Quantum Physics: Timeline and Key Concepts

Quantum Physics

• Mathematical description of matter and energy at the atomic and subatomic levels.
• Studies atoms and their constituent particles.

Quantum Theory Timeline

• 1900: Max Planck proposed energy is quantized.
• 1905: Albert Einstein explained photoelectric effect, introducing photons.
• 1913: Niels Bohr developed atomic model with quantized energy levels.
• 1924: Louis de Broglie proposed matter has wave-like properties.
• 1925-1926: Werner Heisenberg and Erwin Schrödinger independently developed quantum mechanics.
• 1927: Heisenberg articulated the uncertainty principle.
• 1930s: Quantum mechanics applied to nuclear and solid-state physics
• 1940s: Quantum electrodynamics (QED) developed.
• 1950s: Quantum field theory (QFT) further developed.
• 1960s: The Standard Model of particle physics began to take shape.
• Present: Ongoing research in quantum gravity, quantum computing, and quantum information theory.

Key Concepts

• Quantization
  • Energy comes in discrete packets called "quanta."
• Wave-Particle Duality
  • Particles exhibit both wave-like and particle-like properties.
  • Light behaves as both a wave and a stream of photons.
• Superposition
  • Quantum system can exist in multiple states simultaneously.
  • The system "collapses" into one definite state upon measurement.
• Uncertainty Principle
  • There is a fundamental limit to the precision with which certain pairs of physical properties, such as position and momentum, can be known simultaneously.
• Quantum Entanglement
  • Two or more particles linked to share the same fate, no matter how far apart.
  • Measuring one particle instantaneously affects the state of the others.

Core Equations

• Planck's Equation
  • $E = h\nu$
    • E = energy of the quantum
    • h = Planck's constant ($6.626 \times 10^{-34} Js$)
    • ν = frequency of the radiation
• de Broglie Wavelength
  • $\lambda = \frac{h}{p} = \frac{h}{mv}$
    • λ = wavelength
    • p = momentum
    • m = mass
    • v = velocity
• Heisenberg's Uncertainty Principle
  • $\Delta x \Delta p \geq \frac{\hbar}{2}$
    • Δx = uncertainty in position
    • Δp = uncertainty in momentum
    • ℏ = reduced Planck's constant ($\frac{h}{2\pi}$)
• Schrödinger Equation
  • $i\hbar\frac{\partial}{\partial t}\Psi(r, t) = \hat{H}\Psi(r, t)$
    • i = imaginary unit
    • Ψ(r, t) = wave function of the quantum system
    • Ĥ = Hamiltonian operator, corresponding to the total energy of the system

Applications of Quantum Physics

• Quantum Computing
  • Harnessing quantum phenomena like superposition and entanglement for complex calculations.
• Quantum Cryptography
  • Using quantum mechanics to develop secure communication methods.
• Quantum Sensors
  • Developing highly sensitive sensors for measuring magnetic fields, time, and gravity.
• Materials Science
  • Designing new materials with specific properties by understanding and manipulating quantum behavior.
• Medical Imaging
  • Improving medical imaging techniques like MRI and PET scans.

Algèbre linéaire

• Linear Algebra

Vecteurs

• Vectors

Définition

• A vector is an element of a vector space.
• Defined by:
  • A direction (a straight line)
  • A sense (one of the two directions of movement on the line)
  • A norm (a length)

Représentation

• Often represented by an arrow.
  • The direction is given by the line supporting the arrow, the sense by the direction of the arrow, the norm by the length of the arrow.

Opérations sur les vecteurs

• Operations on vectors

Somme de deux vecteurs

• Sum of two vectors
  • To add two vectors $\overrightarrow{u}$ and $\overrightarrow{v}$, place the origin of vector $\overrightarrow{v}$ at the end of vector $\overrightarrow{u}$.
  • The vector $\overrightarrow{u} + \overrightarrow{v}$ is the vector whose origin is the origin of $\overrightarrow{u}$ and whose end is the end of $\overrightarrow{v}$.

Multiplication d'un vecteur par un scalaire

• Multiplication of a vector by a scalar
  • To multiply a vector $\overrightarrow{u}$ by a scalar $k$, multiply the norm of $\overrightarrow{u}$ by $|k|$.
  • If $k > 0$, the direction of $k\overrightarrow{u}$ is the same as that of $\overrightarrow{u}$.
  • If $k < 0$, the direction of $k\overrightarrow{u}$ is opposite to that of $\overrightarrow{u}$.

Produit scalaire

• Scalar product
  • The scalar product of two vectors $\overrightarrow{u}$ and $\overrightarrow{v}$ is a scalar defined by:
    • $\qquad \overrightarrow{u} \cdot \overrightarrow{v} = ||\overrightarrow{u}|| \cdot ||\overrightarrow{v}|| \cdot \cos(\theta)$
    • where $\theta$ is the angle between the two vectors.

Produit vectoriel

• Vector product
  • The vector product of two vectors $\overrightarrow{u}$ and $\overrightarrow{v}$ is a vector $\overrightarrow{w}$ defined by:
    • Its direction is perpendicular to the plane formed by $\overrightarrow{u}$ and $\overrightarrow{v}$.
    • Its direction is such that the trihedron $(\overrightarrow{u}, \overrightarrow{v}, \overrightarrow{w})$ is direct (right-hand rule).
    • Its norm is $||\overrightarrow{w}|| = ||\overrightarrow{u}|| \cdot ||\overrightarrow{v}|| \cdot \sin(\theta)$ where $\theta$ is the angle between the two vectors.

Comparaison de nombres complexes

• Comparison of complex numbers

Définition

• Definition
  • Let two complex numbers $z = a + bi$ and $z' = a' + b'i$ where $a, a', b, b' \in \mathbb{R}$.
  • One cannot compare $z$ and $z'$ if $b \neq 0$ and $b' \neq 0$.
  • If $b = 0$ and $b' = 0$, then $z, z' \in \mathbb{R}$ and in this case, one can compare $z$ and $z'$.

Exemples

• Examples
  • $3 + 2i$ and $1 - i$ are not comparable.
  • $5$ and $7$ are comparable, and we have $5 < 7$.
  • $-2i$ and $4i$ are not comparable.
  • $\sqrt{2}$ and $\pi$ are comparable, and we have $\sqrt{2} < \pi$.

Interprétation Géométrique

• Geometric Interpretation
  • In the complex plane, a complex number $z = a + bi$ is represented by a point with coordinates $(a, b)$.
  • The comparison of complex numbers only makes sense if these points are located on the real axis (x-axis).

Remarque

• Note
  • The notion of order (>,