Quantum Physics | Tale of Space

Max Planck

Niels Bohr

Werner Heisenberg

Quantum Physics

The bizarre world of the infinitely small, where everyday rules cease to apply and particles defy our intuition

1900-1930 - Quantum Revolution

Quantum physics describes the behavior of matter and energy at the atomic and subatomic scale. Born at the beginning of the 20th century, it revolutionized our understanding of nature by revealing a world where particles can be in several places at once, where observation changes reality, and where chance is fundamental.

Wave-Particle Duality

Any object—light, electrons, atoms—can behave like a wave OR like a particle, depending on the experiment performed. This is one of the most confusing concepts in physics.

λ = h / p = h / (m × v)

De Broglie wavelength associated with any particle

Young's double-slit experiment

Electrons pulled one by one toward two slits create an interference pattern, like waves on water. But if we observe which slit each electron passes through, the pattern disappears and we obtain localized impacts like marbles.

What this means

The electron is neither a "wave" nor a "particle" but a quantum object with both properties.
Observation itself changes the behavior of the system.
This duality applies to all matter, not just light.

De Broglie's postulate (1924)

Every particle with momentum p has an associated wavelength. This bold hypothesis was verified experimentally by the diffraction of electrons on crystals (Davisson-Germer, 1927).

De Broglie wavelength: λ = h / p = h / (m × v) Einstein relation (energy-frequency): E = h × f = ℏ × ω Fundamental relations: p = ℏ × k (k = wave vector) E = ℏ × ω

Wave packet

A localized particle is represented mathematically by a superposition of plane waves (Fourier transform). The more localized the particle is (small Δx), the wider the distribution in k (large Δk) — this is the mathematical origin of the uncertainty principle.

Heisenberg's Uncertainty Principle

The fundamental limits of knowledge

Δx · Δp ≥ ℏ/2

It is impossible to know both position and velocity with absolute precision at the same time.

What this really means

It is not a problem of insufficient measurement or technology. The particle simply does not have a precise position and velocity at the same time. This is a fundamental property of nature, not a limitation of our instruments.

Concrete example

If an electron is located very precisely (small Δx), then its velocity becomes very uncertain (large Δp). This is why electrons do not fall onto the atomic nucleus: confined near the nucleus, their velocity becomes enormous and they "escape."

General Form (conjugated observables)

ΔA × ΔB ≥ (1/2) × |⟨[A, B]⟩| where [A, B] = AB - BA (commutator) Position-momentum case: [x, p_x] = iℏ ⇒ Δx · Δp_x ≥ ℏ/2 Energy-time case: ΔE · Δt ≥ ℏ/2

Numerical Examples

Electron in an atom (a₀ = 0.5 Å): Δx = 0.5 × 10⁻¹⁰ m, therefore Δv = 10⁶ m/s (typical velocity in the atom)
Proton in a nucleus (R = 1 fm): Δx = 10⁻¹⁵ m, therefore kinetic energy = 3 MeV (consistent with binding energies)

Consequence: Vacuum fluctuations

The energy-time relationship allows for the creation of short-lived particle-antiparticle pairs if ΔE × Δt is less than or equal to ℏ. These quantum vacuum fluctuations are the source of Hawking radiation from black holes.

Quantum Probabilities

Chance at the heart of nature

|ψ(x,t)|² = probability

The square of the wave function gives the probability of presence.

Wave function

The state of a particle is described by a "wave function" ψ. This function is not directly observable, but its square |ψ|² gives the probability of finding the particle at a given location.

Overlapping and collapse

Before measurement: The particle is in a superposition of all possible states simultaneously.
During measurement: The wave function "collapses" into a single random result.
We can only predict probabilities, never certainties.

Schrödinger equation

Time-dependent equation: iℏ · ∂ψ/∂t = Ĥ · ψ where Ĥ = -ℏ²/(2m) · ∇² + V(r) (Hamiltonian) Time-independent equation (stationary states): Ĥ · ψ = E · ψ

Born's interpretation (1926)

The probability density of presence is given by ρ(r,t) = |ψ(r,t)|². The wave function must be normalized: the integral of |ψ|² over the entire space is 1.

Unitary time evolution

ψ(t) = Û(t) · ψ(0) = exp(-iĤt/ℏ) · ψ(0) Unitary Û ⇒ conservation of probability ⇒ information preserved (quantum unitarity)

Pauli exclusion principle

Why matter does not collapse

The principle

Two identical fermions (electrons, protons, neutrons) cannot occupy exactly the same quantum state. It is as if each quantum "place" could only accommodate a single occupant.

Consequences

Structure of atoms: electrons distributed in layers (K, L, M, etc.)
Periodic table: all chemistry stems from this principle
Stability of matter: prevents atoms from collapsing

Astrophysical application

White dwarfs are stabilized by electron degeneracy pressure: when compressed, electrons cannot all occupy the same state, which creates pressure that resists gravity. For neutron stars, neutron degeneracy pressure plays this role.

Statement (Pauli, 1925)

Two identical fermions (half-integer spin particles) cannot occupy the same quantum state simultaneously.

Formalism: Antisymmetry of the wave function

For N fermions, the total wave function must be antisymmetric under exchange of two particles: ψ(r₁, r₂, ..., rᵢ, ..., rⱼ, ...) = -ψ(r₁, r₂, ..., rⱼ, ..., rᵢ, ...) If rᵢ = rⱼ (same state) ⇒ ψ = -ψ ⇒ ψ = 0 (impossible)

Chandrasekhar mass

M_Ch = 1.44 M_☉ Beyond this mass, the electron degeneracy pressure is insufficient ⇒ collapse into a neutron star or black hole. Tolman-Oppenheimer-Volkoff limit (neutron stars): M_TOV = 2.0 - 2.5 M_☉

Quantum Tunnel Effect

Crossing the impossible

The phenomenon

A particle can "pass through" an energy barrier even if it does not have enough energy to cross it in the conventional way. It is as if a ball could pass through a wall without having enough energy to go over it.

Explanation

Thanks to its wave properties, the wave function of the particle does not stop abruptly at the barrier but decreases exponentially inside it. If the barrier is thin enough, part of the wave emerges on the other side.

Probability of transmission

For a rectangular barrier with height V₀ and width L: T = exp(-2κL) where κ = √(2m(V₀ - E)) / ℏ The higher or wider the barrier, the lower T is.

Application: Stellar nuclear fusion

Coulomb barrier between two protons: V_Coulomb = 1.4 MeV (at r = 1 fm) Thermal energy at the core of the Sun: E_th = k_B · T = 1.2 keV (T = 15 million K) Classically: E_th ≪ V_Coulomb ⇒ fusion IMPOSSIBLE Quantum: tunnel effect with probability T = exp(-E_G/E_th) where E_G = 500 keV (Gamow energy) T = 10⁻¹⁷⁰ per collision, BUT 10³⁸ collisions/second ⇒ Fusion possible, the Sun shines!

Why stars shine

Without quantum physics, stars could not exist. Here's how three quantum principles make the life of stars possible:

Tunnel effect

Allows protons to fuse despite their electrical repulsion. Without it, the Sun's temperature would be insufficient to trigger nuclear reactions.

Pauli principle

Stabilizes white dwarfs and neutron stars against gravitational collapse, creating quantum pressure that resists gravity.

Energy quantification

Determines the spectral lines of stars, allowing us to know their chemical composition, temperature, and velocity billions of light-years away.

Modern Applications

Quantum physics is not just an abstract theory. It forms the basis of many everyday technologies.

Transistors and electronics

All computers, smartphones, and electronic devices rely on semiconductors that operate purely on quantum principles.

Lasers

Stimulated emission, a quantum phenomenon, makes it possible to create coherent light used in medicine, telecommunications, and industry.

medical MRI

Magnetic resonance imaging uses the quantum spin of hydrogen nuclei to visualize the inside of the human body.

Quantum cryptography

The uncertainty principle guarantees communication security: any attempt at interception alters the message and is detected.

Quantum computers

Superposition and entanglement enable certain calculations to be performed exponentially faster than conventional computers.

GPS and atomic clocks

Quantum transitions in cesium or rubidium atoms provide the most accurate time standard in the world.