Quantum Mechanics: History and Key Concepts

Quantum Mechanics

• Quantum mechanics is physics focused on matter and light behavior at the atomic, subatomic level.

History

• 1900: Max Planck introduced quantization to explain blackbody radiation.
• 1905: Albert Einstein explained the photoelectric effect with photons.
• 1913: Niels Bohr developed an atomic model with quantized energy levels.
• 1924: Louis de Broglie proposed matter has wave-like properties.
• 1925: Werner Heisenberg and Erwin Schrödinger created matrix mechanics and wave mechanics.
• 1927: Heisenberg articulated the uncertainty principle.
• 1928: Paul Dirac combined quantum mechanics with special relativity.

Key Concepts

Quantization

• Physical quantities can only take discrete values.
• Energy is the product of Planck's constant and frequency ($E = h\nu$).

Wave-Particle Duality

• Particles act as waves and vice versa.
• Wavelength is Planck's constant divided by momentum ($\lambda = \frac{h}{p}$).

Uncertainty Principle

• Position and momentum cannot both be perfectly known.
• The product of the uncertainties in position and momentum is greater than or equal to the reduced Planck constant divided by two ($\Delta x \Delta p \geq \frac{\hbar}{2}$).

Superposition

• Quantum systems exist in multiple states until measured.
• State vector is a linear combination of component states ($|\psi\rangle = \alpha |\psi_1\rangle + \beta |\psi_2\rangle$).

Entanglement

• Particles are linked, so one particle's state instantly affects the other, regardless of distance.

Core Equations

Schrödinger Equation

• Describes quantum state changes over time.
• $i\hbar \frac{\partial}{\partial t} |\psi(t)\rangle = \hat{H} |\psi(t)\rangle$

Applications

• Lasers: Uses stimulated emission to produce coherent light.
• Transistors: Uses quantum mechanical properties of semiconductors.
• Medical Imaging: MRI relies on nuclear spin.
• Quantum Computing: Computations performed using superposition and entanglement.
• Atomic Clocks: Timekeeping using atomic transition frequency.
• Quantum Cryptography: Data encryption using quantum mechanics.
• Material Science: New materials designed based on quantum mechanical principles.
• Nanotechnology: Matter is manipulated at atomic scale using quantum mechanics.
• Solar Cells: Convert sunlight into electricity using quantum mechanics.
• Chemical Reactions: Electronic structure of molecules are modeled.
• Superconductors: Electricity conducted with no resistance.
• Particle Physics: Fundamental particles are investigated.
• Quantum Sensors: Highly sensitive sensors measure physical quantities with precision.
• Quantum Metrology: Improving measurement precision using quantum techniques.
• Quantum Communication: Confidentiality ensured exploiting quantum entanglement.
• Quantum Simulation: Quantum systems used to simulate other quantum systems.
• Quantum Imaging: Image resolution and sensitivity are enhanced.
• Quantum Radar: Radar performance improved.
• Quantum Biology: Quantum phenomena investigated in biological systems.
• Quantum Optometry: Accuracy of optical measurements improved.
• Quantum Finance: Models and algorithms used to improve financial analysis.
• Quantum Art: Artistic expression explored using quantum concepts.
• Quantum Education: Quantum-related concepts taught through gamification and simulations.
• Quantum Philosophy: Philosophical implications of quantum mechanics explored.
• Quantum Law: Potential legal and ethical implications of quantum technologies examined.
• Quantum Social Science: Models and concepts used to analyze social phenomena.
• Quantum Wellness: Practices inspired to promote emotional well-being.
• Quantum Agriculture: Optimized applying quantum mechanics principles.
• Quantum Ecology: Ecological systems studied using models and techniques.

Algèbre Linéaire et Géométrie Analytique I

Introduction

• A system of $m$ linear equations with $n$ unknowns $x_1, x_2,..., x_n$ is a set of equations of the form:

$\qquad \begin{cases} \begin{aligned} &a_{11}x_1 + a_{12}x_2 +... + a_{1n}x_n = b_1 \ &a_{21}x_1 + a_{22}x_2 +... + a_{2n}x_n = b_2 \ &\vdots \ &a_{m1}x_1 + a_{m2}x_2 +... + a_{mn}x_n = b_m \end{aligned} \end{cases}$

• Where $a_{ij}, b_i \in \mathbb{R}$ for $i = 1,..., m$ and $j = 1,..., n$.
• A solution is an n-tuple $(s_1, s_2,..., s_n) \in \mathbb{R}^n$ that satisfies all equations when $x_1 = s_1, x_2 = s_2,..., x_n = s_n$.

Examples

• System with solution $(x_1, x_2) = ( \frac{2}{5}, \frac{7}{5} )$: $\qquad \begin{cases} \begin{aligned} &2x_1 + 3x_2 = 5 \ &x_1 - x_2 = -1 \end{aligned} \end{cases}$
• System with no solutions: $\qquad \begin{cases} \begin{aligned} &x_1 - x_2 = 1 \ &2x_1 - 2x_2 = 3 \end{aligned} \end{cases}$
• System with infinite solutions: $\qquad \begin{cases} \begin{aligned} &x_1 + x_2 + x_3 = 3 \end{aligned} \end{cases}$
• Examples include $(1, 1, 1)$ and $(5, -1, -1)$.

Matrix Representation

• A system of equations can be represented in matrix form, $Ax = b$
• $A$ is the coefficient matrix, with the unknowns vector as $x$ and of constants as $b$
• Augmented Matrix: The matrix $[A | b]$ is formed by adding column $b$ to $A$

Definition 1.1

• Equivalent Systems: Two systems are equivalent if they share the same set of solutions.

Autovalores y Autovectores

• Discusses linear operators transforming vectors and introduces the idea of vectors retaining direction.

Definition: Eigenvalues and Eigenvectors

• For a linear operator $T: V \rightarrow V$ on a vector space $V$, a nonzero vector $v \in V$ is an eigenvector if $T(v) = \lambda v$ for some scalar $\lambda$
• The scalar $\lambda$ is the eigenvalue of $T$ corresponding to $v$
• An eigenvector is scaled by $\lambda$ when $T$ is applied
• The eigenvalue determines the amount of scaling

Examples

• $T(x, y) = (x + y, x - y)$: Vector (1, 1) is not an eigenvector because T(1, 1) = (2, 0) is not a scalar multiple of (1, 1).
• $T(x, y) = (2x, 2y)$: Vector (1, 0) is an eigenvector with eigenvalue 2, since T(1, 0) = (2, 0) = 2(1, 0).

Observations

• Eigenvectors must be nonzero because T(0) = 0 for any $\lambda$.
• Scalar multiples of eigenvectors are also eigenvectors with the same eigenvalue: T(cv) = cT(v) = c(λv) = λ(cv)

Calculating Eigenvalues and Eigenvectors

• Find the matrix A representing T
• Solve the characteristic equation det(A - λI) = 0, where I is the identity matrix.
• For each λ, solve (A - λI)v = 0 to find eigenvectors $v$.

Example

• If $T(x, y) = (2x + y, x + 2y)$, the matrix is $A = \begin{pmatrix} 2 & 1 \ 1 & 2 \end{pmatrix}$
• $(\lambda - 1)(\lambda - 3) = 0$. The eigenvalues are 1 and 3. Find autovectors for $\lambda_1 = 1$: $\begin{pmatrix} 1 & 1 \ 1 & 1 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 0 \ 0 \end{pmatrix}$ So the autovector is v = (x, -x) with $\lambda_1 =1$
• Repeat for $\lambda_2 = 3$ to find autovectors $v = (x, x)$ with $\lambda_2 =3$

Applications

• Many diverse applications lie in stability analysis of dynamic systems, vibrations, quantum mechanics, image processing, and social networks.

Diagonalization

Definition: Diagonalizable Matrix

• A diagonalizable matrix $A$ of $n \times n$ has $P^{-1}AP$ as a diagonal matrix for some invertible matrix $P$

Theorem

• A matrix $A$ of $n \times n$ is diagonalizable if and only if it has $n$ linearly independent eigenvectors

Diagonalizing a Matrix

• Find the eigenvalues of A.
• For each eigenvalue λ, find a basis.
• If the aggregate bases yields $n$ vectors, then A is diagonalizable.
• Form the matrix $P$ with the linearly independent eigenvectors as columns.
• Form the diagonal matrix $D$ with the eigenvalues corresponding to the eigenvectors in $P$.
• Given the above, $P^{-1}AP = D$.

Example

• Diagonalize the matrix* $A = \begin{pmatrix} 2 & 1 \ 1 & 2 \end{pmatrix}$.
• The eigenvalues are 1 and 3. Because A has linearly independent eigenvectors in $\mathbb{R}^2$, it is diagonalizable.* Matrix $P = \begin{pmatrix} 1 & 1 \ -1 & 1 \end{pmatrix}$

Diagonal matrix $D = \begin{pmatrix} 1 & 0 \ 0 & 3 \end{pmatrix}$

• Calculate to get $P^{-1}AP = \begin{pmatrix} 1 & 0 \ 0 & 3 \end{pmatrix} = D$*