CONCEPTS CONCERNING PHYSICAL THEORIES
SPECIAL RELATIVITY AND QUANTUM PHYSICS
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CONTENTS
*SPECIAL RELATIVITY AND QUANTUM PHYSICS
*THE NON-RELATIVISTIC SCHROEDINGER EQUATION
*THE Special Relativity Energy Equation
*REST MASS IN THE SPECIAL RELATIVITY ENERGY EQUATION
*The Klein-Gordon Equation
*Phase and Group Velocities and the
Klein-Gordon Equation
*TIME-INDEPENDENT AND LENGTH-INDEPENDENT SCHROEDINGER EQUATIONS
*A SCHROEDINGER-LIKE KLEIN-GORDON EQUATION
*EXTENSION OF THE
*LAGRANGIAN CONCEPTS FOR WAVES
*INCLUSION OF ELECTRICAL CHARGE IN THE LAGRANGIAN
*QUANTUM WAVE EQUATIONS INCORPORATING MASS AND CHARGE POTENTIALS
*THE DOPPLER SHIFT AND SPECIAL RELATIVITY
*MASS AND ENERGY IN SPECIAL RELATIVITY AND A GUDERMANIAN HYPERBOLIC VALUE
*POWER OF N SPECIAL RELATIVITY SYSTEMS
*COMPATABILITY BETWEEN HAMILTONIAN RELATIVITY ENERGY AND POWER OF N SPECIAL
RELATIVITY EQUATIONS
*RELATIONSHIPS BETWEEN THE LORENTZ AND THE EINSTEIN TRANSFORMATION EQUATIONS
*RELATIONSHIPS BETWEEN DIFFERENT POWERS OF N
*PHOTON PARTICLE RELATIVITY
*DIMENSIONAL REPRESENTATION OF SPACE
*GAS LAWS AND ENTROPY CONCEPTS KINETIC
THEORY AND THE
*CONSIDERATION OF THERMODYNAMIC PROCESSES
*QUANTUM DISTRIBUTION FUNCTIONS FOR AN IDEAL GAS
*MASS, ANTI-MASS AND ELECTRICAL CHARGES
*THE SPACE-TIME METRIC
*PAIR PRODUCTION AND THE DEFORMABLE METRIC OF SPACE-TIME
SPECIAL RELATIVITY AND QUANTUM
PHYSICS
Max Planck in the year 1900 introduced the concept of quantum photon radiation and Albert Einstein in 1905 published his Special Theory of Relativity. Planck’s quantum frequency supports the fact that mass and electromagnet energy are equivalent. The Lorentz-Einstein transformation equations, based on the constancy of the velocity of light, established new length and time units that are applicable to relatively moving inertial systems. Heisenberg’s quantum matrix mechanics and Schroedinger’s quantum wave equation, along with Max Born’s probability interpretation opened a new deeper insight into the physical world.
The
Klein-Gordon and Schroedinger wave equations and the approach of Dirac are
related, but have not been fully coordinated. Michelson and Morley
established that there was no “ether wind” that influences the velocity of
light prior to the introduction of the special theory of relativity of
Einstein. They did not rule out the existence of an “ether” having
properties other than those that were proposed art the time. The metric
of the space-time continuum of Einstein’s General Theory of Relativity, which
contemplated space being filled with a sort deformable matter, could be a
viable concept even if the general theory is not accepted. It has been
speculated that gravitons are the field particles of the gravitational field,
but they have not been detected. Interaction between mass and anti-mass
is not well understood, nor is it certain why the universe requires
anti-mass.
This document is an attempt to
reconcile some of these seemingly disparate concepts in the hope that this will
lead to a better understanding of the universe.
THE
NON-RELATIVISTIC SCHROEDINGER EQUATION
Maurice de Broglie, in 1924, made a suggestion that led to the modern quantum
theory. He proposed that matter could produce waves and that they would
be governed by the equations 
and
. Erwin Schroedinger, in 1925, formulated a
non-relativistic quantum wave equation that was derived to account for de
Broglie matter waves. A de Broglie matter wave was considered to be a group
wave where the kinetic energy was 
. The Schroedinger equation, including potential energy 
, expressed in one space dimension is: 
. This equation was derived from: 
where 
, which may be based on a
velocity less than the velocity of light. A solution of the above Schroedinger equation, where the
potential energy, , is external potential
energy, can be obtained from the wave function: 
Prior to the development of the Schroedinger equation, Heisenberg, Born
and Jordon described the motion of quantum particles with matrix mechanics that
were subsequently discovered to provide the same results. Because of the
j factor, Max Born suggested that when this wave function and its complex
conjugate wave function, 
, are multiplied together they satisfy the probability
density 
for locating particles where 
, and 
.
If internal potential energy is substituted for external potential very different
consequences result even though they may be presented in similar
equations. The following four equations that relate to the Schroedinger
are also applicable to the wave function 
that is expressed above.
Einstein proposed that the intensity of electromagnetic radiation,
, was
proportional to 
the average value over one cycle of the square
of the electric field strength of the wave. Einstein provided an
explanation of how light may be interpreted as being both a particle and a wave
if
, where
is the average number of
photons per unit time crossing a unit area perpendicular to the direction of
propagation. In essence, the Einstein concept of the density of photons
is equivalent to the probability per unit length in one dimension of finding a
photon. Max Born applied this basic concept to explain that a quantum
“matter” wave could be explained as a probability wave. The
quantum-related probability wave could then be considered to represent a
dimension-less “probability field” equation that identifies the probability of
finding matter in a unit volume of space.
If the terms of the Schroedinger equation are multiplied by
the equation transforms to:
.
Then because
, and 
, this leads to: 
and 
and 
and 
.the potential energy, 
, is then valid when 
.
The assumption made by Schroedinger, namely that the kinetic energy of a
body and an external potential energy of an arbitrary value could be combined
to describe waves, leads to the probability wave interpretation of Born.
Moreover, the potential energy was even assumed to be a function of time and
distance in the presence of a force. Kinetic energy of wave/particle
entities in special relativity is a relative concept, but if the total energy
of an entity is constant the internal potential energy of the object should
also be a relative concept that varies inversely with the kinetic energy as a
function of a constant relative velocity. Therefore, the potential energy
of a real wave-producing entity in a conservative system must be internal to
the entity. The potential energy in the Schroedinger equation
is an external, or force-producing, potential energy for which the
probabilistic interpretation is appropriate. This is not consistent with
the relativistic traveling wave energy equation for conservative “real” waves,
where all of the energy, including the internal relativistic potential energy,
is linked to wave/particle duality.
The Schroedinger equation incorporates the external force-derived potential
energy that defines potential as the work done on a un it mass or charge to
bring it from an infinite distance to appoint in question at which a mass or
charge is situated. Because the definition itself involves infinity this
potential has an inherent indefiniteness, and it is commonly employed only
between two potential levels. For certain applications this type of potential
energy has great value. For example, understanding the conduction of
electrons in the approximate square well potential of metals that is created by
closely spaced positive ions in the metal is made possible by employing this
external potential energy.
The term
is recognized as
appearing in the term
in the original Schroedinger time-dependent equation,
which includes the external potential energy. According to Heisenberg the
term
is representative of the uncertainty that exists when
an attempt is made to measure momentum and position at the same time which is
independent of the measuring instruments that are employed. A similar
procedure using the following operators indicates a similar uncertainty in
measuring total energy and time of measurement.
The external potential energy operator equation is:
The internal
potential energy operator equation is:
The internal potential energy equation does not require an imaginary
term. The external potential energy may have an arbitrary value that is
not dependent on the kinetic energy and the total energy. Although
developed in energy terms, probability waves rather than real waves fit this
format. The
and
terms are not equal to these terms in the internal
potential energy equation which represents an inherent motion-derived potential
energy rather than a force-derived potential energy. In the internal
potential energy equation the potential energy must be equal to, or greater
than, the kinetic energy and consistent with a constant velocity body.
The presence of external potential energy would result in acceleration or
deceleration. With internal potential energycan be defined completely in mass terms: 
, where
represents a definite amount of energy that is
divided between the kinetic energy and the internal potential energy.
In addition to the potential energy issue, 
in the
Schroedinger equation represents the kinetic velocity of a single body, whereas
when the potential energy is
it represents the group velocity of a group of
entities in special relativity. Moreover, if a group velocity can not be
identified no relationship can be established between the group velocity and
the phase velocity of a wave. The existence of a non-dispersive wave can
not be guaranteed without identification of a group wave velocity, even though
a “phase wave velocity” might be assigned from the wave functions employed to
solve the Schroedinger equation.
If a single, low velocity mass body can produce a “group” velocity, then
, where
is much smaller than
. Because the group velocity can be assigned to
the single body, however, this implies that the group velocity of the wave
carries energy. The energy in the de Broglie wave segment for a single
body, therefore, does not appear to be a “construct” for multiple field
particles moving with the body, in a manner analogous to the Einstein photon
concept for electromagnetic waves. The terms
and
may be employed for calculations in classical
equations, but they do not have the same meanings that and 
have in special relativity equations. When
these classical terms are used, it amounts to ignoring that the total energy of
an entity is .
THE SPECIAL RELATIVITY ENERGY EQUATION
The Dirac special relativity energy equation for relativistic mass in the
absence of an external field is: 
This relativistic energy equation was developed from
different considerations than the Lorentz-Einstein transformation equations,
and requires that both the group and phase velocities must be considered when
dealing with photon waves. The velocity in the
term is the kinetic velocity of a quantity of
mass. If the relativistic energy of a mass is equal to, the rest mass, 
, is equal to zero and the total energy is equal to
. In the Einstein special relativity transformation equations
a zero kinetic velocity occurs when two inertial planes are at the same
velocities. In the Dirac equation they have equal masses at rest.
A relativistic mass that comprises a group of photons traveling at a velocity be assigned to both the particle and the wave as the
group velocity of the particle/wave combination. In “free space” photons
travel at a group velocity of
, and the phase velocity, 
, is equal to the group velocity for non-dispersive
waves. The concept of group velocity enters into wave equations when
there are two or more frequencies and there is modulation. Group velocity
is useful only in those circumstances where the frequencies are close enough
together in magnitude so that 
is approximately equal to
. When the group and phase velocities of
electromagnetic waves are equal the rest mass is zero. If the energy
equation is applied to the particle/wave of a photon the following identity
applies when the wave has a group velocity: 
. Classical kinetic
energy for this light wave can be equated to special relativity terms by
rearranging this equation and dividing by 2
. Consequently the
internal relativistic particle/wave potential energy, , as long as the group velocity
is maintained, is: 
,
or expressed in different terms: 
and 
The relativistic particle/wave kinetic energy, 
, then is: 
, and 
. Therefore 
, and 
, where
is the special relativity
Lagrangian for the particle/wave when a group velocity exists and energy is
conserved.
For a mass body we find that at 
we have
and 
so that 
At we have
and 
so that
, where kinetic energy equals potential. A
particle/wave at rest may be considered in special relativity to posses an internal
potential energy equal to.
In the book “QUANTUM PHYSICS, OF Atoms, Molecules, Solids, Nuclei, and
Particles, Robert Eisberg and Robeert Resnick, John Wiley & Sons 1974, page
71, the authors discuss Einstein’s interpretation of radiation intensity.
They state that there are only photons, and that electromagnetic waves
themselves have no energy since they are only a “construct” that determines the
average number of photons per unit volume when intensity is measured. A
group of photons then would supply the total energy of a traveling
electromagnetic wave. In special relativity a mass at rest may be assumed
to have a total energy that is equal to the total energy of a group of photons
that move at the velocity of light. The mass at rest will contain an internal
potential equal to the total energy of the equivalent traveling particle/wave
when it is moving at. This relationship is expressed in the Dirac special
relativity energy equation.
The total energy of a photon group that constitutes an electromagnetic wave is
. The assumption is that all of the “rest energy” of
the group can be converted in to radiation energy. A de Broglie matter
wave for sizeable bodies would travel at the velocity of the body, which would
be much less than the speed of light. This wave would be produced by the moving
body and would have a velocity of
. If a group velocity were provided for this slower
wave the equations that follow would have the same format as the special
relativity equations for light: 
where 
and
, and is less than. This results in a relativistic equation
commensurate with the velocity of the body: 
The available recovery
of the kinetic energy component,
, and the internal potential energy
component, 
, are determined by: 
. This available
energy in the de Broglie wave, however, is limited to the group wave energy of
the single body that supports the wave that accompanies the body, and it is not
equal to .
The following
related relativistic particle/wave equations can also be formed: 
where: 
, and
, and because momentum is: 
, that it is seen: 
.
The
following equation would also apply to this wave: 
. Examining these
equations it is seen that: 
, if
and
. If the kinetic group velocity could exceed the phase velocity would be imaginary.
REST
MASS IN THE SPECIAL RELATIVITY ENERGY EQUATION
The special relativity energy equation for light, (electromagnetic), particle/waves
may be rewritten as follows:
. The wave factors 
and