since it just passes through twice per cycle. We have been dealing all these years with the square of the projection of the r vector on the "real" plane, eliminating the bothersome "unreal" projection on the "imaginary" plane using the complex conjugate trick, and calling the result the probability of finding the "real" coordinate of the particle.

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4.2 The Helix Wave Function in a General Potential V(z,t)

What happens to the solutions of (4.11) in the presence of a general potential V(z,t)? No longer is Y a function of time alone. The particle axis is no longer confined to the z axis. It can pull wheelies and backflips, and it can appear at the same space coordinates multiple times in the course of its travels given the appropriate potential field. But is its energy quantized? If it doesn't account for experimental emission and absorption lines and other observables as does the Standard Model, then this whole exercise illuminates a path going nowhere.

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5. For Further Investigation

We have accounted for the mass of the electron as purely relativistic, and its behavior, at least in inertial frames, with a visualizable model similar to Schrödinger’s equation, but we have not addressed just what solution to Maxwell’s equations, if any, describes such a circulating quantum which exhibits conserved charge and a magnetic moment. It should also be very interesting to explore the helical model in atomic settings and maybe make some animations of what’s really happening when photons are emitted and absorbed, or during pair production and annihilation. Maybe we will find new insight on other particles, the nucleus, antimatter, gravity, and more. There is some reason why particle masses and forces are what they are. Maybe with this visualization tool more progress can be made.

Assuming all particle masses are similarly created, we should be able to express all of physics in terms of momentum. There may be no such thing as mass.

Some Questions

How exactly does the momentum of one quantum interact with another?

Is the electron’s electric field truly spherical and constant? Have any experiments verified this? Does every particle align its spin vector parallel to the direction of travel? What about spin zero particles?

Does the helix model explain the two-slit experiment, Compton scattering, the photoelectric effect, etc?

How does a proton capture an electron and become a Hydrogen atom? The corresponding Compton wavelength and thus the radius of a proton is ~1/2000 that of the electron. Does the electron bag the proton in its field as it passes through and get zapped back and forth like the lasso of a lasso expert jumping back and forth through his lasso? And if the proton doesn’t hit dead center, do the resulting asymmetric ringing contortions look like the beautiful quantum states we have plotted for the various H energy levels?

Are the quarks in a proton three quanta following bizarre paths that forbid separation, or maybe just one following a really bizarre path, or maybe its the same thing. More likely, the Maxwellian dynamics in atomic and nuclear domains leave the simple helical path we have envisioned for the free electron (which may itself be simplistic) behind for more complex geometries and dynamics. The quantum may not need to be a particle at all.

What does the relativistic helix equation look like? This configuration begs us to find out.

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Appedix A - Derivation of the Relativistic Helix Equation

A1 Rules and Substitutions

For clarity the following differentiation rules and substitutions are posted and numbered here for reference later:

Rules
1                                     2      
3      

Substitutions
4                                                  5      
6                 (2.7)                     7                   (2.8)
8                       9      

10             (4.2)

A2 Derivatives
10,3             (A2.1)
                                             
1                         9      
                        10
3        (A2.2)

8,10
6
7             (A2.3)

Combining (A2.3) and (A2.2)

      (A2.4)

Which reduces to (4.3) for constant vz or v_z << c. Remember in (A2.4) that r and w are no longer constant, but functions of v_z as per (2.7) and (2.8). Thus (A2.4) really means

₆,₇             (A2.5)
On to the second derivative: