Green Physics Magic

Green Physics Magic!

June 5, 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
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 Schrodinger equation is time-dependent telling about the location of the particle on the axes, being represented by the time-dependent 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 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|.  z= the complex conjugate= the modulus= probability amplitude that one will have to square for the wave equation.  |Z²|= probability, and if you compute 1 squared = 100% that position there for the electron for the axes.

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.  

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 are P₁ =|φ₁|² and for the second slit 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.

A wave mathematically shown is 1) Ae^ix + 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 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 e^kt, or, if you want to e^ix 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 to another (3x+4) for curves or lines that can take the form on the axes as snakes.  Also, one can graph arcs, ellipses, and Bessel Functions or the like which can form, circles of equations (x-h) ² + (y-k) ²= r²and their circumferences, 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.  The 3-axes when you find the volume for these with the three dimensions for, say, a circle will show the limits of what humans can perceive in the natural world with our eyes, and maybe we can imagine more a priori though maybe not a posteriori from mathematical proofs.

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 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!!!  Bye!