Note: This is an old treatment. The early section on the Fine Structure Constant might be of interest. The rest not so much.

Some implications of viewing the electron as an orbiting quantum are pursued to reveal interesting visualizations of electron spin, the Compton Wavelength, the Fine Structure Constant, the Heisenberg Uncertainty Principle, and possibly Schrödinger’s Equation. The Helix Wave Function describing the absolute path of the quantum is derived and explored for the non-relativistic case, and for the relativistic case of a constant velocity free electron. A full relativistic treatment is begun and carried to the first time derivative. The second derivative remains unexplored as yet.

**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

If we specify the orbit, which we identify as the electron’s spin orbit, as circular with a circumference of one wavelength, λ

[3.86e-11 cm] (1.3)

and the magnitude of the spin angular momentum is

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.

Consider an electron moving in the z direction at constant velocity, v

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

(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.”

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.

** 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

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

or in primed notation

(4.2)

Taking time derivatives assuming v

(4.3)

and (4.5)

Defining substituting

(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

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

For non-relativistic v

(4.11)

The free particle solutions, Y(z,t) = re

(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.

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

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

What happens to the solutions of (4.11)

**5. For Further Investigation
**

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.

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.

**Appedix A - Derivation of the Relativistic Helix Equation
**

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: