Green Physics Magic

Green Physics Magic!

April 16, 2024

 

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 amazing textbooks and many, many more science books that are read more like novels!

Precalculus by James Stewart
Elements of Statistics by Mode
Calculus by James Stewart
College Physics by Randall Knight
College Physics for Scientists and Engineers by Randall Knight
Differential Equations by William Boyce
Linear Algebra by Gilbert Strang
Thermal Physics by Daniel Schroeder
Lectures on Physics Volumes 1-3 Richard Feynman
Nanostructures and Nanotechnology by Douglas Natelson

The following mathematical methods described in those textbooks can be simplified into a The Little Mermaid Arc:

We find that for linear algebra, solving augmented linear matrices are for systems of linear equations, and the operations are the 1) Delta, 2) Del Operator, and 3) Addition Laws- [+ (is adding)- in other words is adding and subtracting vectors or is solving systems of linear equations through the Gauss-Jordan reduction method or the different methods, [x for vectors (is the Del Operator, or multiplying)- is multiplying vectors through the cross product to get 90-degree, perpendicular angles for solutions to linear algebra equations, like for Maxwell's Equations (which are used to find the x-, y-, and z-axis directions of the electric, magnetic, and propagating direction components of electromagnetic waves- they go at the speed of light, by the way)], [Also, a triangle, or the delta graphic, expresses other vector, or angle quantities, than the cross product's standard 90-degree angle, and it is used in time-dependent (which often involves movement) equations].

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 general and particular solutions of the Schrodinger Equation for finding a "particle-in-a-box" is, similarly, mathematically shown in a wave 1) Ae^ix + Be^-ix; this 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; also, this 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 wave Ψ= Acoskx + Bsinkx, which represents the chances from 0 percent to one hundred percent- the |Ψ|² absolute values that show this percentage will range 0 to 1 --0 to 100 percent-- for detecting a particle at a certain place along the length, like, of a big glass aquarium tank with cool goldfish in it!

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

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).
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.
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 Daphne for my great gift.  My female elemental (like with the druids in fantasy myths) and nymph I love so much and must always in my whole life love her for happiness given.  I dedicate this science abstract to the loving child Andrea!

Chapter 2

Discovery and Summary

All Matrix Overlap Elements involving (i.e. double quantum waves), Wave Packets, and Waves.  When there is a moving wave thing heading forwards meeting another wave heading towards it (like with the Double Quantum well with the interior of the well where the interference is) and with the interference of any waves of wave packets traveling.  All answers will be scalars for MOE’s and similar wave-similar elements, following the Ψ polarization-state trends of the <012> +-34 bra-ket and kroenecker delta δ=0 (for rest) δ=1 (for the kroenecker delta is for a definite state only for a unique value, or constant; the other states will all have δ=0).

All Scalar (non-bold lettering k, for example) or Vector (a bold K).  Answers from Sine and Cosine waves is, in this instance, k-space of the form e^kt, or, if you want e^ix, e^-ix, you will be able to employ a complex vector space, which is regular space or k-space with all negative y-values as negative i’s, and, since the square complex conjugate of an imaginary number is positive, all the normalizable |Ψ|² states will come out correct.

In short the female protectors curve is Stewart textbook, Randall Knight College Physics.  Here I first learned of the properties or coolness of derivatives in velocities of, say, cars and airplanes in College Physics.  A velocity quantity is the first derivate of the equation or function (f(x).  The acceleration is the second derivative.  Also, I learned about integrals in Calculus by Stewart and how you can orangeshade this is a new verb- orangeshade or purpleshade or aquamarineshade or redshade to describe various colors you can employ for shading the trapezoidal area under the graph of the integral to give the value of the area and hence the scalar answer for the integral, whether the form of the area is a definite or an indefinite (what terms!- confusing) integral.

Also, I learned about integrals in Calculus by Stewart and how you can orangeshade this is a new verb- orangeshade or purpleshade or aquamarineshade or redshade to describe various colors you can employ for shading the trapezoidal area under the graph of the integral to give the value of the area of the trapezoid and hence the scalar answer for the integral, whether the form of the area is a definite or indefinite (what terms!- confusing) integral.  Fourier transforms can be used for infinite-space- another idea like k-space but different.

Solutions of Differential Equations can be any general and all elemental differential equations that, when added, will form the general equations in form.  A differential equation can have as many powers of the y main variable as possible, like the y-value can even be y’’’’’’’’’’, so you could have y’’’’’’ + 8y’’’’’ + 29y’’’’’ + 3y’’ = 33- this is the main equation; you solve for y, and any combination of differential equations will also add up to the first general equation you have to solve to find all the graphing values in our case, or regime, of math.  Y⁶  + 8y⁵ + 29y⁵

Sigma ⁱ_i 3X+4, --à extending the summation idea, you could remember trapezoidal areas of and under the integral and take the Sigma as a paradigm even of limits as that can be extended to be transforms that are almost exactly the true value of the integrated area.  Transforms are like the Fourier (0 [I made this up for a vertical z-axis line of any length for circular [also my transform has three levels of freedom for diffraction of light through a cardboard hole, where light looks the best]), the ½ Fourier series, and the 1 Fourier transform).  Maxwell’s Equations for a forces population density for lim-à infinity of the curve-integral x-axis is the asymptote, asymptotal for x-, y-, z- axes values as x approaches infinity for a curves.

For the Kids: Essay About Physics to Highly Simplify Mathematical Equations

 

A simple curve can represent the equations’ ideas from some branches of mathematics and science:

 

Algebra I

Algebra II

Pre-Calculus

Statistics

Calculus

Linear Algebra

Quantum Physics

Nanotechnology

Linear Algebra Can Be Represented by a Mom’s Curve:

 

Augmented Linear Matrices- For Systems of Linear Equations (From Linear Algebra)

Operations (Delta, Del, and Commutative Laws)

Also, Remove the Confusing Terms from Linear Algebra (Rank, Dimension, Basis)- (Continued…)

And the Banal Ones from Statistics (Mean, Median, and Mode)

Bloch Equations for The Periodic Arrangement of Atoms in Crystals From Nanotechnology (From Nanotechnology)

Schrodinger Differential Equation for a Particle-in-a-Box (From Quantum Physics)

Time-Dependent Hamiltonians (Introduced in Quantum Physics)

Nonlinear Waves (Wave Packets)

All Matrix Overlap Elements and for Double and Triple And More Quantum Wells and Similar Interferences for Waves

All Scalars and Vectors  Answers from Sine and Cosine Waves of the form e to the kt,  or e to the ix and e to the �"ix (For Complex Vector Space) From the Aforementioned  Concepts

The Mom’s Curve is Like an Isotherm in the Calculus Textbook

Dear Reader, I had the curve drawn, but it did not appear- just visualize a simple curve for this drawing!  And I dedicate this physics abstract to Andrea, my visitor from black and very deep outer space, thanking her most heartily!