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Harmonic Oscillator

Harmonic Oscillator - Physics Forum

Harmonic Oscillator - Physics Forum. Discuss and ask physics questions, kinematics and other physics problems.


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Old 07-15-2005, 08:43 PM
Ron Poteet
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Default Harmonic Oscillator



I would like to present the example of the Harmonic Oscillator as a means to
demonstrate the importance of matter waves. This is a continuation of MEaT,
the theory of Matter, Energy and Time. In this theory matter has two states,
particles which are stationary and exist in real time, and matter waves
which travel at the speed of light and exist in imaginary time. The
multidimensionality of time is necessary for the conservation of Matter,
Energy and Time. Time dilation of the particle state is the natural
evolution of multidimensionality of time. The motion of matter occurs only
as waves of matter in imaginary time.

An object of mass travels at an apparent speed v by having a percentage of
its mass in wave state. The normalized velocity is given by b = v/c, and the
amount of mass in wave state is given by the relation m = M*b, where M* is
the rest mass. The amount of mass remaining in the particle state is M*(1 -
b^2)^.5. This relation for the particle mass is a result of the
multidimensionality of time. An expression of this mass as a complex number
is

M = M*{(1-b^2)^.5 - ib}. The -i in this equation implies that the wave state
lags the particle state by a phase of pi/2 of its conversion cycle in time,
and that the wave state exists in imaginary time. Time has two factors, rate
and duration. Time dilation implies that it is the rate of time which is
conserved, not its duration: w = w*{(1-b^2)^.5 - ib}, where w is defined by
the relation

w*h-bar = M*c^2, where h-bar is Plank's constant divided by 2pi. It is seen
that M^2 and w^2 are constants of the motion. When a particle is converted
to a wave its mass is transferred to imaginary time, and its rate of time is
transferred with it. The rate of time for the remaining particles is thereby
reduced. This is the cause for time dilation, and the reduced mass in the
particle state for a body in motion. I would also like to point out that the
term b is also b = n/N, where n is the number of particles which have been
converted to waves, and where N is the number of particles for the rest
mass. It can be seen that b = v/c represents a time average for the apparent
speed of the body, while b = n/N is an ensemble average. MEaT is the only
representation for Matter, Energy and Time that allows for the proof that
time averages are equal to ensemble averages; until now this equality has
only been assumed, and never proved. We are now at a point where an analysis
of the Harmonic Oscillator can be understood.

The Hamiltonian for the Harmonic Oscillator is given by

H = M*((1-b^2)^.5)(w^2)(x^2)/2 + (p^2)/2M*

where the mass term reflects the stationary particle state and the momentum
refers to the wave momentum, p=mc, where m is the mass of the wave state. It
is instructive at this point to factor the Hamiltonian into Creation and
Annihilation operators as follows:

H = {((M*((1-b^2)^.5)/2)^.5)wX + iP/(2M*)^.5}{((M*((1-b^2)^.5)/2)^.5)wX -
iP/(2M*)^.5}

+ i{X,P}(w/2)((1-b^2)^.25)

H = aa* - (wh-bar/2)((1-b^2)^.25)

where a is the Annihilation operator and a* is the Creation operator. The
Hamiltonian operator and its commutator operate on a wave function on its
right (not shown). The Creation operator operates first, raising the energy
state of the system by one energy level, wh-bar, by causing a particle to
change state into a wave, then the Annihilation operator reduces the energy
state of the system by one level, by causing a wave to change state into a
particle, obtaining the original system energy. The commutator represents
the energy emitted by the wave state when it is converted to a particle; the
term (1-b^2)^.25 represents the time dilation of the energy rate due to the
apparent motion of the body. This commutator has always previously been
considered to be the ground state energy of the system. This is seen to be
incorrect. The ground state energy of the system is the energy of the
particle state, (M*((1-b^2)^.5)c^2)/2. Note that this isn't Einstein's
relation E=M*c^2. Einstein's relation for E is the energy of the wave state
for the body traveling at the speed of light; E=mc^2 is the energy of the
wave state. Also note that the term for the ground state energy is divided
by 2; the other half of the energy for the wave state comes for the
potential energy field. The conservation of the energy of the wave state
comes from the application of the commutator energy being put back into the
system by causing another particle to convert to a wave, etc. This is the
mechanism by which energy is conserved. A change of the energy state of the
system is conducted by the potential energy field. The observant reader
should realize that the motion of the body can be altered by shining photons
with the frequency given by the commutator onto the body ( you might call
this the principle for artificial gravity, or a new principle for mass
engines.) The reader should also note that MEaT provides an explanation for
the quantization of energy (i.e. the energy of the wave states are
quantized.) MEaT describes reality accurately with one theory, instead of
requiring many theories which don't agree with each other.

I would like to include in this paper another example of the wave nature of
matter. It is known that the two states of charge, the stationary particle
state and its wave state, emit fields whose characteristics are particular
to its state of motion; Particles of charge emit E fields and their waves
emit B fields. It is rather naive to think that other types of matter don't
have similar characteristics. Particles of mass attract other particles of
mass, and waves of mass attract other waves of mass; this is called Gravity.
Until now the wave state of matter has been ignored and about 96% of the
gravity within a Galaxy has gone unaccounted for. The part of gravity which
has gone missing is the Lorentz force. It is obvious that the gravity
generated by the Black Hole at the center of a Galaxy emits an axially
symmetric gravitational field; this should lead someone to think that the
Black Hole just might be made of matter waves, and not by a singularity of
particles. The E and B fields of mass are solved by Maxwell's equations with
a negative source of mass for the E field, and a positive source of mass for
the B field. Some questions about Faraday's Law for mass should arise which
can be settled measurements of gravity waves. Do you remember Joe Weber? I
think that if he had used gyros instead of a block mass he might have had
better luck in finding gravity waves. Anyway, there are some nice pictures
from the Hubbell telescope of gravity waves which should be enlightening. To
continue, consider the orbit of stars, planets and other matter about the
center of a galaxy. The motion of these objects imply that they have a
percentage of their matter in wave state. The closer you get to the center
of the galaxy the faster this matter moves (conservation of angular
momentum) and the greater will be the percentage of its mass in wave state.
There will be a radius from the galactic center where all objects will be
completely in wave state. This is the radius where the Black Hole begins. A
Black Hole is made entirely of matter waves which exist in imaginary time. A
Black Hole is black (invisible) because it exists in imaginary time. Dark
matter (matter waves) is dark because it also exists in imaginary time.
Matter waves constitute a vector potential, and the curl of this vector
potential defines a matter (mass) wave B field centered at the galaxy's
center, the center of the Black Hole. You can imagine that this is a B field
of incredible magnitude. Strong enough to influence alignment and motion of
any rotating (spinning) matter within its domain. The Black Hole and all the
spinning stars and planets within the Galaxy act like magnets of matter
(mass) whose structure and order are dictated by the Black Hole at its
center and the motion of the rest of the Galaxy. The Sun and all the planets
of the Solar system, and all other star systems, have their axis of rotation
aligned to the Black Hole, and not to the Sun. For the three planets which
are in question one must consider the possibility that they have rotating
cores, which we know do exist. One can also imagine that the B fields of the
Galaxy and its Black Hole might have an effect on the motion of weather, and
in particular Hurricanes. I would like to note that the Lorentz force does
not tell us anything about the forces between particles and waves, if they
do exist. Also, the mutual attraction of particles is a potential energy
field which affects the transition of particles into waves, which then come
under the influence of other waves.

So, Physics has gone wrong somewhere, sometime, when Physicists decided to
ignore the wave nature of matter. Modern Physics will only begin when the
wave nature of matter waves is finally recognized.

Ron Poteet

7/15/05


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