Vectors and Lines

Prerequisites

• A vector is defined by its direction, sense, and magnitude.
• A vector $\overrightarrow{AB}$ is represented by an arrow from point A to point B.

Components

• In an orthonormal coordinate system, a vector $\overrightarrow{AB}$ is defined by its components: $\overrightarrow{AB} = \begin{pmatrix} x_B - x_A \ y_B - y_A \end{pmatrix}$

Magnitude

• The magnitude(norm) of a vector $\overrightarrow{AB}$ is the length of the segment [AB].
• $|| \overrightarrow{AB} || = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2}$

Collinear Vectors

• Two vectors $\overrightarrow{AB}$ and $\overrightarrow{CD}$ are collinear if and only if: $\begin{vmatrix} x_B - x_A & x_D - x_C \ y_B - y_A & y_D - y_C \end{vmatrix} = 0$

Lines

• A line is defined by a point and a direction vector.

Parametric Representation

• A line passing through point A with direction vector $\overrightarrow{u}$ is represented by the parametric equation: $\begin{cases} x = x_A + t \cdot x_u \ y = y_A + t \cdot y_u \end{cases}$

Cartesian Representation

• A line is represented by the Cartesian equation: $ax + by + c = 0$

Normal Vector

• A normal vector to a line with Cartesian equation $ax + by + c = 0$ is $\overrightarrow{n} = \begin{pmatrix} a \ b \end{pmatrix}$.

Slope

• The slope of a line with Cartesian equation $ax + by + c = 0$ is $m = -\frac{a}{b}$.

Point-Line Distance

• The distance between a point $P(x_P, y_P)$ and a line with Cartesian equation $ax + by + c = 0$ is: $d(P, line) = \frac{|ax_P + by_P + c|}{\sqrt{a^2 + b^2}}$

Energy Bands in Solids

Summary of Kronig-Penney Model

• Electrons move within a periodic potential
• Solutions are Bloch functions: $\psi(x) = u(x)e^{ikx}$, where $u(x)$ is periodic with the lattice period
• Energies are derived from the condition $P\frac{sin(\alpha a)}{\alpha a} + cos(\alpha a) = cos(ka)$, where $P = \frac{mV_0 ba}{\hbar^2}$, $\alpha^2 = \frac{2mE}{\hbar^2}$, $V_0$ is the potential barrier height, $b$ is the barrier width, and $a$ is the well width
• Energy bands are allowed and forbidden due to the constraint $|cos(ka)| \le 1$

Number of States in a Band

• For a 1D crystal with $N$ atoms and periodic boundary conditions, $k = \frac{2\pi n}{Na}$, where $n = 0, \pm 1, \pm 2,...$ and $k$ is the allowed value
• The number of states in the 1st Brillouin zone equals the number of atoms $N$.
• Each band contains $N$ states.
• Each state accommodates 2 electrons considering spin.
• Each band accommodates $2N$ electrons as a result.

Metals, Insulators, and Semiconductors

• Metals: Highest occupied band partially filled, allowing electrons near the Fermi level to easily move when an electric field is applied
• Insulators: Highest occupied band (valence band) is full; next band (conduction band) is empty; large energy gap $E_g$, preventing electron movement unless energy $\ge E_g$ is gained
• Semiconductors: Similar to insulators with a small energy gap $E_g \approx 1 eV$; behave as insulators at $T = 0$, but become conductive at room temperature as electrons are thermally excited from the valence to the conduction band

Comparison of Complex Numbers

Algebraic Form

• Given $z = a + bi$ and $z' = a' + b'i$, where $a, a', b, b' \in \mathbb{R}$, $z = z'$ if and only if $a = a'$ and $b = b'$.

Modulus and Argument

• Given $z = [r, \theta]$ and $z' = [r', \theta']$, where $r, r' \in \mathbb{R}^+$ and $\theta, \theta' \in \mathbb{R}$, $z = z'$ if and only if $r = r'$ and $\theta = \theta' [2\pi]$.

Comparison

• Order relations do not exist in $\mathbb{C}$; complex numbers cannot be compared.

Bernoulli's Principle

• States that an increase in fluid's speed occurs simultaneously with a decrease in pressure or the fluid's potential energy, for an inviscid flow of a nonconducting fluid.
• Pressure in a fluid decreases as speed increases.
• Fast-moving fluids exert less pressure than slow-moving.
• Fluid moving from wide to narrow area causes changes in speed.

Formula

• $P_1 + \frac{1}{2} \rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho g h_2$
  • $P$ = pressure
  • $v$ = velocity
  • $\rho$ = density
  • $h$ = height above reference point
  • $g$ = acceleration due to gravity

Examples

Airplanes

• Wings are designed to make air flow faster over the top than under the wing.
• Higher speed equals lower pressure, so pressure above is less than pressure below, creating lift.
  • Air flows faster over curved top (low pressure).
  • Air flows slower under flat bottom (high pressure).

Curveball

• Ball thrown with spin creates a thin layer of air dragged around the ball.
  • Airflow direction relative to spinning surface in the same direction.
  • Airflow direction relative to spinning surface in the opposite direction.
• Pressure difference deflects the ball.

Chemical Principles

Atomic Structure

Development of Atomic Theory

• Ancient Greece: Matter consists of indivisible particles (atomos).
• Dolton's Atomic Theory (1803):
  • All matter consists of indivisible particles called atoms
  • Atoms of the same element have the same mass and properties
  • Compounds are formed by combining two or more elements in a fixed ratio
  • Chemical reactions are merely the rearrangement of atoms
• Thomson's Cathode Ray Experiment (1897):
  • Cathode rays are a flow of negatively charged particles $\rightarrow$ discovery of the electron
  • Atoms are spherical particles with a positive charge, with electrons embedded in them (plum pudding model)
• Rutherford's $\alpha$ Particle Scattering Experiment (1911):
  • $\alpha$ particles fired at gold foil $\rightarrow$ most passed through, with some deflected
  • Confirmation of the atomic nucleus: a positively charged nucleus exists at the center of the atom, with electrons orbiting around it (solar system model)
• Bohr's Hydrogen Atom Model (1913):
  • Electrons can only exist at specific energy levels
  • Electrons absorb or emit specific energy to move between energy levels
  • Successful explanation of the hydrogen atom's line spectrum

Atomic Constituents

• Atomic nucleus: protons, neutrons $\rightarrow$ comprise most of the atom's mass
• Atomic number (Z): Number of protons in the atom = number of electrons (for neutral atoms)
• Mass number (A): Number of protons + neutrons in the atomic nucleus
• Nuclide: Atom with a specific number of protons and neutrons
  • $^{A}_{Z}X$ (X: element symbol)
• Isotope: Elements with the same number of protons but different numbers of neutrons
  • Chemical properties are mostly identical, but physical properties differ slightly
• Atomic mass: Average mass considering the abundance ratio of the isotopes
  • The mass of $^{12}C$ is set at 12 amu, based on which the masses of other atoms are expressed
  • Atomic mass = $\sum$ (mass of the isotope $\times$ abundance ratio)

Molecules, Ions, and Chemical Formulas

• Molecule: Particle formed by the chemical bonding of two or more atoms
  • Molecular formula: Formula showing the type and number of atoms comprising a molecule (e.g., $H_2O$, $CO_2$)
  • Empirical formula: Formula showing the simplest integer ratio of atoms comprising a molecule (e.g., $H_2O$, $CH$)
  • Structural formula: Formula showing the bonding pattern of atoms comprising a molecule (e.g., H-O-H)
• Ion: Charged particle
  • Cation: Ion losing electrons, having a (+) charge (e.g., $Na^+$, $Ca^{2+}$)
  • Anion: Ion gaining electrons, having a (-) charge (e.g., $Cl^-$, $O^{2-}$)
• Chemical formula: Formula showing the type and number of atoms comprising a compound
  • Ionic compounds: Expressed as empirical formulas (e.g., $NaCl$, $MgCl_2$)

Chemical Reaction Equations

Quantitative Relationships in Chemical Reaction Equations

• Chemical reaction equation: Equation showing the relationship between reactants and products, using chemical formulas and symbols to express chemical reactions
  • Stoichiometric coefficient: Number expressing the moles of each chemical species in a chemical reaction equation
  • Balancing chemical reaction equations: Adjust the number of each type of atom to be the same for the reactants and products
• Mole, mol: Unit expressing the amount of a substance
  • $1 mol = 6.022 \times 10^{23}$ items (Avogadro's number, $N_A$)
  • Molar mass, M: Mass of 1 mol (g/mol) = atomic mass, molecular mass, or formula mass with g/mol units added
• Quantitative relationships in chemical reaction equations: Coefficient ratio in a chemical reaction equation = mole ratio = (in the case of gases) volume ratio
  • for a reaction where reactants A and B produce products C and D: $aA + bB \rightarrow cC + dD$
  • $n_A : n_B : n_C : n_D = a : b : c : d$ (n: number of moles)

Reaction Yield

• Reaction yield: (amount of product actually obtained / amount of product theoretically obtainable) $\times$ 100%
• Limiting reactant: Reactant that is consumed first, limiting the extent of the entire reaction
• Excess reactant: Reactant present in greater amount than the limiting reactant

Sentiment Analysis

• Process of detecting emotion (positive, negative, neutral) in text.

Importance

• Brand image: Understanding market perception.
• Customer understanding: Insights into needs/expectations.
• Product improvement: Identifies areas needing improvement.

Methods

• Lexicon-based: Uses a lexicon of pre-defined words and phrases with sentiment scores.
• Machine learning (ML): Algorithms trained on labeled text datasets.
• Deep learning: Complex models like RNN/CNN learn patterns for high accuracy.

Applications

• Social media monitoring: Tracking mentions of brands.
• Customer feedback: Analyzing reviews and survey responses.
• Market analysis: Understanding public sentiment about products/services.
• News analysis: Determining the sentiment of news articles.

Challenges

• Irony/sarcasm: Difficult to detect for algorithms.
• Context: Word meaning changes based on context.
• Subjectivity: Varies from person to person.

Tools

Quantum Mechanics

Definition

• Deals with the behavior of matter and light on the atomic and subatomic scale

Classical vs Quantum Physics

• Classical physics works well at macroscopic level and assumes variables are determistic and continuous
• Quantum mechanics needed for microscopic world and assumes variables can be probabilistic and discrete

Key Concepts

• Quantization: Restrictions to discrete values ​​for energy, momentum, and other quantities
• Wave-Particle Duality: Particles like electrons can exhibit wave-like behavior, and waves like light can exhibit particle-like behavior
• Uncertainty Principle: There is a fundamental limit to the precision with which certain pairs of physical properties can be known
• Superposition: System can exist in multiple states simultaneously until measured
• Entanglement: Link between quantum systems such that they share fate no matter distance

Mathematical Formalism

Wave Function

• Represented by $\Psi(r, t)$, location at r and time at t
• $|\Psi(r, t)|^2$ is the probability density of finding the particle

Schrödinger Equation

• Time-dependent Schrödinger equation: $i\hbar \frac{\partial \Psi}{\partial t} = \hat{H} \Psi$ - where: $i$ = imaginary unit, $\hbar$ = reduced Planck constant, $\Psi$ = wave function, $\hat{H}$ = Hamiltonian operator
• Time-independent Schrödinger equation: $\hat{H} \Psi = E \Psi$ - where: $E$ is the energy of the system

Operators

• Operators correspond to physical observables
  • Momentum operator: $\hat{p} = -i\hbar \nabla$
  • Energy operator (Hamiltonian): $\hat{H} = -\frac{\hbar^2}{2m}\nabla^2 + V(r)$, $m$ = mass, $V(r)$ = potential energy

Core Principles

Heisenberg's Uncertainty Principle

$\Delta x \Delta p \geq \frac{\hbar}{2}$, where $Δx$=position ($Δx$) of a particle, $Δp$=momentum ($Δp$)

Superposition and Measurement

• Quantum system exists in a superposition of states until a measurement is made.

Quantum Entanglement

• When two or more particles are entangled, their fates are intertwined, and measuring the state of one particle instantaneously influences the state of the other, regardless of the distance between them.

Applications

• Quantum Computing
• Quantum Cryptography
• Materials Science
• Medical Imaging
• Laser Technology

Key Experiments

• Double-Slit Experiment
• Stern-Gerlach Experiment
• Photoelectric Effect

Introduction to Differential Equations

Definitions and Terminology

Differential Equation

• A differential equation (DE) is an equation containing the derivatives of one or more variables with respect to one or more independent variables.

Notation

• An ordinary differential equation (ODE) contains derivatives of a dependent variable with respect to a single independent variable. Examples:
  • $\frac{dy}{dx} + 5y = e^x$
  • $\frac{d^2y}{dx^2} - \frac{dy}{dx} + 6y = 0$
  • $\frac{dx}{dt} + \frac{dy}{dt} = 2x + y$
• A partial differential equation (PDE) contains partial derivatives of one or more dependent variables with respect to two or more independent variables. Examples:
  • $\frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} = 0$
  • $\frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}$

Order

• The order of a DE (ODE and/or PDE) is the order of the highest derivative in the equation.

Linearity

• A DE of order n is linear if it has the form:
  • $a_n(x) \frac{d^ny}{dx^n} + a_{n-1}(x) \frac{d^{n-1}y}{dx^{n-1}} +... + a_1(x) \frac{dy}{dx} + a_0(x)y = g(x)$
• A differential equation is nonliniear if not in this form. Examples:
  • non-linear: $(\frac{dy}{dx})^2 + y = 0$
  • non-linear: $y\frac{dy}{dx} = sen(x)$

Solution

• A solution: Any function $\phi$ defined on an interval I and possessing at least n derivatives that are continuous on I, which when substituted into a DE of order n reduces the equation to an identity.

Example

• $y = e^{0.1x^2}$ is a solution of $xy' - 2y = 0$ on $(-\infty, \infty)$. Showing the solution steps:
  • Solving gives: $x(e^{0.1x^2})' - 2e^{0.1x^2} = x(0.2x e^{0.1x^2}) - 2e^{0.1x^2} = 0$

Comparison of Complex Numbers

Definition

• $z = a + bi$ and $z' = a' + b'i$ are two complex numbers with $a, a', b, b' \in \mathbb{R}$. $z = z'$ if and only if $a = a'$ and $b = b'$.
  • $z = (x + y) + i(x - y)$ and $z' = 3 - 5i$. $z = z'$ if and only if $\begin{cases} x + y = 3 \ x - y = -5 \end{cases}$
• On obtient $x = -1$ et $y = 4$.

Properties

• Two complex numbers are equal if and only if they have the same modulus and the same argument.
• Soient $z = [r, \theta]$ et $z' = [r', \theta']$, alors $z = z' \Leftrightarrow r = r'$ et $\theta = \theta' + 2k\pi$, $k \in \mathbb{Z}$. -eg. $z = \left[ 2, \frac{\pi}{3} \right]$ et $z' = \left[ 2, \frac{7\pi}{3} \right]$ sont égaux car $\frac{7\pi}{3} = \frac{\pi}{3} + 2\pi$.

Remark

• There is no order relation in $\mathbb{C}$. It cannot be said that one complex number is greater or less than another.

Thermodynamics

Spontaneous Processes

• Spontaneous processes occur without external intervention.
  • Examples: downhill waterfalls, sugar dissolving in coffee, iron rusting, gas expanding.
• Opposite of spontaneous process is nonspontaneous, which requires continuous external intervention.

Spontaneity

• Tendency of process to occur without external intervention.
• Spontaneity indicates nothing about the rate of the process.

Reversible and Irreversible Processes

• Reversible: Reversed by infinitesimal change.
  • Example: Melting ice at 0°C; system in equilibrium with surroundings.
• Irreversible: Unable to be reversed; spontaneous processes are irreversible as system is not in equilibrium and process has direction.

Entropy

• $S$: Measure of disorder/randomness
• Higher disorder/randomness = higher entropy.

Entropy Change

• $\Delta S = S_{final} - S_{initial}$
  • If $\Delta S > 0$, disorder increases.
  • If $\Delta S < 0$, disorder decreases.

Entropy and Probability

• Related to number of possible microstates.
• Microstate: Specific arrangement of positions/kinetic energies.
• Higher microstates = higher entropy.
• Expansion illustrates this, as gas molecules have more arrangements in a larger volume.

Entropy Increases With

• Number of gas molecules, volume, and temperature.

Entropy Change and Heat

• For an isothermal process: $\Delta S = \frac{q_{rev}}{T}$
  • $q_{rev}$ is heat absorbed during reversible process
  • $T$ is absolute temperature in Kelvin.

Second Law of Thermodynamics

• Entropy of the universe increases for any spontaneous process: $\Delta S_{univ} = \Delta S_{sys} + \Delta S_{surr} > 0$
• If a reversible process: $\Delta S_{univ} = \Delta S_{sys} + \Delta S_{surr} = 0$

Third Law of Thermodynamics

• Entropy of perfect crystalline substance is zero at absolute zero (0 K): $S(0 K) = 0$
• 0K serves as reference point to find absolute entropy of any substance.