Hopf Fibration: Mapping $S^3$ to $S^2$

The 3-Sphere ($S^3$)

• $S^3$ is the set of points (x, y, z, w) in $\mathbb{R}^4$ such that $x^2 + y^2 + z^2 + w^2 = 1$.
• $S^3$ can be represented as pairs of complex numbers $(z_1, z_2)$ in $\mathbb{C}^2$ where $\lVert z_1 \rVert^2 + \lVert z_2 \rVert^2 = 1$.
• $S^3$ is visualized as sitting inside $\mathbb{C}^2$, which is isomorphic to $\mathbb{R}^4$.

The Hopf Fibration

• The Hopf Fibration is described as $S^1 \hookrightarrow S^3 \xrightarrow{\pi} S^2$.
• The mapping $\pi : S^3 \rightarrow S^2$ is defined as $(z_1, z_2) \mapsto (2z_1\bar{z_2}, \lVert z_1 \rVert^2 - \lVert z_2 \rVert^2)$.
• This mapping projects pairs of complex numbers in $S^3$ to a point in $\mathbb{C} \times \mathbb{R}$, isomorphic to $\mathbb{R}^3$.
• It can be verified that $\lVert 2z_1\bar{z_2} \rVert^2 + (\lVert z_1 \rVert^2 - \lVert z_2 \rVert^2)^2 = (\lVert z_1 \rVert^2 + \lVert z_2 \rVert^2)^2 = 1$, confirming that $\pi$ maps $S^3$ to $S^2$.

Fibers of π

• The pre-image of a point $p$ under $\pi$, $\pi^{-1}(p)$, is isomorphic to $S^1$ for all $p \in S^2$.
• For $p = (0, 1) \in S^2$, the pre-image is $\pi^{-1}(0, 1) = {(z_1, z_2) \mid 2z_1\bar{z_2} = 0, \lVert z_1 \rVert^2 - \lVert z_2 \rVert^2 = 1 }$.
• This means $z_1 = 0$ and $\lVert z_2 \rVert = 1$, so $\pi^{-1}(0, 1) = {(0, z_2) \mid z_2 \in S^1 } \cong S^1$.
• Given an open set $U \subset S^2$, $\pi^{-1}(U) \cong U \times S^1$.
• $S^3$ can be thought of as a collection of circles, where each circle maps to a point on ( S^2 ), and these circles do not intersect.

Visualizing the Hopf Fibration

• Visualizing the Hopf Fibration in 4D is difficult so a stereographic projection from $S^3$ onto $\mathbb{R}^3$ can be applied.
• This stereographic projection is given by $S^3 \hookrightarrow \mathbb{R}^4 \xrightarrow{\phi} \mathbb{R}^3$, where $(x, y, z, w) \mapsto (\frac{x}{1-w}, \frac{y}{1-w}, \frac{z}{1-w})$.
• The projection $\phi : S^3 \setminus {(0, 0, 0, 1)} \rightarrow \mathbb{R}^3$.
• The inverse map is $\phi^{-1} : \mathbb{R}^3 \rightarrow S^3 \setminus {(0, 0, 0, 1)}$ defined as $(x, y, z) \mapsto (\frac{2x}{1 + x^2 + y^2 + z^2}, \frac{2y}{1 + x^2 + y^2 + z^2}, \frac{2z}{1 + x^2 + y^2 + z^2}, \frac{-1 + x^2 + y^2 + z^2}{1 + x^2 + y^2 + z^2})$.
• The stereographic projection visualizes the Hopf Fibration as:
  • Nested tori family
  • Ccircle along the z-axis
  • Circle in the xy-plane
• Each torus maps to a circle on $S^2$, while the z-axis circle and xy-plane circle map to the north and south poles of $S^2$, respectively.

Partial Differential Equations

Key Concepts

• Partial Differential Equations (PDEs) are categorized based on their order and properties
• Second-Order Linear PDEs have the form $A \frac{\partial^2 u}{\partial x^2} + B \frac{\partial^2 u}{\partial x \partial y} + C \frac{\partial^2 u}{\partial y^2} + D \frac{\partial u}{\partial x} + E \frac{\partial u}{\partial y} + Fu = G$
• For PDEs, the discriminant is defined as $\Delta = B^2 - 4AC$
  • If $\Delta > 0$, the PDE is Hyperbolic
  • If $\Delta = 0$, the PDE is Parabolic
  • If $\Delta < 0$, the PDE is Elliptic

PDE examples

• Wave Equation: Describes wave phenomena, $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$
• Heat Equation: Describes heat distribution, $\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$
• Laplace Equation: Appears in various contexts: $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$

Boundary Conditions

• Dirichlet conditions specify the value of the function $u$ on the boundary.
• Neumann conditions specify the normal derivative of the function, $\frac{\partial u}{\partial n}$, on the boundary.
• Robin conditions specify a linear combination of $u$ and its normal derivative on the boundary.

Methods of Solution

• Separation of Variables involves assuming a solution of the form $u(x, t) = X(x)T(t)$
  • The equation is substituted in the PDE and separating variables to solve the resulting ODEs and applying boundary/initial conditions.
• Fourier Series represent periodic functions as an infinite sum of sines and cosines.
  • A function f(x) can be represented as $f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos\left(\frac{n\pi x}{L}\right) + b_n \sin\left(\frac{n\pi x}{L}\right) \right)$
  • The coefficients $a_n$ are calculated as $a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos\left(\frac{n\pi x}{L}\right) dx$
  • The coefficients $b_n$ are calculated as $b_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) dx$
• Laplace Transforms transform a PDE into an algebraic equation and the solution is obtained by inverting the Laplace transform.
• Numerical Methods approximate solutions using numerical techniques.
  • The Finite Difference Method approximates derivatives using difference quotients.
  • The Finite Element Method divides the domain into smaller elements and approximates the solution within each.

Key Equations and Formulas

• Wave Equation: The Solution (D'Alembert's Formula): $u(x, t) = \frac{1}{2} [f(x + ct) + f(x - ct)] + \frac{1}{2c} \int_{x-ct}^{x+ct} g(s) ds$ where $u(x, 0) = f(x)$ and $\frac{\partial u}{\partial t}(x, 0) = g(x)$.
• Heat Equation: The solution using Separation of Variables is $u(x, t) = \sum_{n=1}^{\infty} B_n \sin\left(\frac{n\pi x}{L}\right) e^{-\alpha (n\pi/L)^2 t}$
• Laplace Equation: Solution in Polar Coordinates $u(r, \theta) = A_0 + \sum_{n=1}^{\infty} r^n (A_n \cos(n\theta) + B_n \sin(n\theta))$

Important Theorems

• Uniqueness Theorem: Under specific boundary and initial conditions, solutions to PDEs are unique.
• Maximum Principle: For elliptic equations, the maximum and minimum values of the solution occur on the boundary.

Applications

• The Heat Transfer: Models temperature distribution
• Wave Propagation: Describes the wave phenomena
• Fluid Dynamics: Is used for analyzing fluid flow
• Electromagnetism is used to solve the electromagnetic problems

Key Notes

• PDEs are fundamental in engineering and science
• Understanding solution methods and classifications is crucial
• Numerical methods can be necessary for complex issues

Applications of Derivatives: Maximums and Minimums

Definitions

• Given a number $c$ in $f$'s domain, $f(c)$ is defined:
  • As an absolute maximum of $f$, if for all $x$ within $f$'s domain, $f(c) \geq f(x)$.
  • As an absolute minimum of $f$, if for all $x$ within $f$'s domain, $f(c) \leq f(x)$.
  • As a local maximum of $f$, if when $x$ is near $c$, $f(c) \geq f(x)$.
  • As a local minimum of $f$, if when $x$ is near $c$, $f(c) \leq f(x)$.
• The Extreme Value Theorem: $f$ reaches an absolute maximum value $f(c)$ and an absolute minimum value $f(d)$ for some number $c$ and $d$ in $[a, b]$, if $f$ is continuous on a closed interval $[a, b]$.
• Fermat's Theorem: If $f'(c)$ exists, and if $f$ has a local maximum/minimum at $c$, then $f'(c) = 0$.
• Critical Number: A critical number for function $f$ is a number $c$ in $f$'s domain where $f'(c) = 0$, or $f'(c)$ is nonexistent.

Steps For Finding Absolute Minimums and Maximums

• How to find a continuous function $f$'s absolute values between a closed interval $[a, b]$:
  • Step 1: Find $f$'s values at its critical numbers in $(a, b)$.
  • Step 2: Find $f$'s values at the closed interval's end points (a and b).
  • Step 3: Absolute maximum value is the largest value from steps 1 and 2. Absolute minimum value is the smallest value from steps 1 and 2.

The First Derivative Test

• Give that $c$ is a critical number for continuous function $f$:
  • $f$' has a positive to negative change at $c$. Then $f$ has a local maximum at $c$.
  • $f$' has a negative to positive change at $c$. Then $f$ has a local minimum at $c$.
  • $f$' doesn't change signs at $c$. Thus, $f$ has no local minimum or maximum at $c$.

The Second Derivative Test

• Assume that $f''$ is continuous near $c$.
  • If $f'(c) = 0$ and $f''(c) > 0$, then $f$ has a local minimum at $c$.
  • If $f'(c) = 0$ and $f''(c) < 0$, then $f$ has a local maximum at $c$.

Concavity

• A $f$'s graph which sits atop all its tangents in a certain interval is referred to as "concave upward" for that interval. A $f$'s graph which sits below all its tangents in a certain interval is referred to as "concave downward" for that interval.

Concavity Test

• $f''(x) > 0$ for all values of $x$ in $I$: The graph of $f$ is concave upward in $I$.
• $f''(x) < 0$ for all values of $x$ in $I$: The graph of $f$ is concave downward in $I$.

Point of Inflection

• A point $P$ on a curve is termed an inflection point wherever concavity is changed.
• Example: To find the absolute extreme values of the equation, $f(x) = x^3 - 3x^2 + 1$ over the interval $[-\frac{1}{2}, 4]$:
  • Numbers Critical: Find equations critical numbers in $(-\frac{1}{2}, 4)$:
    • $f'(x) = 3x^2 - 6x$
    • $3x^2 - 6x = 0$
    • $3x(x - 2) = 0$
    • $x = 0, x = 2$
  • Critical Numbers: find $f$'s values:
    • $f(0) = 1$
    • $f(2) = -3$
  • Interval Points: The values for $f$ at the intervals endpoints:
    • $f(-\frac{1}{2}) = \frac{1}{8}$
    • $f(4) = 17$
  • Conclusion: The absolute maximum value is $17$ and the absolute minimum value is $-3$.

Regulation of Gene Expression

Introduction:

• Gene Expression: The process where genetic information is used to synthesize a functional gene product, often proteins or functional RNA.
• Gene expression is regulated to occur:
  • In the correct cell type
  • At the correct development stage
  • In response to the correct environmental cues.

Levels of Gene Regulation

• Transcriptional Control: Determines the timing and amount of RNA transcription with regulatory proteins like activators and repressors. Other components include enhancers, silencers and chromatin remodeling.
• RNA Processing Control: The regulation of processes such as splicing, capping and polyadenylation within a RNA transcript.
• RNA Transport and Localization Control: Regulates the transportation and localization of RNA molecules from the nucleus to the cytoplasm.
• Translational Control: Determines when and volume of an RNA molecules translation into a protein with factors including ribosomes, initiation factors and RNA-binding proteins.
• Protein Activity Control: Regulates protein function post-production, such as post-translational modifications, degradation and feedback mechanisms.

Importance of Gene Regulation

• Cellular Differentiation: Enables specialization and specific functions in cells.
• Responding to the Environment: Allows organisms to adapt to environmental changes by altering gene expression.
• Development: Coordinate the development and growth of organisms
• Disease: Gene expression dysregulation leads to diseases such as cancer.

Mechanisms Of Regulation

• Regulatory Proteins:
  • Activators: Proteins that increase gene transcription by binding to DNA.
  • Repressors: Proteins that decrease gene transcription by binding to DNA.
  • Transcription Factors: Proteins that regulate gene expression by binding to DNA sequences.
• Enhancers and Silencers:
  • Enhancers: DNA sequences increasing gene transcription.
  • Silencers: DNA sequences decreasing gene transcription.
• Chromatin Remodeling:
  • Histone Modification: Chemical modifications to affect chromatin structure and gene expression- acetylation and DNA methylation.
• Non-coding RNAs:
• MicroRNAs (miRNAs): RNA molecules binding to mRNA to inhibit translation or promote degradation.
• Long Non-coding RNAs (lncRNAs): RNA molecules regulating gene expression.

Gene Regulation Examples

• Lac Operon in E. coli:
  • Genes in lactose metabolism.
  • Is repressed in the absence of lactose.
  • In the presence of lactose, transcription happens.
• Steroid Hormone Receptors:
• Steroid hormones bind to receptor proteins.
• A complex which can enter the nucleus and alters gene expression is formed.

Conclusion

Control of gene expression is pivotal for organisms in a variety of pathways. There are also multiple tiers of control and the importance of understanding these processes.

• Regulation is a pivotal essential process.
• It involves multiple control levels.
• It is crucial to understand development, disease and response to the environment.

Liner Algebra - Definitions

Space vector definition

Vector space is the same $E$ non-set supplied with the following dual law

• An internal composition law; LCI +: $E * E -> E$ such as $(x,y) -> x + y$
• An external composition law; LCE *: $K * E -> E$ such as $(L,x) -> L * x$. When K is the body(R or C)

The properties that the laws must satisfy

• $(E, +)$ is an abelien group, therefore
• Assosiativity: E, (x+y)+z = x+(y+z).
• Neutral element: $0_E$ E, x, x+0 = 0+x = x
• Symetric: Ex, -x in E wherex+(-x) = -x+x = $0_E$
• Commutivity: Ex, x + y = y + x
• Distributity of the ICE relating to the addition of K: L Mu in K, Ex in E, (L+M) * x = L * x + M * x
• Distributity of the ICE relating to the addition of E: L in K, Ex in E, L * (X+Y) = L * x + L * y
• Mixe associativity: L M in K: Ex in E: (LM) * x = L * (M * x)
• Neutral element for the ICE: Ex in E: $1_K*x = x$

Sub-vector space Definition

Call it $E$, the vector space from a body $K$ . A Sub-assembly $F \subseteq E $ is the sub-vectorial space from E if:

• F is not empty.
• F is unstable with the linear combination, like the 3D world
• Ex, y in F. L Mu in K with L* X + M * y in F

Theorem

F is asub-vectorial space from E, only if:

• $0_E$ in F
• Ex, y in F, x+y in F
• EL in K, Ex in F, L * x in F

Linear combination definition

For $x_1 + x_2 + ⋯ + x_n $ the vectors out of vector space E in body K.

A vector's linear combination is expressed as. $L_1{x_1} + L_2{x_2} + ⋯ + L_n{x_n}$, where $L_1 + L_2 + ⋯ L_3$ in K

Generating family definition

A family of vectors $x_i$ of a vectorial space E are said to be generate if the vector E is able to express itself like an ending linear combination of the vector of this family. We remember E= Vect $(X_i)i \in I$

Free family definition

A family of $(x_i + x_2 + ⋯ + x_n)$ of a vectior space E that is said to be free if the linear combination oh these vectors that gives vector zero is that of a value when all coefficents are zero. By other meaning

$L_ix_1 + L_2x_2 + ⋯ + L_nx_n = 0_E -> L_1 = L_2 = ⋯ = L_n = 0 $

Bound family Definition

A vector family is described as bound or linear dependance if its not free. This signification that here is a linear non trivalial cornbination oh these vectors that give the vector zero, meaning:

There is $(L_1 + L_2 + ⋯ + L_n) \neq (0,0,0)$ that $L_ix_1 + L_2x_2 + ⋯ + L_nx_n = 0_E $

Base Definition

The base of vektorial space is what is free and what is generating

Size Definition

The size of a vectorial space E is the vectors on the base side in E. If a vector space ends then its size is ending If a vector space does not end then size is unending

Linear application

Liner application Definition

Leave Ei out Then f: E -> F is said to be liner if:

FX, y in E: f(x+Y)= f(x)+f(y) FL in K, Ex in E: f(lx) = Lf(x) In an equal way, f is liniar if: FL Mu in K: Ex y in E; f(LX MuY)= Lf(x) + Mf(y)

Nucleus Definition

The liner Aplication kernel: f: E -> F which is said as Ker(f). Is the amount of vector E that are send it ower to F’s zero factor: KER F = EXin E, Fx=0 F

Image Definition

A liniar application: f:E -> F. And is referd to as. IS the amount of vector F who are on lesses the result one of vector E: Im f = Y in F: Ex in E, fx =y = FYIX en E

The size theorem

Call $f : E \to F$, a lineal application, when E is a vectior space with ending size. Then: size E = size Ker(F) + Size Im(F) -size IM F is calld the range of F: Known Rgf

Matrix

Matix definition

A Matrix A that is M*N in size. with coefficents in body K is a rectangle that is M rows with N columns that every element is a element from K. we remember like A= (aij) 1<=i<=m, 1<=j<= n where Aij in K’

Matrix operations definition

Addition: If A=Aij and B=Bij are 2 matrixes that have the same size. Then there sum is matrix C = A+B defined by Cij+Aij+Bij for al I J.

Scale multiple If A=Aij is a matrix with size m*N and l in k them the multiple product is a matrix B by B=laij for al I J.

Matricing Muptiple: IA=Aij the size MN matrix and B= the Bjk of size N=P the rproduct C=AB of MP define the cik by

Cik= sum = 1 to n - aij * bjk to al I k

Transpos Definition

The transpos from matrix A to aij is the matrix. A=aji

I other word the riws from Matrix come the Columns and vise verca

Transpos Definition

A square matrix AN of NxN is said to be inversable. If there is matrix B with a size that the product is AB=BA=In is called the matrix identity form size NN. the matrix B is the invers of A * and is notated Ai

Determinant Definition

The definition of a squar matrix A that known =DetA or the absolute value of A. It s s scaler that can the calculator froms differnt meaning for a 2*2 matrix. AB=CD is the determent = ads-bc

Determinant Properties

Detat = dett

Detat = det det B

A is inversible is the same a def a is not the same a 0

If A has all zore as riws or columns then the Def a is the same as 0

If A-s is obtained form A by exchagign all riws and colimns then . Data = -det a

Proper values nad proper vectors

• Proper values Definition* If A is square it has size number equal N N . A number L in K is proper values if exists a factor is K1 that A is the same as lv
• Proper vectors definition* A no zero factor, V is vector and a valuse number, AI-V= LV
• Proper size Definition* The size on eagen value number is L (E) and the set size number in is the augmented size EX IKK 2 ( AVAL= V ) = Ker L
• Polar characteristics definition* polar size L is the a AAI of the values A A’s

Capitla 7 - Krylov Iterations

• Objectives*
• Inroductions to krylov itertive methood
• Poer iterrative mrthhods / Lanczos method / method of conjugate gradient
• Inroducion* Iterative krylov methods are numerical methods being used to slovestions o linear equations form Ax=B, Where A = is NN size matix,x = inkbow vectyor size N,B= vector side size A

Those methods are useful when the Matrix A = is a bit size or un full which direct methods is so memory expensive and time consumintg

• Kylove Space* Krylor space is the space matrix A and B

A B AB A”K-1B

Which pan deoties that a liner cominbations of gievn vector. krylov metjods look fr a solutioon approsiamto $x_k$

KAXB that aproches the excat solution

• Power iterative method* The power method is method to find the large values proper f a matrix and proper vectors
• Basic Aulgorithym* 1/ Chosse abinital 9, if size 1 from 1,2.. 3/ in $Z_k = Aq(k-1)$ 3/ in $Q_k = Z_k$ with the absolute value signs 3/ In $1 = qTAq$

From inside this alogorithim 9K approsimates the vector aorrespodning go the ahrhe value proper LK approsimates to the large value Proper conergance

This method conerges in size and aorrespondijm vector so when matric has a ahre property value to the vector proper when intiual values havnt any the part that is 0 value

EXample So Matrix can use this to be found on a lange porperty value

Lanczos method, This is the method to find proper value of a matrix a symmetix

Besix Principle Lanxos builds a base of Krylov space in gran Schmidt process, the base is callen lanczos vectors

• Algorithm*

chosse initial $\mathcal{G}$, of size 1

initatiale $ \alpha_0 = 0$ and $q_0 = 0$ f $J = 1,2$ un till you convernces

$\mathcal{G} z_j = Aq_j$ $\alpha_j =Q^t_jz_j$

$Z{j+1} = Z_j - \alpha_jQ_j - Bjq{-1}$

$Q{j+1} = z{ j}+1, bj+1

the values are $9i \infty j \alpha_j Q_j B_j $

This is the auefficnr to oarhtogmize vectiotos

• Triagonal Matrix*

Thsi s an algorthytm matrix. The Matrix A transforms and a toragonal matrix is. The deogonal elements and the

$T_k = \begin{bmatrix} B_2 & d & & \ b & d. B3 & L & & \ & B2 & d & B_3 & \ && Bd3 & \ddots & 0\ && && d

\end{bmatrix}$

methos og conjugate grdient is to solve a system of linear eauqtion A. B when matrics A is posititvedffinte

Besxi princlpie: search the solution of AXB by minzing equationa

PHIX / = 12 xT Ax - BTS

Conjugate vectors that warants the quic convergence to olution are generate by the algorthim

• Krylov Iterations - Important Theorems*

• A is hermition Matriz A is equal A transpose

• Chebyshey Plynomial

• Guía de inicio rápido de Kontent.ai*

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Bernoulli's Principle

• There is an inverse correclation between pressure and the velocity of a fluid. Faster fluids have decreased pressure.
• The principle was formulated in the 18th century.
• Denoted: $v_1$ = fluid velocity at point 1. $p_1$ = pressure at point 1. $\rho$ = fluid density. $v_2$ = fluid velocity at point 2. $p_2$ = pressure at point 2.
• The primary formula for Bernoulli's principle is $\frac{v_1^2}{2} + \frac{p_1}{\rho} = \frac{v_2^2}{2} + \frac{p_2}{\rho}$.

Applications

• The wings of an airplane.
• The diffuser on a race car.
• The Venturi meter.

The Big Bang Theory

• The prevailing cosmological model for the early development of the Universe.
• All matter was contained at a single point in the distant past.
• The universe expanded from an extremely dense state and at an incredibly high temperature.
• It cooled and formed the subatomic particles and atoms.
• Due to gravity, particles coalesced and formed stars and galaxies.

Evidence

• Objects Receding: objects emit lower frequency light when further.
• Universe expanding: The further the object is, the greater the redshift and the universe is expanding.
• CMBR permeated at 2.7K: faint electromagnetic radiation permeates the universe in the microwave region and it has the characteristics of blackbody radiation.
• Early helium: The early universe consisted of 75% hydrogen and 25% helium
• The abundance of this element is in good agreement.

Timeline

• t = 0 - Big Bang
• t = 10^{-43} s - Planck time dominates-Quantum gravity
• t = 10^{-36} s - Exponential Expansion (Inflation)
• t = 1 second- Atomic Nuclei(Nucleosynthesis)
• t = 3 minutes - End of nucleosynthesis
• t = 380,000 years - Atoms Are Formed (Recombination starts)
• t = 150 million years-First stars are created.
• t = 1 billion years -First Galaxies form
• t = 9 billion years -The formation of solar systems
• t = 13.8 - Present day

Unresolved

• Root of explosion/cause for Big Bang
• Nature of matter, dark energy
• What happened before explosion.
• Collapsing question (Big Crunch)

Quantum Mechanics

Core Meaning

• It provides a descrition of nature's physical properties.
• It involves subatomic particles.
• It descrite mattar and energys actions.

Quantum Key Concepts

• Particles can have wave actions.
• There are accuracy constraints (Heisenberg).
• A item can be everywhere at once (Superposition state)
• Items stay linked at distance (Entanglement).
• It has discrete values.

Quantum Physics Formulas

• Schrödinger =How quantum states change.
• Planck = Connection E.
• Heisenberg= unremovable value in item properties

Quantum Functions

• MRI utilizes concepts in medine
• Helps grasp materais
• Comps that are impossible
• Creates secure messages

Key Quantum Names

• Bohr
• Planck
• Einstein
• Schrodinger
• Heisenberg

Quantum Challenges

• Quantum Decoupling
• Understanding theory
• Reconciling theory with relitivis

States and Areas With Quantum Physics

• Spin; Quantum numbers.
• Superposition; amplitude, measurement
• Approximation.
• Particles are identical

Quantum Physics Equations

• Schrodinger equation is i h / t Y
• Heisenberg's formula is ( X delta ) (P D = h/2
• Plancks forumla is E = hV