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QUANTUM MECHANICS

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

INTRODUCTION:
The universe is an assembly of radiation photons, discrete quantum particles, nuclei, atoms, molecules, ions, liquids and solids. One of the main means of observing the behaviour of these species is via the electromagnetic radiation that they emit or absorb under various circumstances. In order to make sense of this wealth of data it is essential to have a good mathematical model of discrete particles and their interaction with radiation. This mathematical model can then be extended to explain more complex asemblies.

Quantum mechanics is mainly concerned with the behaviour of quantum charged particles, including absorption and emission of radiation. This web page summarizes some of the major issues of quantum mechanics. It is shown that stable charged particles are in reality static solutions to well known electromagnetic equations.

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 quantum states. These states occur in part due to the properties of prime numbers. For a particular particle at any instant in time there is no certainty as to which quantum state it will adopt. However, a large number of particles will behave in a macroscopicly repeatable manner.

Energy is stored in charged particles 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, which together form the particle's rest mass.

A spheromak is formed when a filament of net charge Q circulates at the speed of light C around a closed path of length Lh with characteristic frequency:
F = C / Lh.

The filament of circulating charge follows a complex closed spiral path having Np poloidal turns around the spheromak toroid's main axis of symmetry and Nt toroidal turns around the spheromak minor toroidal axis. This filament traces out a quasi-toroidal shaped surface known as a spheromak wall. The spheromak wall geometry is stable because the electric and magnetic forces precisely balance each other everywhere along the filament path and because the spheromak forms a potential energy well. Application of a strong external magnetic field causes spheromak energy state splitting where the energy difference between the split states is proportional to the external magnetic field strength.

Understanding quantum mechanics requires understanding of the concept of a spheromak wall and the constraints on the filament path that defines the spheromak wall. The multiplicity of potential filament paths causes much of the complexity of quantum mechanics.

All isolated quantum charged spheromaks in free space share the same relative geometrical shape. As the characteristic radius Ro of a spheromak decreases its associated stored energy Ett and frequency F both increase. Spheromaks gain or lose energy by absorption or emission of electromagnetic photons. The energy contained in an emitted photon is equal to the energy difference between the spheromak's initial and final states. At any particular energy a spheromak can have multiple quantum states.

Assemblies of free quantum charged particles as in a plasma or a solid electricity conductor also tend to form spheromaks. Atomic electrons form spheromaks. A spheromak exists when a quantized net charge forms a net current which circulates at the speed of light along a stable closed spiral 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 in solids at low temperatures the positive charges hardly move at all. Conduction electrons in a metal, semi-conductor or plasma 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.

In a stable spheromak the number of poloidal filament turns Np and and the number of toroidal filament turns Nt can only be whole number positive integers with no common factors. Hence the ratio (Np / Nt) is a rational number

Each prime number P has two associated potential families of quantum states Np, Nt. A quantum particle's energy is inversely proportional to its filament length Lh. In interactions with electromagnetic radiation a spheromak absorbs or emits a quantum of radiant energy known as a photon. The effect of the photon emission or absorption is to change the length Lh of the particle filament, which defines the particle energy.

The spiral filament path and hence the spheromak wall shape is characterized by the spheromak dimensions with respect to the spheromak characteristic radius Ro. The maximum spheromak wall radius from the main axis of symmetry is specified by:
(Rs / Ro),
the minimum spheromak wall radius from the main axis of symmetry is specified by:
(Rc / Ro)
and the spheromak wall maximum height above the equatorial plane at R = [(Rs=Rc) / 2]Ro is specified by:
(Hm / Ro)

Each spheromak constantly tries to maximize its spiral path length Lh Which corresponds to minimizing its total energy.

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 mechanics applies to any system that can form circulating current spheromaks. Examples include nuclear particles, atomic particles, atoms, molecules, solid matter, liquid matter and plasmas. Most real matter consists of a large number of weakly 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.

Free electrons are spheromaks. Chemical binding is by weakly interacting spheromaks. Atomic nuclei are composed of strongly interacting spheromaks.
 

HISTORY
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, Fp is the emitted radiation frequency and h is a natural physical constant.

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
where:
h = an apparent physical constant
and
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. This representation fails for single electrons.

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. Each physical system is characterized by a set of discrete quantum states. Each energy Eigenvalue can randomly adopt one of multiple quantum states.

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.

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
or
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.

Hence:
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
where:
Fx / C = F V / C^2
or
Fx = F (V / C)

(h F)^2 - (h Fo)^2 = C^2 P^2
or
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
whereas:
Fh = C / Lh
where:
Ro = nominal electron spheromak radius
and
(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 energy component is many orders of magnitude smaller than the electromagnetic energy component.
 

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

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

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

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

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

PHILOSOPHY OF QUANTUM MECHANICS
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 achieve stability by forming a quasi-toroidal shaped structure known as a spheromak which when isolated in free space has a characteristic geometry and a radius dependent amount of contained field energy. Each spheromak has a characteristic frequency Fh related to its radius and contained energy. 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 F from Fa to Fb 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 = Fb - Fa.

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 filament path poloidal turns Np and by its number of closed filament path quasi-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 F. 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 can 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)
where:
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
or
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 quantum states (possible real solutions) at each energy level.

Atomic charged particles superficially exhibit stationary periodic wave like qualities. However, the existence of multiple discrete 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 his 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 / dF)
where dE is the change in potential energy of the particle that emits or absorbs radiation and dFAt the = 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 elecrtomagnetic 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.
 

ORIGIN OF QUANTIZATION OF ELECTROMAGNETIC RADIATION:
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.
7. Np and Nt are whole integers. A small change in energy is likely associated with a matching incrementation-decrementation in Np and Nt. A larger change in energy may be caused by a jump change in system prime number P.
8.Np and Nt are typically related by the formula:
P = Np + 2 Nt.
9. For a spheromak at its most stable operating point:
Np = 2 Nt + 1.
At the point of stabiity he net result is:
P = 4 Nt + 1
Hence the potential operating point of a spheromak is set by a certain prime number P.
 

CHARGED PARTICLE ENERGY STATES:
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:
Up + Ue
where:
Ro indicates spheromak linear 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.
Upor = magnetic field energy density at the center of the spheromak

4. Inside the spheromak wall the toroidal magnetic field energy density is of the form:
U = Uto (Ro / R)^2
where:
Uto = Ut|(R = Ro)

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

6. For R >> 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 spheromak's net charge Qs;

13. In an atomic particle the circulating net charge Qs 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 approximate equation:
[Lh / 2 Pi]^2 = = {[(Rs - Rc) / 2]^2 [1 / 2] + (Hm^2 / 2)} Nt^2 + {[(Rs - Rc) / 2]^2][1 / 2] + [(Rs + Rc) / 2]^2} Np^2
where Np and Nt are positive integers Rs is the spheromak outer diameter, Rc = spheromak inner diameter and Hm is the spheromak half height.

16. Spheromak solutions are further governed by Np and Nt being integers with no common factor. In combination with prime number theory this constraint requires that:
P = Np + 2 Nt
where P is a prime number.

17. Quantum sysems are further governed by the requirement that quantum systems are probablistic and that the greatest probability occurs when:
d[(Lh / 2 Pi Ro)^2] / dNt = 0

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

19. Maxwell's equations set the magnetic field at the center of a round current ring of radius Ro having Np turns as Bpor = Muo Np I / 2 Ro.

Maxwells equations set the linear magnetic field Bto in a long straight solenoid as:
Bto = Muo Nt I / (2 Pi Ro)

Hence:
Bpor / Bto = Pi Np / Nt = 3.14159265 (Np / Nt)

20. However, in a spheromak:
Np = 2 Nt + 1

21. A spheromak relies on the relationship:
Bpor / Bto = 2 Pi

22. A spheromak achieves this relationship by setting Np = 2 Nt + 1

23. Then: Bpor / Bto = [2 Pi + Pi / Nt]

25. Recall that prime number theory caused:
P = Np + 2 Nt.
At Np = Nt + 1
the choice of prime number P values is somewhat restricted by the requirement that:
P = 4 Nt + 1

26.______

27. 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

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

NMR RADIATION EMISSION-ABSORPTION:
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
and
Fp = change in particle electromagnetic characteristic frequency
= emitted or absorbed radiation frequency
 

RADIATION AND THE UNIVERSE:
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 average absorbed thermal power from solar radiation equals the avereage emitted thermal infrared radiation power.
 

ATOMIC SPECTROSCOPY AND CHEMISTRY:
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:
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 August 23, 2024.