Intro to Quantum Mechanics

Quantum Mechanics

• Deals with matter and energy at atomic and subatomic levels.
• Underlies atomic structure, chemical bonding, solid-state, nuclear, and particle physics

Key Concepts

• Quantization: Physical quantities take on discrete values.
• Wave-particle duality: Quantum objects behave as both waves and particles.
• Superposition: Quantum system exists in multiple states simultaneously, with probabilities.
• Uncertainty principle: Limit to precision of knowing certain pairs of physical quantities simultaneously. Also known as the Heisenberg uncertainty principle.
• Entanglement: Two or more quantum systems linked, even when separated by distances.

Mathematical Formulation

• Predictions rely on mathematical equations.
• Schrödinger equation: $i\hbar \frac{\partial}{\partial t} \Psi(r,t) = \hat{H} \Psi(r,t)$ describes quantum system evolution.
  • $i$ is the imaginary unit.
  • $\hbar$ is the reduced Planck constant.
  • $\Psi(r, t)$ is the wave function.
  • $\hat{H}$ is the Hamiltonian operator.

Applications

• Lasers: Based on stimulated emission.
• Transistors: Based on quantum properties of semiconductors.
• MRI: Medical imaging using quantum properties of atomic nuclei.
• Quantum computing: Uses quantum phenomena for calculations.
• Quantum cryptography: Secure communication using quantum mechanics laws.

Challenges and Future Directions

• Interpretation of quantum mechanics not agreed upon.
• Quantum gravity attempts unifying quantum mechanics and general relativity.
• Quantum technologies are under development.

Linear Algebra and Geometry

Determinants

• For $A \in M_n(\mathbb{K})$, the determinant of $A$, denoted det $A$ or $|A|$, is defined by:
• $\operatorname{det} A=\sum_{\sigma \in \mathcal{S}n} \varepsilon(\sigma) \prod{i=1}^n a_{i, \sigma(i)}$.
  • Where $\mathcal{S}_n$ is the set of permutations of ${1, \ldots, n}$, and $\varepsilon(\sigma)$ is the signature of the permutation $\sigma$.
• If $A=(a) \in M_1(\mathbb{K})$, then $\operatorname{det} A=a$.
• If $A=\left(\begin{array}{ll}a & b \ c & d\end{array}\right) \in M_2(\mathbb{K})$, then $\operatorname{det} A=a d-b c$.
• If $A=\left(\begin{array}{lll}a & b & c \ d & e & f \ g & h & i\end{array}\right) \in M_3(\mathbb{K})$, then $\operatorname{det} A=a e i+b f g+c d h-c e g-b d i-a f h$ (Rule of Sarrus).
• Determinant is an alternating $n$-linear form of the column vectors of $A$.
• $\operatorname{det} A=\operatorname{det} A^t$.
• $\operatorname{det}(A B)=\operatorname{det} A \cdot \operatorname{det} B$.
• $A \in M_n(\mathbb{K})$ is invertible if and only if det $A \neq 0$.
  • If $A$ is invertible, then $\operatorname{det}\left(A^{-1}\right)=(\operatorname{det} A)^{-1}$.
• If $A \in M_n(\mathbb{K})$ is triangular, then $\operatorname{det} A$ is the product of diagonal elements of $A$.
• Swapping two columns of $A$ multiplies the determinant by $-1$.
• Multiplying a column of $A$ by a scalar $\lambda \in \mathbb{K}$ multiplies the determinant by $\lambda$.
• Adding a multiple of one column of $A$ to another does not change the determinant.
• For $A=\left(a_{i j}\right) \in M_n(\mathbb{K})$, denote $A_{i j}$ as the matrix obtained by removing the $i$-th row and $j$-th column of $A$.
  • Cofactor of $a_{i j}$ is $\Delta_{i j}=(-1)^{i+j} \operatorname{det} A_{i j}$.
  • For any $i, j \in{1, \ldots, n}$ :
    • $\operatorname{det} A=\sum_{j=1}^n a_{i j} \Delta_{i j}$ (expansion with respect to the $i$-th row).
    • $\operatorname{det} A=\sum_{i=1}^n a_{i j} \Delta_{i j}$ (expansion with respect to the $j$-th column).
• To calculate a determinant practically, use elementary operations to triangularize the matrix and then compute the product of the diagonal elements.
  • You can also use the determinant expansion with respect to a row or a column.

Linear Equation Systems

• Consider a system of $m$ linear equations with $n$ unknowns:
• $A X=B$, where $A \in M_{m, n}(\mathbb{K}), X \in M_{n, 1}(\mathbb{K}), B \in M_{m, 1}(\mathbb{K})$. -Transform the system into an equivalent system using elementary row operations on the augmented matrix $(A \mid B)$. -Solve the system by going back up the equations

Rouché-Fontené Theorem

• Let $A X=B$ be a system of $m$ linear equations with $n$ unknowns. Note $r=\operatorname{rank} A$ and $r^{\prime}=\operatorname{rank}(A \mid B)$.
  • The system admits at least one solution if and only if $r=r^{\prime}$.
  • If $r=r^{\prime}$, then the set of solutions is an affine subspace of $\mathbb{K}^n$ with dimension $n-r$.
  • If $r=r^{\prime}=n$, then the system admits a unique solution.
• If $A \in M_n(\mathbb{K})$ is invertible (if $\operatorname{det} A \neq 0$ ), then the system $A X=B$ admits a unique solution given by $X=A^{-1} B$.
  • In this case, the $i$-th component of $X$ is given by Cramer's formula:
  • $x_i=\frac{\operatorname{det} A_i}{\operatorname{det} A}$, where $A_i$ is the matrix obtained by replacing the $i$-th column of $A$ by the vector $B$.

Vector spaces

• A vector space over a field $\mathbb{K}$ is a set $E$ with two operations:
  • Addition: $+: E \times E \rightarrow E$.
    • Associativity
    • Commutativity
    • Exists a neutral element
    • Exists an opposite
  • Multiplication by a Scalar: $\cdot: \mathbb{K} \times E \rightarrow E$
    • $\forall \lambda, \mu \in \mathbb{K}, \forall u \in E, \lambda \cdot(\mu \cdot u)=(\lambda \mu) \cdot u$
    • $\forall \lambda \in \mathbb{K}, \forall u, v \in E, \lambda \cdot(u+v)=\lambda \cdot u+\lambda \cdot v$
    • $\forall \lambda, \mu \in \mathbb{K}, \forall u \in E, (\lambda+\mu) \cdot u=\lambda \cdot u+\mu \cdot u$
    • $\forall u \in E, 1_{\mathbb{K}} \cdot u=u$
• Examples
  • $\mathbb{K}^n$ is a vector space over $\mathbb{K}$
  • $M_{m, n}(\mathbb{K})$ is a vector space over $\mathbb{K}$
  • $\mathbb{K}[X]$ is a vector space over $\mathbb{K}$
  • The set of continuous functions from $\mathbb{R}$ to $\mathbb{R}$ is a vector space over $\mathbb{R}$
• Vector Subspaces
  • A subset $F$ of a vector space $E$ is a vector subspace of $E$ if:
    • $0_E \in F$
    • $\forall u, v \in F, u+v \in F$
    • $\forall \lambda \in \mathbb{K}, \forall u \in F, \lambda \cdot u \in F$
• Linear Combinations
  • Let $u_1, \ldots, u_n \in E$. A linear combination of $u_1, \ldots, u_n$ is a vector of the form $\lambda_1 u_1+\cdots+\lambda_n u_n$, where $\lambda_1, \ldots, \lambda_n \in \mathbb{K}$.
• Vector Subspace Spanned
  • Let $u_1, \ldots, u_n \in E$. The vector subspace generated by $u_1, \ldots, u_n$, denoted $\operatorname{Vect}\left(u_1, \ldots, u_n\right)$, is the set of all linear combinations of $u_1, \ldots, u_n$.
  • It is the smallest vector subspace of $E$ containing $u_1, \ldots, u_n$.
• Free Families, Generating Families, Bases
  • A family $\left(u_1, \ldots, u_n\right)$ of vectors from $E$ is said to be free if $\forall \lambda_1, \ldots, \lambda_n \in \mathbb{K}$, $\lambda_1 u_1+\cdots+\lambda_n u_n=0_E \Rightarrow \lambda_1=\cdots=\lambda_n=0$.
  • A family $\left(u_1, \ldots, u_n\right)$ of vectors from $E$ is said to be generating if $\operatorname{Vect}\left(u_1, \ldots, u_n\right)=E$.
  • A family $\left(u_1, \ldots, u_n\right)$ of vectors from $E$ is a basis of $E$ if it is free and generating.

Dimension

• If $E$ admits a finite basis, then all bases of $E$ have the same number of elements.This number is called the dimension of $E$, denoted $\operatorname{dim} E$.

Theorem of the Incomplete Basis

• Let $E$ be a vector space of finite dimension. Then any free family of $E$ can be completed into a basis of $E$.

Rank of a Family of Vectors

• The rank of a family of vectors $\left(u_1, \ldots, u_n\right)$ from $E$ is the dimension of the vector subspace generated by $u_1, \ldots, u_n$.

Linear Applications

• An application $f: E \rightarrow F$, where $E$ and $F$ are vector spaces over $\mathbb{K}$, is linear if:
  • $\forall u, v \in E, f(u+v)=f(u)+f(v)$.
  • $\forall \lambda \in \mathbb{K}, \forall u \in E, f(\lambda u)=\lambda f(u)$.
• Examples
  • The null application $f: E \rightarrow F$ defined by $f(u)=0_F$ is linear.
  • The identity $\operatorname{id}_E: E \rightarrow E$ defined by $\operatorname{id}_E(u)=u$ is linear.
  • The derivation $D: \mathbb{K}[X] \rightarrow \mathbb{K}[X]$ defined by $D(P)=P^{\prime}$ is linear.
  • The integration $I: \mathbb{K}[X] \rightarrow \mathbb{K}[X]$ defined by $I(P)(X)=\int_0^X P(t) d t$ is linear.
• Kernel and Image
  • The kernel of $f$ is $\operatorname{Ker} f=\left{u \in E \mid f(u)=0_F\right}$. It is a vector subspace of $E$.
  • The image of $f$ is $\operatorname{Im} f={f(u) \mid u \in E}$. It is a vector subspace of $F$.
• Rank Theorem
  • If $E$ is of finite dimension, then $\operatorname{dim} E=\operatorname{dim} \operatorname{Ker} f+\operatorname{dim} \operatorname{Im} f$.
• Injective, Surjective, Bijective Linear Applications
  • $f$ is injective if and only if $\operatorname{Ker} f=\left{0_E\right}$.
  • $f$ is surjective if and only if $\operatorname{Im} f=F$.
  • $f$ is bijective if and only if $f$ is injective and surjective. In this case, $f^{-1}: F \rightarrow E$ is linear.
• Isomorphisms
  • A bijective linear application is called an isomorphism. If $f: E \rightarrow F$ is an isomorphism, then $E$ and $F$ are said to be isomorphic.
• Endomorphisms
  • A linear application from $E$ to $E$ is called endomorphism.
• Automorphisms
  • A bijective endomorphism is called automorphism.

Euclidean vector spaces

• Scalar Product
  • A scalar product on a real vector space $E$ is an application $\langle\cdot, \cdot\rangle: E \times E \rightarrow \mathbb{R}$ that satisfies the following properties:
    • Symmetry
    • Linearity on the left
    • Positive Defined
• Euclidean Vector Space
  • A Euclidean vector space is a real vector space provided with a scalar product.
• Norm
  • The norm of a vector $u \in E$ is defined by $|u|=\sqrt{\langle u, u\rangle}$.
• Cauchy-Schwarz Inequality
  • $\forall u, v \in E,|\langle u, v\rangle| \leq|u| \cdot|v|$.
• Orthogonality
  • Two vectors $u$ and $v$ are said to be orthogonal if $\langle u, v\rangle=0$. We denote $u \perp v$.
• Orthogonal Families, Orthonormal Families
  • A family $\left(u_1, \ldots, u_n\right)$ of vectors from $E$ is said to be orthogonal if $\forall i \neq j, u_i \perp u_j$.
  • A family $\left(u_1, \ldots, u_n\right)$ of vectors from $E$ is said to be orthonormal if it is orthogonal and if $\forall i,\left|u_i\right|=1$.
• Gram-Schmidt Process
  • Let $\left(u_1, \ldots, u_n\right)$ be a basis of $E$. We can construct an orthonormal basis $\left(e_1, \ldots, e_n\right)$ of $E$ as follows:
    • $e_1=\frac{u_1}{\left|u_1\right|}$
    • $e_2=\frac{u_2-\left\langle u_2, e_1\right\rangle e_1}{\left|u_2-\left\langle u_2, e_1\right\rangle e_1\right|}$
    • $e_3=\frac{u_3-\left\langle u_3, e_1\right\rangle e_1-\left\langle u_3, e_2\right\rangle e_2}{\left|u_3-\left\langle u_3, e_1\right\rangle e_1-\left\langle u_3, e_2\right\rangle e_2\right|}$
• Orthogonal Projections
  • Let $F$ be a vector subspace of $E$. The orthogonal projection of $u \in E$ on $F$ is the vector $p_F(u) \in F$ such that $u-p_F(u) \perp F$.
  • If $\left(e_1, \ldots, e_k\right)$ is an orthonormal basis of $F$, then $p_F(u)=\sum_{i=1}^k\left\langle u, e_i\right\rangle e_i$.
• Hermitian Vector Spaces
  • A Hermitian vector space is a complex vector space provided with a Hermitian scalar product. The properties are similar to the Euclidean vector spaces, but one has to pay attention to the complex conjugation.

Statistical Annex

• The statistical data relates to populations, economies, and labor markets in several countries for the years 2019-2022.
• Countries include: Albania, Bosnia and Herzegovina, North Macedonia, Montenegro, and Serbia.

Population

• Population is recorded in thousands of people.
  • All countries recorded small drop in population between 2019 and 2022

GDP

• GDP is recorded in millions of current US dollars.
  • All countries recorded GDP growth between 2019 and 2022

GDP Growth Rate

• GDP growth rate is recorded as an annual percentage.

Unemployment Rate

• Unemployment rate is recorded as an ILO estimate.

Introduction

Definition 1.1.1

• Linear equations with $n$ unknowns $x_1, x_2, ..., x_n$ take the form $a_1x_1 + a_2x_2 +... + a_nx_n = b$, where $a_1, a_2,..., a_n, b$ are given constants.

Definition 1.1.2

• A system of $m$ linear equations with $n$ unknowns is a set of $m$ linear equations, each with $n$ unknowns.

Definition 1.1.3

• A solution to a system of linear equations is a list of $n$ numbers $s_1, s_2,..., s_n$ that satisfy each equation in the system when substituted for $x_1, x_2,..., x_n$ respectively.
  • The set of all possible solutions is called the solution set.

Definition 1.1.4

• A system of linear equations is compatible if it has at least one solution, and incompatible if it has no solutions.

Solving Systems of Linear Equations

Definition 1.2.1

• Two systems of linear equations are equivalent if they have the same solution set.
• To solve a system of linear equations, the goal is to transform it into a simpler equivalent system where the solution is evident.

Proposition 1.2.2

• The following operations transform a system of linear equations into an equivalent system:
  1. Swapping two equations.
  2. Multiplying an equation by a non-zero constant.
  3. Adding a multiple of one equation to another equation.