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QUANTUM MECHANICS

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

INTRODUCTION:
This web page summarizes the major issues of quantum mechanics that are relevant to this web site.
 

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 the emitted photon energy and Fp is the emitted photon frequency.
Today h is known as the Planck constant, where:
h ~ 6.62606957 X 10^-34 J-s.

In 1905 Albert Einstein explained the photoelectric effect by assuming that the energy Ep carried by a photon followed the equation:
Ep = h Fp
where:
h = physical constant
and
Fp = photon frequency.

About 20 years later deBroglie used the average behavior of a beam of electrons passing through two nearby 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. This so called wave-particle duality is one of the least intuatively understood aspects of quantum mechanics. This authore rationalizes the experimentally observed behaviour by saying that at the quantum level there are multiple real solutions so what we experimentally observe is a weighted superposition of these solutions. This weighted superposition is valid for many particles but not for a single particle which will follow a particular discrete solution which is only one member of the family of real solutions.

Generally the different real solutions are separated by energy incrments:
dE = h Fp
where Fp is the frequency of a photon with energy dE.

Particle Parameter Definitions:
P = particle linear momentum;
M = particle mass;
V = particle velocity
C = speed of light;
E = total particle energy;
h = Planck Constant;
F = particle characteristic frequency.

Lamda = wavelength of electron stream wave;
Lamdax = apparent wavelength of electron in the dual slit experiment
Fo = rest mass characteristic frequency
Lamdao = electron beam wavelength as the beam velocity approaches zero.
 

Then:
Lamda = C / F
and
Lamdao = C / Fo
and
Lamdax = C / Fx

Assume the basic hypothesis of quantum mechanics:
E = h F

In this relationship:
E is the total particle energy and:
F = C / Lamda.

Assume special relativity:
E^2 = P^2 C^2 + Mo^2 C^4
= P^2 C^2 + Eo^2

Hence:
(h F)^2 - (h Fo)^2 = C^2 P^2

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)

and

(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

Hence:
F^2 (1 - (V / C)^2) = Fo^2

Hence a particle is characterized by two frequencies, Fo associated with its rest mass and Fx associated with its linear momentum.

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

The origin of h is:
h = (dE / dF)
where:
dE = 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 = dE / dF

In order to determine h one can do a highly accurate verion of the deBroglie dual slit experiment or one can use a different electro-magnetic methodology, as is done on this web site. In order to implement the electro-magnetic methodology we implicitly assume that h is the same for all particle structures. The structure that we analyze in detail is a spheromak.

Thus:
E = (Eelectromagnetic) + (Enon-electromagnetic)

Hence a change in particle energy dE is:
dE = d(Eelectromagnetic) + d(Enon-electromagnetic)
= (he dFh + hn dFn)
where:
he = dE / dFh
and
hn = dE / dFn

If we change total particle energy E by application of an external magnetic field Bx and if:
d(Enon-electromagnetic) / dBx = 0
then:
dE / dBx = (dEelectromagnetic) / dBx

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

We then assume that he = hn = h

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

Different techniques of measuring h give slightly different values due to various error sources. On this web site we will demonstrate determination of h both from theoretical electrodynamic analysis and by experimental measurement. However, both of these determinations of h are dependent on the assumption that:
d(Enon-electromagnetic) / dBx = 0.
Comparison with experimental data shows that this assumption is incorrect at an error of a few parts per million.
 

PHILOSOPHY OF QUANTUM MECHANICS
For reasons unknown the net charges of atomic particles are quantized in exact integer multiples of 1.60217657 X 10^-19 coulombs. A charged particle can form a stable toroidal shaped structure known as a spheromak which when isolated in free space holds a specific amount of field energy. Each electromagnetic spheromak has a characteristic frequency Fh related to internal charge circulation. When a spheromak is in an external magnetic field the total spheromak field energy depends on the orientation of the spheromak symmetry axis with respect to the external magnetic field axis.

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 inner radius Rc, its outer radius Rs, 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 Rc and Rs 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.

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 = 6.62606957 X 10-34 m^2 kg / s

Quantum mechanical uncertainty also introduces uncertainty into projections regarding both the past and 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 do 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 mankind has control over his own future.

An important quantum mechanical issue in modern electronics is that some materials, such as pure silicon, exhibit a free electron energy band gap. In pure silicon with suitable doping there are both low energy free hole states and higher energy free electron states that are separated by an energy band gap containing no available energy states. This band gap enables the formation of transistors and hence bistable electronic circuits known as flip-flops. Bistable electronic circuits form the basis of modern computers. This band gap also enables formation of solar cells.

There is another energy gap known as the work function between the conduction electrons in a metal and electrons in free space.
 

Different methods of measuring h electronically give slightly different values 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 a photon and dFh = Fp = photon frequency.

Numerous experimentally observed atomic spectra, chemical bonding and electronic phenomena have been successfully explained by assuming that h is a physical constant. The entire branch of physics known as Quantum Mechanics was formulated based on that assumption.

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 he, the electromagnetic formulation 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. However, 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.
 

ORIGIN OF QUANTIZATION OF ELECTROMAGNETIC RADIATION:
1. Assume that our local universe is partially composed of the stable particles known as electrons and protons that have quantized charge;
2. Assume that atoms are in essence aggregations of protons and electrons;
3. During the particle aggregation process electromagnetic radiation is emitted and/or absorbed;
4. Conservation of energy requires that the energy carried by a photon be precisely equal to the change in energy of the particle or system of particles that emits or absorbs the photon;
5. Hence the origin of h as it affects radiation lies in the relationship between energy and frequency in electrons, protons and other charged particles. Interacting particles will randomly attempt to adopt their lowest available energy state and in so doing they will emit photons.
 

CHARGED PARTICLE ENERGY STATES:
The stable energy states of a charged particle spheromak can be found by assuming that:
1. The particle energy partially consists of the electric and magnetic field energy components associated with a spheromak;
2. The field energy density U outside the spheromak wall is of the form:
U = Uo [Ro^2 / (Ro^2 + R^2 + H^2)]^2
where:
Ro indicates spheromak size where Rc < Ro < Rs;
R = radius of a point from the main axis of spheromak symmetry and
H = distance of a point from spheromak's equatorial plane.
Inside the toroidal spheromak wall the energy density is of the form:
U = Uc (Rc / R)^2
It can be shown that:
(Rs Rc) = Ro^2
These expressions for U allow the existence of a stable charged particle in the form of a spheromak with outside spheromak wall radius on the equatorial plane Rs and inside spheromak wall radius on the equatorial plane Rc.
For R >> Ro the expression for U simplifies to classical electrodynamics.
3. The electric and magnetic field energy held by a charged particle spheromak forms part or all of the 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;
4. The observed net particle charge is the difference between quantized amounts of circulating positive charge and negative charge;
5. The charge quantization process is not known to this author;
6. The electric and magnetic field energies integrated out to infinity are constants for an isolated free particle but change as particle fields overlap causing emission of photons;
7. 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;
8. The electric field energy arises from the radial electric field caused by the net distributed charge;
9. In an atomic particle the moving charge has no mass and hence is not subject to inertial forces. Part of the particle energy (and hence rest mass) is contained in the electric and magnetic fields;
10. 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.
 

NMR RADIATION EMISSION-ABSORPTION:
When the above described charged particle stable minimum energy configuration is placed in an external magnetic field the original single energy state takes on a range of values depending on the particle's spheromak axis 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. This effect is known as ESR (electron spin resonance) or 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 certain geometrical mathematical relationships. 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
= photon frequency
 

RADIATION AND THE UNIVERSE:
The local universe is full of radiation photons that result from quantized energy transitions that occur within aggregations of stable charged particles. In a high radiation density environment unexcited charged particles absorb photons 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 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 matter and surrounding space thermal photons are constantly being emitted into space and are 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 photons in the 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.
 

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 March 30, 2018.

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