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 Schrödingers 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 Planks 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:
the corresponding wavelength
[2.43e-10 cm] (1.2)
is the Compton wavelength of the electron.
If we specify the orbit, which we identify as the electrons spin orbit, as circular with a circumference of one wavelength, λe, then the spin radius is
[3.86e-11 cm] (1.3)
and the magnitude of the spin angular momentum is
The Bohr radius for the Hydrogen ground state is
[5.3e-9 cm] (1.5)
where e is the electron charge. The ratio
is the Fine Structure Constant. An interesting physical correspondence I have not seen elsewhere.
For constant (2.1)
Since a quantums velocity is always c, we have
The de Broglie postulate E = hf specifies E as the total relativistic energy of the particle, which in the observer frame is
and from (2.5)
We see that r = |r| decreases with vz, which is to be expected since E = hf means higher energies correspond to shorter wavelengths and thus to smaller r. Interestingly, as vz approaches c, r approaches 0 and the helix approaches the straight line path of a quantum with infinite energy.
3. Schrödingers 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
where h is Planks 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
For em radiation v = c, the velocity of light, so
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, mo, is
Since the rest mass of a quantum is zero we have for its momentum
3.1 De Broglies 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
and that the motion of the particle is governed by the wave propagation properties of the pilot waves.
3.2 Schrödingers Requirements
Schrödinger picked up from there searching for the corresponding propagation equation. Changing de Broglies pilot waves terminology to wave function to describe both the waves and the mathamatical function Y(x,t), he adopted de Broglies postulates, but not de Broglies definition of E as the total relativistic energy,
choosing instead the classical definition of total energy,
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
Solutions to Schrödingers equation for a free (constant velocity) particle moving on the z axis are:
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 Schrödingers x is our z and our x and y become the complex plane, we write:
Which describes a quantum traveling at constant velocity vz on a helical path around the z axis in terms of the complex plane coordinates. This is not a solution to Schrödingers 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
Taking time derivatives assuming vz and thus w and r constant
Defining substituting from (2.7) and from (2.8) gives
Multiplying both sides of (4.8) by the total relativistic energy
which in full notation gives
Which is the full relativistic equation for the helix model free particle (constant vz). Of course in a non-zero potential V(z,t) vz 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 Schrödingers 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), vz = vz(z,t) will not be constant.
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 Schrödingers equation, but we have not addressed just what solution to Maxwells 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 whats 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.
How exactly does the momentum of one quantum interact with another?
Is the electrons 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 doesnt 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
A1 Rules and Substitutions
For clarity the following differentiation rules and substitutions are posted and numbered here for reference later:
6 (2.7) 7 (2.8)
Combining (A2.3) and (A2.2)
Which reduces to (4.3) for constant vz or vz << c. Remember in (A2.4) that r and w are no longer constant, but functions of vz as per (2.7) and (2.8). Thus (A2.4) really means
On to the second derivative: