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By Charles Rhodes, P. Eng., Ph.D.

Quantum mechanics is generally concerned with the available energy states and absorption and emission of radiation by free particles, particles in an external magnetic field, bound electrons, nuclear particles, nuclei, atoms, molecules and crystals.This web page summarizes the major issues of quantum mechanics that are relevant to this web site.

In the macroscopic world people are accustomed to physical problems that have unique solutions.Quantum systems are intuatively more complicated because they have multiple discrete real solutions known as energy states. In quantum mechanics each particle adopts one of a set of discrete energy values. For a particular particle there is no certainty as to which energy state it will adopt. However, a large number of particles will behave in a macroscopicly repeatable manner.

Energy is stored in matter via quasi-toroidal electromagnetic structures known as spheromaks. A spheromak has associated radial electric, poloidal magnetic and toroidal magnetic fields. These fields store electric and magnetic field energy, commonly known as particle rest mass.

A spheromak is formed when net charge circulates at the speed of light around a closed path. The spheromak current path within the spheromak wall, which involves both poloidal and toroidal turns, is stable because everywhere along that path the net electric and magnetic force on the current filament is zero.

Spheromaks in free space all have the same relative geometrical shape. The spheromak's closed current path length together with the speed of light sets the spheromak's natural frequency. As the radius of a spheromak decreases its associated stored energy and frequency both increase. Spheromaks gain or lose energy by absorption or emission of elecromagnetic photons. The energy contained in an emitted photon is equal to the energy difference between the spheromak's initial and final states.

Quantum charged particles are spheromaks. Assemblies of quantum charged particles also tend to form spheromaks. Atomic electrons form spheromaks. A spheromak exists when a quantized net charge forms a current which circulates at the speed of light around a stable closed spiral path before the current retraces its previous path. Note that in many cases the actual physical charge movement rate is much less than the speed of light because the net charge is the difference between two larger numbers of quantized positive and negative charges. Freqently at low temperatures the positive charges hardly move at all. Conduction electrons in a metal circulate within a sea of positive charge. If the numbers of quantized positive and negative charges are large but only slightly different the relatively small net charge acts as if it circulates at the speed of light.

The charge circulation follows a complex closed spiral path having Np poloidal turns around the main axis of symmetry and Nt toroidal turns which traces out a quasi-toroidal shaped surface known as a spheromak wall. The numbers Np and Nt are positive integers.

The spiral current path is further characterized by the maximum radius from the axis of symmetry Rs, the minimum radius from the axis of symmetry Rc and the maximum height from the equatorial plane Hm.

In a stable spheromak Np and Nt can only be whole number positive integers with no common factors. Hence the ratio of Np to Nt is of the form:
P = Np + 2 Nt
where P is a prime number.

Hence the ratio (Np / Nt) is a rational number in which Np is odd and P is prime.

Hence any quantum state is characterized by a prime number P and an an odd integer Np.

In atomic chemistry atoms with larger atomic numbers adopt larger primes P which gives these atoms the appearance of electron "shells". In interactions with electromagnetic radiation a spheromak absorbs or emits a quantum of radiant energy known as a photon.

In crystals the field energy density does not go to zero at large distances from each atomic nucleus. The result is the formation of gaps in the available spheromak electron energy states. This phenomena is central to modern solid state electronics.

Quantum states apply to any system that can form circulating current spheromaks. Examples include nuclear particles, atomic particles, atoms, molecules, solid matter, liquid matter and plasmas. Hence real matter consists of a large number of interacting spheromaks.

Application of an external magnetic field to a spheromak causes it to adopt one of two closely spaced energy states. When a spheromak transitions between these two energy states it emits or absorbs a photon of electromagnetic radiation.

Nucleons are strongly interacting spheromaks.

During the first few years of the 20th century Planck explained the spectral behavior of thermal radiation by making the assumption that:
Ep = h Fp
where Ep is a quantum of energy emitted by a particle via radiation and Fp is the emitted radiation frequency.

In 1905 Einstein explained the photoelectric effect by assuming that the energy Ep absorbed by an electron from incident light (electromagmetic radiation) obeyed the equation:
Ep = h Fp
h = an apparent physical constant
Fp = radiation frequency.

Today h is known as the Planck constant, where in new metric system units:
h = 6.636070150 X 10^-34 J-s.

During the 1920s deBroglie used the average behavior of a beam of electrons passing through two adjacent slits to conclude from the resulting interference patterns that a stream of electrons can be mathematically represented as a propagating wave with wavelength Lamdax. This mathematical representation is only valid for a stream of electrons which indicates average electron behavior.

Single electrons behave as discrete particles with path uncertainty related to their quantum state. A large number of sequential single particles exhibits wave like behaviour. This wave-particle duality is one of the least intuitively understood aspects of quantum mechanics. It appears that a physical system is characterized by a set of discrete quantum states which exist independent of their occupancy. Each of these quantum states has a characteristic energy Eigenvalue.

In the dual slit experiment the dual slits establish a set of quantum states. Each electron in an electron beam randomly adopts one of the available quantum states. When there are a large number of electrons the electrons appear to interfere with each other whereas in fact it is the set of quantum states that creates the appearance of wavelike interference. A single slit creates a different quantum state configuration than a dual slit.

Electromagnetic particles and available quantum states are both the result of highly non-linear spheromak equations. At the microscopic level spheromak interactions have multiple real solutions known as quantum states. At any instant in time each particle adopts only one of these quantum states but different identical particles can adopt different states. What we experimentally observe with a large number of particles is a weighted superposition of particles with different quantum states. This weighted superposition is only valid for many particles. A single particle in a single trial adopts a particular discrete energy state. Hence a large number of particles is required to obtain an averge solution instead of a discrete solution.

Generally in quantum mechanics the different real solutions (available energy states) are distinguished by energy incrments:
dE = h Fp
where Fp is the frequency of emitted or absorbed radiation transferring amount of energy dE in a transition between the two energy states.

Particle Parameter Definitions:
P = particle linear momentum;
C = speed of light;
E = total particle energy;
Eo = particle rest energy;
h = Planck Constant;
F = particle natural frequency.

Lamda = wavelength of electron beam wave as observed in a dual slit experiment

Special relativity gives:
E^2 = P^2 C^2 + Mo^2 C^4
= P^2 C^2 + Eo^2

Note that the momentum component of total energy is orthogonal to the rest mass component of total energy.

When an electron beam is directed at a dual slit it forms an interference pattern consistent with an electron beam wavelength Lamda that conforms with:
P^2 C^2 = (E V / C)^2
= (h C / Lamda)^2
P = h / Lamda
= h Fp / C

Thus the apparent electron beam wave frequency is proportional to the electron linear momentum. This is one of the fundamental experiments of quantum mechanics. From this experiment we can conclude that the set of available quantum states depends on the incident electron momentum.

P = M V = (E V) / C^2
= (h F V) / C^2
= (h C V) / (Lamda C^2)
= (h / Lamda)(V / C)
= (h / Lamdax)
= (h Fx) / C
Fx / C = F V / C^2
Fx = F (V / C)

(h F)^2 - (h Fo)^2 = C^2 P^2
F^2 - Fo^2 = (C / h)^2((h / Lamda)(V / C))^2
= (C / Lamda)^2 (V / C)^2
= (C F / C)^2 (V / C)^2
= (F)^2 (V / C)^2
= Fx^2
= C^2 P^2 / h^2

Hence the dual slit experiment responds to two frequencies, Fo associated with the electron rest mass and Fx associated with the electron's linear momentum. In effect Fx is a beat frequency between F and Fo similar to the way that a photon frequency is a beat frequency between two spheromak Fh values.

There is a fundamental issue here in that Fo is the frequency associated with the electron rest mass whereas Fh is the frequency associated with the electron spheromak. As determined by electron spin resonance measurements, Fh is about two orders of magnitude less than Fo. It appears that the explanation may be that:
Fo = C / 2 Pi Ro
Fh = C / Lh
Ro = nominal electron spheromak radius
(Lh / 2 Pi Ro)
is a spheromak geometric constant closely related to the Fine Structure constant.

Note that as the electron velocity V increases Lamdax, which is the apparent electron wavelength as determined from the dual slit interference pattern, decreases, while Fo is the frequency of the standing wave inside a stationary electron. In a practical experimental apparatus:
Fo >> Fx

Recent work by this author on the Fine Structure Constant indicates that familiar stable charged particles have spheromaks that are really non-propagating solutions to Maxwells equations under the conditions of:
a) Charge quantization;
b) Local charge circulation at the speed of light;
c) Np and Nt have no common factors other than unity;
d)The field energy density is identical on both sides of a spheromak wall.

Hence quantum mechanics appears to be a study of local non-propagating solutions to Maxwells equations under the above conditions.

During the 1890s Lorentz pointed out that there was an apparent inconsistentcy between mechanics and electromagnetism. That inconsistentcy was revealed by the Lorentz Transform, which later became part of Special Relativity.

Gravity appears to be the long range interaction between potential energy wells that have no net charge.

This author does not know the source of charge quantization. However, if charge was not quantized we would not exist to discuss the subject.

In most inter-particle interactions the gravitational force is many orders of magnitude smaller than the electromagnetic force.

The origin of h is:
h = (dE / dF)
dE = the change in particle potential energy
dF = change in particle natural frequency.
The constant h appears in many different physical relationships so it is desireable to accurately determine:
h = dE / dF

In order to determine h we implicitly assume that h is the same for all free particles. The structure that we analyze in detail is a spheromak.

Hence a change in particle energy dE is:
dE = h dFh
h = dE / dFh

If we change total particle energy E by application of an external magnetic field Bx then the change in particle energy dE is given by:
dE = [dE / dBx] dBx

However, if the only means of the particle changing energy is emission or absorption of electromagnetic quanta of energy Ep then:
dE = h dFh = Ep = h Fp

h Fp / dBx = dE / dBx
(Fp / dBx) = (1 / h)[dE / dBx]
(Fp / dBx) = accurately measureable parameter
[dE / dBx] = value that can be found by mathematical analysis
h = value of the Planck Constant to be determined.

On this web site we demonstrate determination of h from theoretical electrodynamic analysis.

For reasons unknown the net charges of atomic particles are quantized in exact integer multiples of Q = 1.602176634 X 10^-19 A-s. A charged particle can form a physically stable quasi-toroidal shaped structure known as a spheromak which when isolated in free space has a characteristic geometry and a radius dependent amount of static field energy. Each electromagnetic spheromak has a characteristic frequency Fh related to its radius. When a spheromak is in an externally applied magnetic field the total spheromak static field energy depends on the orientation of the spheromak axis of symmetry with respect to the axis of the external magnetic field.

When such a spheromak absorbs or emits energy its axial orientation with respect to the external magnetic field axis changes. The equations for spheromak energy show that there is a fixed proportional relationship between the change in spheromak characteristic frequency Fh from Fha to Fhb and the change in spheromak total energy from Etta to Ettb. This same proportional relationship also applies to radiation photons which have an electromagnetic wave frequency Fp given by:
Fp = Fhb - Fha.

The magnetic field of a particle affects the energy states of other nearby particles. In many cases, on a microscopic scale, there are many real solutions for a multi-particle system's stable energy. When large numbers of spheromaks (particles) are involved the fraction of the particles that adopt each energy state can be determined via a probabilistic analysis.

The source of these multiple real energy states is in part the structure of atomic particle spheromaks. A spheromak can be characterized by its nominal radius Ro, by its number of closed path poloidal turns Np and by its number of closed path toroidal turns Nt. Changes in spheromak energy related to photon emission/absorption cause small changes in Ro which in turn affect the particle spheromak's characteristic frequency Fh. In a large cluster of particles at any instant in time there will be a temperature dependent fraction of the particles in each energy state.

Quantization of energy occurs because in a stable charged particle Np, Nt, dNp and dNt must all be integers. Hence energy, which is a real quantity, is only stable in integer based quantities. Changes in stable energy occur as a result of changes in particle radius or changes in the integers Np and/or Nt. In multi-particle systems the particles may adopt different Np and Nt values so the situation quickly becomes very complicated.

Any measurement of a particle's energy state involves a photon emission or absorption which will change the particle's energy state. Due to ongoing photon absorption/emission an observer is uncertain as to a particular particle's actual energy at any instant in time. This phenomena is known as quantum mechanical uncertainty. When this uncertainty is expressed as:
[(position uncertainty) X (momentum uncertainty)] ~ (h / 4 Pi)
or as:
[(energy uncertainty) X (time uncertainty)] ~ (h / 4 Pi)
h = 6.62607015 X 10-34 m^2 kg / s

Quantum uncertainty introduces uncertainty into records regarding the past and projections regarding the future.

Recall that a change in kinetic energy is given by:
dEk = (dP / dT).dX
dEk dT = dP.dX ~ (h / 4 Pi)

However, quantum mechanical solutions reliably model the behaviour of statistically large groups of particles on the basis of statistical fractional occupancy of available energy states (possible real solutions) at each energy level.

When viewed quantum mechanically atomic charged particles exhibit stationary periodic wave like qualities. There is no intention to pursue quantum mechanics on this web site other than to show the origin of the Planck constant and mention that the cause of quantum mechanical behaviour is multiple real solutions (energy states) to the governing physical equations. The existence of multiple real solutions allows life forms a limited degree of free will in decisions regarding their immediate future. Hence to a limited degree our future is not deterministic and mankind has some control over the future.

An important quantum mechanical issue in modern electronics is that some materials, such as pure silicon, exhibit an electron energy band gap. A band gap is a range of energies that free electrons cannot take. This band gap enables the formation of transistors and hence bistable electronic circuits which form the basis of modern digital computers. The band gap also enables formation of solar cells and electronic cameras.

There is another important electron energy step known as the work function between the conduction electrons in a solid metal and isolated electrons in free space.

Different methods of measuring h electronically give slightly different values in part due to kinetic energy associated with the recoil momentum of the photon emitting or absorbing particle. The origin of h is:
h = (dE / dFh)
where dE is the change in potential energy of the particle that emits or absorbs radiation and dFh = Fp = radiation frequency.

Numerous experimentally observed atomic spectra, chemical bonding and electronic phenomena have been successfully explained by assuming that h acts like a physical constant.

Shortly after WWII the phenomena of proton magnetic resonance was experimentally observed. Proton magnetic resonance also led to a much better understanding of the physical origin of h. It turns out that h is a frequently reoccurring composite of other constants that arise from the stationary solution of the electromagnetic equations that describe a free charged particle. The Schrodinger formulation of quantum mechanics, which treats h as an independent physical constant and which treats a stream of charged particles as pseudo wave like objects is widely used because it allows relatively easy practical solution of many physical problems. However, the Schrodinger methodology does not convey a good understanding of physical reality to the end user.

An advantage of the Schrodinger methodology is that it results in second order differential equations that with a modest amount of work can be manually solved. That was important in the 1960s when personal computers simply did not exist and central main frame computers were difficult to program. Today, with a modest amount of programming effort, a personal computer can easily find solutions to the higher order equations that are frequently encountered in quantum mechanics.

1. Our local universe is partially composed of the stable particles known as electrons and protons that have quantized charge;
2. Atoms are in essence aggregations of protons and electrons bound in mutual potential energy wells;
3. During the particle aggregation process there is net emission of electromagnetic radiation;
4. Conservation of energy requires that the change in radiant energy be precisely equal to the change in energy of the particle or system of particles that emits or absorbs the radiant energy;
5. Hence the origin of h as it affects radiation lies in the relationship between energy and frequency in charged particles. Interacting particles will form mutual potential energy wells that in a cold environment will randomly emit radiant energy in quantum amounts in order to adopt the lowest available energy state.
6. If the surrounding environment contains a high radiation density, the charged particle assembly will absorb radiation until the radiation power absorption balances the radiation power emission.

The stable energy states of a charged particle spheromak can be found by assuming that:
1. The particle energy consists of the electric and magnetic field energy components associated with a spheromak;

2. A spheromak has a quasi-toroidal shaped wall that acts as the boundary between the inside region and the outside region.

3. The static field energy density U outside the spheromak wall is of the form:
U = Uo [Ro^2 / [(K Ro - R)^2 + Z^2)]^2
Ro indicates spheromak size;
R = radius of a point from the main axis of spheromak symmetry and
Z = distance of a point from the spheromak's equatorial plane.
K = 0.5 + D D ~ 0.0007

4. Inside the spheromak wall the static field energy density is of the form:
U = Uto (K Ro / R)^2
It can be shown that:
Uto = Uo

5. These expressions for U allow the existence of a stable charged particle in the form of a spheromak.

6. For collections of particles where R >> K Ro the expression for U simplifies to classical electrodynamics.

7. The electric and magnetic field energy held by a charged particle spheromak forms particle rest mass. For a particle at rest this energy is almost constant except during matter-antimatter interactions. Changes in electric and magnetic field energy due to photon absorption or emission are usually small compared to the rest mass energy. Changes in gravitational field energy are extremely small as compared to the rest mass energy;

8. The observed net particle charge is the difference between quantized amounts of circulating positive charge and negative charge;

9. The charge quantization process is not known to this author;

10. The electric and magnetic field energies integrated out to infinity are constants for an isolated free particle but change as particle fields overlap causing kinetic energy and emission of photons;

11. The toroidal and poloidal magnetic field energy arises from the movement of distributed quantized charge along a closed spiral path at the speed of light;

12. The electric field energy arises from the radial electric field caused by the net distributed charge;

13. In an atomic particle the circulating charge has no mass and hence is not subject to inertial forces. The static electric and magnetic fields contribute to the particle's rest mass energy;

14. Maxwells equations are satisfied. At every point on the spheromak wall the total field energy density on both sides of the spheromak wall is equal so that the the spheromak has a stable geometrical configuration. Viewed another way, the charge and charge motion together form a stable minimum energy geometric configuration. Absent an external field any deviation from this minimum energy configuration increases the total electromagnetic energy. The total electromagnetic energy is proportional to the charge circulation frequency Fh.

15. The spheromak geometry gives the equation:
[Lh / 2 Pi Ro]
= [Np P]^0.5

16. Spheromak solutions are further governed by the Np and Nt no common factor restriction and that:
P = Np + 2 Nt
where P is a prime number.

17. The energy content of a spheromak is governed by the equation:
Ett = [(Uo Ro^3) Pi^2 ____________ where:
Uo = energy density at the center of the spheromak.
Pi = 3.14159265

18. The frequency Fh of a spheromak is:
Fh = Lh / C
Lh = current filament path length
C = speed of light

19. The energy density at the center of a spheromak in free space is given by:
Upo = [Q^2 B^4 / (32 Ro^4 Epsilono Pi^2 K^4)]

20. The Planck Constant is:
dEtt / dFh =______________________

21. The spheromak geometric constant is:
[Lh / 2 Pi Ro] = [Np P]^0.5

27. The Fine Structure Constant Alpha is provided by the equation:
[1 / Alpha] = ________________________

When the above described charged particle is placed in an external magnetic field the original single energy state takes on a range of values depending on the particle's orientation with respect to the external magnetic field. The energy difference between the different particle orientations is proportional to the applied external magnetic field. An individual charged particle can transition between two different orientations (energy states) by emission or absorption of a photon of electromagnetic radiation. With protons this effect is known as NMR (nuclear magnetic resonance). The relationship between the photon energy Ep and the emitted or absorbed radiation frequency Fp is given by:
Ep = h Fp
where h is the Planck constant. In reality h is a composite of other physical constants including:
quantized charge, permiability of free space, speed of light and a geometrical ratio known as the Fine Structure Constant.

When an atomic nucleus contains multiple particles with quantized charges the poloidal magnetic fields associated with these quantized charges tend to cancel each other. The NMR signal strength is strongest when the sum of protons + neutrons) is odd although H-2, Li-6, B-10 and N-14 are sometimes used in NMR studies. NMR signal analysis is further complicated by the shielding effect of atomic electrons which act to reduce the externally applied magnetic field in the vicinity of the atomic nucleus.

Note that:
Ep = photon energy ~ change in particle energy between the magnetically aligned and unaligned states
Fp = change in particle electromagnetic characteristic frequency
= emitted or absorbed radiation frequency

The local universe is full of radiation that results from quantized energy transitions that occur within aggregations of stable charged particles. In a high radiation density environment unexcited charged particles absorb radiation and thus adopt a higher average energy state and hence a higher temperature. Similarly in a low radiation environment excited charged particles emit radiation and thus adopt a lower average energy state and hence a lower temperature. All substances absorb and emit thermal radiation to some degree, although molecules with electrostatic bonding and hence charge separation couple much more strongly to electromagnetic radiation than do molecules without such charge separation.

In warm matter charged particles are constantly absorbing and emitting radiation photons. At the boundary between the warm matter and surrounding space radiation is constantly being emitted into space and is constantly being absorbed from space. When the rate of photon energy absorption equals the rate of photon energy emission the matter is at the same temperature as the radiation in the surrounding space.

Similarly Earth absorbs a fraction of incident solar radiation and emits infrared thermal radiation.

Steady State Emission temperature is the temperature at which the steady state absorbed thermal power from solar radiation equals the steady state emitted thermal infrared radiation power.

Persons involved in analysis of atomic spectra, chemical reactions and solid state electrical phenomena usually don't care about the physical origin of h. They simplify their work by treating h as an independent physical constant. Further, many quantum mechanical calculations are done assuming Newtonian mechanics to make the equations simple enough for practical closed form solution.

Nuclear particle interactions often occur at particle kinetic energies that are a significant fraction of the particle rest mass energy. Under these circumstances special relativity must be taken into account. Most nuclear calculations by engineers are done using simple cross section models and tabulated experimental results. Accurate quantum mechanical analysis of nuclear particle interactions tends to be the domain of high energy particle physicists.

This web page last updated October 19, 2022.

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