Green Physics MagicA Story by Jonathan Failla After all this time: I just finished my abstract! Green Physics Magic! October 23, 2025 Summary and Discovery An upward arc, a simple curve can
show the basic equations' ideas from some branches of mathematics and the
sciences: For my discovery I read these textbooks: Precalculus by James Stewart Elements of Statistics by Mode Calculus by James Stewart
College Physics by Randall Knight Nanostructures and Nanotechnology by Douglas Natelson The following mathematical methods
described in those textbooks can be simplified into a The Little
Mermaid Arc: Also, the confusing terms from
linear algebra are rank, dimension, basis, span, linear independence and
dependence, and homogeneous and nonhomogeneous equations and from statistics
are the boring three m's in mean, median, and mode. The solutions to the linear
equations from our first paragraph can be described in form as y=mx+b, for
which the general form is in an augmented linear matrix is |33 |= |3,4,5 |
|x,y,z|. Also, in nanotechnology, the
so-called Bloch Equations portray both the periodic arrangement of atoms and
the electron potentials (an electron potential is the energy in electron volts
(eV) of the electron in the valence energy orbital, and the potential dies off
or increases predictably in a linear fashion with increasing space from the
original, starting energy state of the atom of the electrons in the crystals-
like diamond, citrine, or calcite). One can visualize a printed Bloch wave
above a crystal whose atoms are formed like this cool, repeating, periodic
pattern of, say, citrine The Schrodinger equation is dependent on time (but can also be represented as time-independent to find the location at any specific time) telling about the location of the particle on the axes, being
represented by the Hamiltonian H21 and H12. The general and particular solutions of the Schrodinger
Equation for finding a "particle-in-a-box" or at a specific location
in space for the one-particle system of the hydrogen atom, has to do with the
probability density of the Psi2 (the wave equation), which is a
complex number whose modulus squared gives the probability density at a
particular location. It refers to an electron cloud orbitals about the
nucleus 1s. Psi describes the amplitude of an electron wave as it travels
about the nucleus in 3 dimensions. The probability amplitude, however,
has no physical significance. Also, for higher energy orbitals, the
probability of finding an electron in the energy orbitals does not fall off
linearly as the distance from the nucleus increases, but there are local minima
or nodes for the higher levels and more than one local maxima! One can
find the graphs for these higher electron energy orbitals' shells on the
Internet with a search of the higher energy orbitals of an atom for the
electrons encircling. Also, the Heisenberg Uncertainty Principle states
that the electron's position is unknown. The wave equation is a second-order
differential equation, and it is self-explanatory that when Psi squared is low
or high the probability will also be low or high that a particle will be
discovered. The wave equation is normalized, so all the values along the axis where the height is gotten from will show that all the heights's values added up and squared will all equal the value of 1, which is a constant (1, and the one will equal the 100% probability that one will find the particle in one of the places in the box along the axis where the target is that is the detector to measure where the location of the electron hits). The modulus squared equals the wave equation. Modulus
|z|= the square root of a squared + b squared. The modulus is also called
the absolute value of the complex number, and the modulus squared |Z2|=|Ψ2 |=|φ2|. The modulus will phase through all the value on the x-axis for the probabilities of the electron hitting the detector like I mentioned before, and all will be normalizable with all the individual values adding up to 1, or 100%. z=
the complex conjugate= the modulus= probability amplitude that one will have to
square for the wave equation. The Schrodinger equation finds the probability of a particles being in a particular space within a volume, when one expands the talk of one-dimension-type x-axis values from the experimental detector measurements along a length of the wall to three dimensions. Now for the probability
with φ being the probability amplitude of water waves or electrons
coming at a wall with two holes, the probability amplitudes of both the waters'
intensity and the electrons hitting along the wall's edge that contains the
measured far-detector points are P = |φ1+ φ2|2. After many electrons resembles the interference pattern of light. There is the square of the absolute
value of a complex number Phi for each of the two slits the wave or the
electrons from the electron gun goes through. Each individual slit alone
letting the waves or electron with the electron gun graph is similar for
water waves and electrons coming at the two slits. The probability
amplitudes for the first slit alone are P1 =|φ1|2 and
for the second slit alone for the wave/particles P2=|φ2|2 The paradigm for probability and
intensity does not follow the commutative paradigm of adding for when two ocean
waves meet (i.e. when A amplitude of first wave + A the amplitude of the second
wave= 2A), but the energy detector detects the rate at which energy of the
waves and wave/particle duality is carried to the detector. The interference of water waves and electrons is different with water waves exhibiting diffraction and bouncing off obstacles and with changes visible on the surface of the waves in the water medium, while with electron waves they create interference based on a probability situation (the probability amplitudes are measured along the length of the wall that one finds the detector on with the phi wave equation being squared related to the graph of the two-slit experiment and related to the quantum mechanical act of measuring a particle). A wave mathematically shown is 1) Aeix +
Be-ix for a complex vector space, where the y-axis maximum and
minimum values are 1 and -1, which represent the square of -i as 1 with the
square of i as -1. This is the equation is for electrostatic potential
how it drops off from an electron or set of electrons and then picks up with
the adjacent electron of the crystal lattice structure of each different
element, or crystal, as it is coming closer in the frame, or in the periodic
wave Ψ= Acoskx + Bsinkx. Also, the solutions for Ψ- this time for the energy values of the Matrix Overlap Elements, involving multiple quantum wells (and, in extension, for the addition or subtraction [ebb and flow] heights of all like ocean waves that bump and grind into and out of each other- ha!) for time-dependent Hamiltonians of the form CH12 and CH21. For all waves, as you journey across the x-axis the changing, time-dependent radian values are from 0 radians to 2π radians (a conventional way of writing the radians is as four values 0, ½π, π, 3/2π, and 2π) and the heights along the y-axis are 1 to -1 (this along the period, or total length, of the wave), and they will show the different heights of the curves (waves) as you go along the radian k-space values. One gets the y-values of the functions cycling in the periods through inputting the changing SOHCAHTOA Theta-angle values in the triangle for your Wavies, I will name them. Reference: SOH Sine: Opposite/Hypotenuse CAH Cosine: Adjacent/Hypotenuse TOA: Opposite/Adjacent Also, you
find 90-degree angles for the perpendicular natures of Maxwell's Equations for
Electromagnetic Waves (for B [magnetic wave], E [electric wave], and Z
[direction of propagation at the Z-axis])- note all three are at a 90-degree
angle, or are perpendicular to each other. The electric wave is produced by a
potential in a crystal or bulk material, which makes an electric field that
produces a force incident on a test charge (it can physically move a charge,
like the charge moved can be an electron). A magnetic wave is formed from
the Tri-Source (I made this term up!) of own-axis spinning of
the protons in the nucleus and the valence electron spinning around its own
axis as also it goes around this third value contributing to the total
magnetism, the angular acceleration: the electron's fast moving speed in a
circular path from the valence energy orbital and about the nucleus of the
atom. Hamiltonians (H11,
H22, H12, H21 An
upside-down triangle with a small upside-down triangle graphed within it (I
made this up also!)= There is a perpendicular 90-degree-angle value for a
situation that is portrayed by H11 H22, and
the little triangle inside the Del 90-degree triangle is for when you have with
the different angles (i.e. with radians other than 0, π, π/2, 3π/2,
and 2π) for conformations of things other than just with the 2 values. A
2-value conformation would be a cis-trans isomer, a chiral molecule, and a
two-state NH3 Ammonia molecule that has two shapes with
its atomic form that I read about in the discussion of Hamiltonians recently. Also, the commutative-style regime
extends in visualizibility and vulnerability to Polarizing light (with
elliptical polarizibility, circular, and regular polarization [regular
polarized light you have for Stern-Gerlach apparatuses, and you find the final
interference-filled light result in which all the light was changing through
each polarizing filter, or apparatus, from bra-ket notation as the Psi final energy
value with appropriate visual wavelength for the lightbeams; Polarizing is
following the Kronecker delta (the following lower case δ is a Greek letter;
no, not the delta, silly, which is also a Greek letter D but upper case!) δ's
|1||0|and the H22, H11, H21,
and H12 commutative laws, which extend to Matrix Overlap
Elements for multiple quantum wells and to waves (like standing waves for the
differing notes on violin and guitar strings). The general paradigm for derivatives
for the quantity x squared is y'(x)= the limit as h heads towards zero for
f(x+h) 2- f(x) 2/h. Then it proceeds in the
equation's actions to be x2+ 2xh + h2- x2/h.
Then it is boiled down to h(2x+h)/h. Then it is 2x+h and finally 2x,
which is number for the derivative. One takes the midpoint of the secant
line (the limit heading towards 0 of the points about the midpoint of the
secant line) for being between .99 and 1.01. The slope represented by the
variable m of the secant line for x1= .9 and x2=1.1 is y2
- y1/x2 - x1 = 1.12 - .92/1.1-.9
= 1.21-.81/.2=2. This (2x) is the value of the derivative of x2. The slope of the tangent line of x
squared is f'(1) = 2, which is m or the slope of x2 at x=1, which is
the same slope (rise/run) of the secant line on the parabola at x=1 for the
limit at 0, which is portrayed in the definition of the derivative. This
slope, or derivative, at the points I just mentioned is the instantaneous rate
of change of a function for the function's x- and y-values. Also, the probabilities for the
electron being in certainly defined orbitals can be represented by electron
clouds of different colors representing negative and positive signs (diagrams show probability as a density of a cloud, and the brighter areas show a higher probability, similar to the node/antinode paradigm in the standing waves) and can be
visualized internally as standing waves that have an integer number of
wavelengths around the 2 pi circumference of the nucleus (and can be seen as
waves forming around the circle, or the portrayal of the nucleus' orbital, for
graphical representation). I like the three-dimensional graphical
pictures with different colors, and how the nodes are recognized for the s
orbitals. Standing waves are formed by musical
strings and with the De Broglie matter-wave wavelengths can be a good
representation of the wave/particle duality found in all of nature and
shown by the 2-slit experiment for photons and how they can be seen as acting
like waves and particles and same with the larger-wavelength particles that
have more mass and are expressed as energy also like photons via e=mc2
from Albert Einstein's, which is making the same matter and energy
analogously. The De Broglie matter waves are a way of showing an electron
as a wave instead of just as a particle. Diagrams would be useful to
portray these motions. The fact that stringed instruments have standing
waves shows that physics is fun when musical instruments and music take center
stage for the material one reads about and writes of. All Scalar (non-bold lettering k,
for example) or Vector (a bold k)
answers gotten from Sine and Cosine waves is, in the instance, k-space of the
form ekt, or, if you want to eix or e-ix.
One can also play with interference, changing the equations' frequency and
amplitudes and having the equation i2= -1 that is the same
commutative laws with real waves and that follow the same commutative
paradigm. You will always be able to employ a complex vector space, which
is regular space of K-space with all negative values as negative i's, and,
since the complex conjugate of an imaginary is positive, all the |Psi|2
states will come out normalizably correct when measured as the positive 1, or
100 percent chance of locating each particle! In short my female protection curve
is College Physics by Randall Knight, the Calculus textbook
by James Stewart, and Nanotechnology and Nanosystems by
Douglas Natelson. Here I first learned of the properties of derivatives
in velocities of, say, cars and airplanes in College Physics.
A velocity quantity is the first derivative of an equation, or f(x). The
acceleration is the second derivative, which I designate with a V
(karat). Also, I learned about integrals in Calculus by
Stewart and how you can orangeshade, this is a new verb- orangeshade- or
purpleshade or aquamarine or redshade (or a combination of all the colors!) to
describe the various colors you can employ for shading the trapezoidal area
under the graph of the integral on your sheet of paper or your computer and
hance the scalar answer, whether the form is a "definite" or an
"indefinite" along the x-axis (what confusing terms!) integral. Also, for commutative oceanwaves I
do summarize the math for series and say how many of your or my own transforms
or the popular ones like the Fourier can be used for infinite-spaces - another
general idea like k-space but mine for series is different than the given
regular eigenvalues (k-space) x-axis terms as having my own unique terms.
Solutions of differential equations can be the one general and all the
particular differential equations that, when added, will equal the general
equation (the general solution will be in the form y=mx+b, there will be
integer powers of x and constants in front of them that can take the form on
the axes as black or pink snakes)! The derivative of y is always there
for the unchanging differential equations form so that you can solve for
derivative equations, as the name concerning derivatives is differential
equations. A differential equation can have as
many powers of x as possible, and the values of y' can even be dy6/
dx, so you could have 14dy6/dx + 8 dy5/dx + 28 dy3/dx
+ 4dy2/dx=38: this is the main equation; you solve the differential
equations for y, and you can have different graphing values for this regime of
math (to make things more interesting you can put in a polynomial function for
the initial y's for a different level, like 6y6 + 13y5 +
3y5 after you solve for the differential equation to spice things
up). The "particular"
y-equations will be written- with no immediate derivative- as, like I mentioned
before, y= mx+b that will equal constants (like 33) and are related to via a
commutative paradigm of division and multiplication of the constants to a
"general equation" of a form of y=mx+b also (but the y will apply to
the whole equation and will be entered in to all the y's in the differential
equation for solving this differential equation!) Solutions of
differential equations can be any general and all particular differential
equations that, when added under the commutative paradigm, will form the
general equations in form. You can use with summations Sigma
from i to i from one number o another (3x+4) for curves or lines that can take
the form on the axes as snakes. Also, Bissel Functions or the like can
form), circles of equations (x-h) 2 + (y-k) 2= r2and
their circumferences, arcs, or ellipses, and it would be like I mentioned
before the constants with Infinite-Spaces as the limit heads towards infinity
for the sums and transforms lim heading towards infinity for the sigma i to i
as outer space planets (!), and, extending the summation idea, you can start
with the Sigma as the trapezoidal paradigm 3-axis description (as representing
our world's shapes) of a The Little Mermaid arc as a curvy poisonous snake like
the cobra as the limit heads towards infinity. Also you can extend the dimensions
of the graphs of planes and cubes to a transformed extra dimensions for
higher-dimensional study like with gravity studies for braneworlds or the event
horizons of black holes or a search for a hypothetical gravity force carrier
(the graviton) in the lone, serene, peaceful depths of outer space. A
Jonathan Taylor Failla transform on the a-axis would always be some value of an
integrated area for a 3-axis scalar values' height, and one can have all of the
axes' areas and volumes (like of a cube or a circle or a plane or the analog,
like thinking of the ideal gas law) translated and transcribed with an arc on the a-axis with a personal meaning for your equations of the combination of shapes on your own graphs on
graphing papers from the regular 3 dimensions, and one can get creative through the twists of the figures, and a lovely ineffable moment (our time-dependent derivative is
like me going on a fun carousel ride with the girl of my dreams at Goodwin Park
in Hartford, this wondrous "green physics magic" philosophy applies
to derivatives also!). Nonlinear cool stuff I like that is not described as like vacuous are fractals, snowflakes (both of all kinds of shapes), ocean/river vortices (from chaos theory), EPR (Einstein, Podolsky, Rosen) photon nonlocality and photon entanglement of photons for quantum computing (one distant photon knows what the other is doing, which is useful for non-binary <1 <2 quantum bits [binary bits 0,1 are used for regular computers in a Boolean Logic sense of a simple true/false or question and answer query]). Quantum computers are good for cryptographic secret communications between parties in the fiber optic networks, which provides great safety to both financial and personal data nests. Public-key encryption is when there is a public key where one has a message that is to be encrypted and where one has this message unencrypted. Then one changes with this public key the encrypted message into an encrypted form and then sends the message to one's associate, and the person who receives the message will decrypt the message with one's private key. Quantum computers can be used to do encryption using symmetric key encryption, where the parties share a common, secret key for decryption and encryption between the receiver and sender (of the text and the ciphertext) respectively. Quantum computers might try to hack into their message, but it can be safe because the quantum computers rely on photons sent over fiber optic cables that protect the keys (if there is an interceptor, the photons will be affected by the laws of quantum physics and there will be an equivalent of a measurement taken causing the interceptor to be caught). But recent quantum-resistant post-quantum cryptography (QPC) will protect against these quantum computers' threats. Diagram for the movement of charge
density in a changing magnetic field from one location to another location (the
density is made up of upward-arrow North and downward-arrow South charges of an
electron, manifested as spin-up and spin-down particles in space, namely, by my
appellation of Half-Valence Diagram, where with the diagram you see
the V, or population density shift from being more of like the spin down than
the spin up or vice versa. Magnetism has to do with the three
values of spin-orbit coupling [i.e. angular acceleration and spin of an
electron about its own axis] and the charge from the nucleus where are the
neutrons and the protons. Jon can form a colorful crayon arc with curvilinear
coordinates. The terms of spin-orbit coupling should be written as integers for
its values, I think, not as the confusing 1/2, 3/2, 5/2 for its values).
The arc can be described by simple, optimistic-curve, constant integers like 1,
2, 3, 4, and 5... This arc can be like an isotherm, can be used in all of
the mathematical equations in physics, and can be imagined with all colors of
your imagination! I would like to thank my female
guiding light who has protected and watered me to grow as a plant does,
improving my social ability (I love going out to talk), intelligence and
strength (my arm strength and my visual scanning perception in editing my books
are so valuable to me). I thank her for my great gift. My female
elemental (like with the druids in fantasy myths), sprightly nymph I love so
much and must always in my whole life love her for happiness given.
Finally, thank you to my family and friends and kids from the universe!!! This highly exciting physics abstract discovery is for conceptualizing in our mind's eye extra dimensions past the normally perceived three dimensions. I was just cogitating how if there are three equal transforms on the three axes that all head to an equal number on the x-, y-, and z-axes, then you can like I did for the page before shown in the box as an arc introduce and then visualize well thanks to imaginary numbers and their fantasy realm extending the real and reality a fourth dimension, and I visualized and imagined for this a delta operator in a complex conjugate for an additional transform in the 4th dimension (that is not near the speed of light due to Lorentz invariance in complex vector space and shown with different fashionable colors). I included a drawing of a complex conjugate with the two axes displaying real and imaginary numbers (with the y-axis for the imaginary values and the x-axis for the real values). Drawing with a crayon an upward line, arc, but to better portray this is a curve tensors up with its own delta and the upside-down triangle dal curl, as it were, for the Tri-Source values can be for the values of the axes for the electric, magnetic, and direction of propagation following the commutative paradigm, and you can make yours into a colorful spaceship navigation compass with straight and arced lines of different sizes and color for the fourth dimension arc, which then takes on different sizes and colors as I drew for the page before. (In my drawing here, I drew two arcs of different sizes and colors with Green with a 6 length and Blue with an 8 length for sizes of the arcs). The dimensions can also be represented thusly for the dimensions spread out over space. All values are recognized under a general, commutative paradigm for the Tri-Source as limit of X>>> infinity, the limit of y >>> infinity, and the limit of z>>> infinity. The complex conjugate z in complex vector space for perceiving the spaceship's compass gotten from the identical two transforms' arcs, and the X- and y- axes of the complex conjugate are set at 90 degrees to each other. The 3's represent the uniform tensors (Jxy + Jyz + Jxz), and all four tensors are with the arc representing "a"- Jxyz(a). 3-dimensional shown as three cubes from the 3-axis transforms employed to get to the 4th dimension via three 2-d tensors that represent the planes of the page (this fits). Each cube is the answer for 1 transform's area in each dimension. This means one cube left for something new and further from a transform. Three dimensions for real and three for imaginary are represented in each of the first three cubes representing the three dimensions and the 9 and 9 represents 9 dimensions each for the sums. Arc suggests new graphics to represent areas (answers) for transforms and graphics for drawing and hope for more exciting shapes and colors for math and physics appears. I did some extrapolating for the conclusion of the abstract of theoretical physics, as one may see appears. Extrapolating from experience: the 4th dimension arc represents the 4-tensor arc, and, since the tensors are doubled, the dimensional component should be doubled as well. In a plane we found etherial possibilities for further dimensions beyond a plane, which is 2-dimensional. I hope I have not made too many presumptions or extrapolated too much to obtain the harmony numbers, and I think this is my final version of my science abstract with no more corrections. Description of Drawing:
I have one large square with four boxes in it, and three of the boxes have diagonal lines for the x-, y-, and z-axes. Then the fourth box contains an arc. Real Imaginary 3rd Dimension 9 9 4th Dimension 18 18 5th 36 36 6th 72 72 7th 144 144 Most answers add up to 9, which is fascinating. There were 3 real dimensions and 6 imaginary plus real dimensions in each of the three boxes with diagonal (straight lines). Then for the 4th dimension and above I doubled the number of dimensions! There is harmony between counting imaginary numbers for the values of all these numbers adding up to 9.
There was a young innocent man there from Harvard, and he looked at my equations and could not make heads or tails of them. Then a different young man looked at them and said I was a genius. The equations had a lot of cosines or whatever in them, and they were really complicated and were written on a sheet of paper. © 2026 Jonathan Failla |
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Added on December 24, 2022 Last Updated on January 4, 2026 |
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