Vectors: Plane and Space

Vectors in the Plane and Space

• A vector $\overrightarrow{AB}$ is defined by its direction and length, going from point A to point B.
• Two vectors are equal if they share direction, sense, and length.
• A unit vector has a length of 1.

Coordinates

• Coordinates $(v_1, v_2)$ of vector $\overrightarrow{v}$ are found by projecting onto coordinate axes.
• In space, $\overrightarrow{v} = (v_1, v_2, v_3)$.
• Length is denoted as $|\overrightarrow{v}|$:
  • Plane: $|\overrightarrow{v}| = \sqrt{v_1^2 + v_2^2}$
  • Space: $|\overrightarrow{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}$
• Example: $\overrightarrow{v} = (3, 4)$ gives $|\overrightarrow{v}| = \sqrt{3^2 + 4^2} = 5$.

Vector Operations

Addition and Subtraction

• For $\overrightarrow{u} = (u_1, u_2)$ and $\overrightarrow{v} = (v_1, v_2)$:
  • $\overrightarrow{u} + \overrightarrow{v} = (u_1 + v_1, u_2 + v_2)$
  • $\overrightarrow{u} - \overrightarrow{v} = (u_1 - v_1, u_2 - v_2)$
• Applies similarly in space using $\overrightarrow{u} = (u_1, u_2, u_3)$ and $\overrightarrow{v} = (v_1, v_2, v_3)$.

Scalar Multiplication

• Given vector $\overrightarrow{v} = (v_1, v_2)$ and scalar $c$:
  • $c\overrightarrow{v} = (cv_1, cv_2)$

Scalar Product

• Given $\overrightarrow{u} = (u_1, u_2)$ and $\overrightarrow{v} = (v_1, v_2)$:
  • $\overrightarrow{u} \cdot \overrightarrow{v} = u_1v_1 + u_2v_2 = |\overrightarrow{u}| |\overrightarrow{v}| \cos(\theta)$, where $\theta$ is the angle.
• In space, $\overrightarrow{u} \cdot \overrightarrow{v} = u_1v_1 + u_2v_2 + u_3v_3$.

Vector Product (in Space)

• For $\overrightarrow{u} = (u_1, u_2, u_3)$ and $\overrightarrow{v} = (v_1, v_2, v_3)$, $\overrightarrow{u} \times \overrightarrow{v}$ is orthogonal to both.
• Direction follows right-hand rule.
• $\overrightarrow{u} \times \overrightarrow{v} = (u_2v_3 - u_3v_2, u_3v_1 - u_1v_3, u_1v_2 - u_2v_1)$
• Parallelogram area is $|\overrightarrow{u} \times \overrightarrow{v}|$.

Geometric Applications

Equation of a Line

• Defined by $P_0(x_0, y_0)$ and direction vector $\overrightarrow{v} = (a, b)$.
• Parametric equation:
  • $x = x_0 + at$
  • $y = y_0 + bt$, where $t \in \mathbb{R}$.
• Cartesian equation: $ax + by + c = 0$.

Equation of a Plane

• Defined by $P_0(x_0, y_0, z_0)$ and normal vector $\overrightarrow{n} = (a, b, c)$.
• Plane equation: $a(x - x_0) + b(y - y_0) + c(z - z_0) = 0$, or $ax + by + cz + d = 0$ where $d = -ax_0 - by_0 - cz_0$.

Fundamentals of Quantum Mechanics

• Describes physical properties at the atomic and subatomic level.
• Provides a mathematical framework to predict experimental outcomes.

Core Quantum Concepts

• Quantization: Limits physical quantities to discrete values.
• Wave-Particle Duality: Particles can exhibit wave-like behavior.
• Superposition: System exists in multiple states.
• Uncertainty Principle: Limit to precision of knowing pairs of physical properties.
• Quantum Entanglement: Linked particles share the same fate regardless of distance.

Schrödinger Equation

• Describes the time evolution of a quantum system.

Time-Dependent Equation

• $i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r},t) = \hat{H}\Psi(\mathbf{r},t)$

Time-Independent Equation

• $\hat{H}\psi(\mathbf{r}) = E\psi(\mathbf{r})$

Operators in Quantum Mechanics

Examples

• Position Operator: $\hat{x} = x$
• Momentum Operator: $\hat{p} = -i\hbar\frac{\partial}{\partial x}$
• Hamiltonian Operator: $\hat{H} = -\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r})$

Applications of Quantum Mechanics

• Atomic Physics
• Condensed Matter Physics
• Quantum Computing
• Nuclear Physics

Dirac Notation

• 'Bra-ket' notation describes quantum states.
• Ket: Denotes a quantum state.
• Bra: Denotes the dual vector corresponding to ket.
• Inner Product: Represents the inner product between states.

Spin

• Intrinsic angular momentum of particles.
• Electrons have spin of 1/2, with spin up or spin down.

Quantum Harmonic Oscillator

• Describes particle motion in a quadratic potential.

Hamiltonian

• $\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega^2\hat{x}^2$

Energy Levels

• $E_n = \hbar\omega(n + \frac{1}{2}), \quad n=0,1,2,...$

Perturbation Theory

• Approximates solutions to quantum mechanical problems.

Time-Independent

• $H = H_0 + \lambda H'$

Time-Dependent

• Calculates transition rates given a time-dependent perturbation.

Identical Particles

• Indistinguishable in quantum mechanics.

Key Concepts

• Symmetric Wave Function: Bosons.
• Antisymmetric Wave Function: Fermions.
• Pauli Exclusion Principle: No two fermions in the same state.

Linear Algebra

Definition of Linear Application

• $f: E \rightarrow F$ is linear if ( f(x + y) = f(x) + f(y) ) for all ( x, y \in E ) and ( f(\lambda x) = \lambda f(x) ) for all ( x \in E, \lambda \in \mathbb{K} ).
• ( \mathbb{K} ) is a field (( \mathbb{R} ) or ( \mathbb{C} )).
• E and F are vector spaces over ( \mathbb{K} ).

Example

• $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ where $f(x, y) = (x + y, x - y)$

Important Properties

• $f(0_E) = 0_F$
• $f(-x) = -f(x)$
• $f(\sum_{i=1}^{n} x_i) = \sum_{i=1}^{n} f(x_i)$
• $f(\sum_{i=1}^{n} \lambda_i x_i) = \sum_{i=1}^{n} \lambda_i f(x_i)$

Vocabulary

• Linear form: f is an linear application with F = $\mathbb{K}$
• Endomorphism: f is an linear application with E = F
• Isomorphism: f is bijective
• Automorphism: f is an endomorphism bijective

Operations on Linear Applications

Linear Combination

• $(\lambda f + \mu g)(x) = \lambda f(x) + \mu g(x)$

Composition

• $g \circ f : E \rightarrow G$ where $(g \circ f)(x) = g(f(x))$

Image and Kernel

Definition

• Kernel: $\operatorname{Ker}(f) = {x \in E \mid f(x) = 0_F}$
• Image: $\operatorname{Im}(f) = {y \in F \mid \exists x \in E, f(x) = y}$

Properties

• Ker(f) is a subspace of E, Im(f) is a subspace of F.
• Injective if and only if Ker(f) = {0}.
• Surjective if and only if Im(f) = F.

Rank Theorem

• If E is finite dimensional: $\dim(E) = \dim(\operatorname{Ker}(f)) + \dim(\operatorname{Im}(f))$
• Dimensionality of Im(f) is called the rank of f and is denoted rg(f).

Consequences

• If dim(E) = dim(F): f injective if and only if f surjective if and only if f bijective
• If dim(E) = dim(F) and f injective: f is an isomorphism

Matrix of a Linear Application

Definition

• Matrix's columns are coordinates of $f(e_1),..., f(e_n)$ in base $\mathcal{B}_F$.

Properties

• $M_{\mathcal{B}_E, \mathcal{B}G}(g \circ f) = M{\mathcal{B}_F, \mathcal{B}G}(g) M{\mathcal{B}_E, \mathcal{B}_F}(f)$
• $M_{\mathcal{B}_E, \mathcal{B}F}(f^{-1}) = (M{\mathcal{B}_E, \mathcal{B}_F}(f))^{-1}$

Change of Base

• If X are coordinates of x in ( \mathcal{B} ) and X' are coordinates of x in ( \mathcal{B'} ), then X = PX'.

Rank of a Matrix

Definition

• Number of linearly independent columns.

Properties

• Rank is less than or equal to minimum of rows or columns.
• If A is a square matrix, A is invertible if and only if rank(A) = n.

Determinant

Definition

• (\det(A))

Properties

• If A has two identical rows or columns, then (\det(A) = 0).
• Determinant changes sign when swapping rows.
• A is invertible if and only if (\det(A) \neq 0).

Eigenvalues and Eigenvectors

Definition

• ( \lambda ) is a value if there exists a vector such that ( \det(f - \lambda id_E) = 0 )

Properties

• (\det(f - \lambda id_E) = 0)

Diagonalization

• Exists if ( P^{-1} A P ) is a diagonal matrix

Trigonalisation

• Exists if ( P^{-1} A P ) is a triangular matrix

Straight Lines

Slope of a line

• ( m = \tan\theta )
• If $P(x_1, y_1)$ and $Q(x_2, y_2)$ are any two points on a line, then its slope is given by $m = \frac{y_2 - y_1}{x_2 - x_1}$.

Angle between two lines

• ( \tan\theta = |\frac{m_2 - m_1}{1 + m_1m_2}| )
• Two lines are parallel iif ( m_1 = m_2 ).
• Two lines are perpendicular iif ( m_1m_2 = -1 ).

Equation of a line

• Slope-intercept form: ( y = mx + c )
• Intercept form: ( \frac{x}{a} + \frac{y}{b} = 1 )
• Normal form: ( x\cos\omega + y\sin\omega = p )

General equation of a line: ( Ax + By + C = 0 )

Distance

• Distance of a point from a line: ( d = |\frac{Ax_1 + By_1 + C}{\sqrt{A^2 + B^2}}| )
• Distance between two parallel lines: ( d = |\frac{C_1 - C_2}{\sqrt{A^2 + B^2}}| )