Gabriel

... y Café ☕

[Libertario 🟨⬛ @QuotesforGoal]

“Se empieza por la idea que el Estado debe dirigir todo; que es mejor que haya una corporación que dirija las cosas, y no que todo «quede abandonado al caos, o a circunstancias individuales»; y se llega al nazismo o al comunismo, claro”

Jorge Luis Borges

[Estado Mínimo @xestadominimo]

“El fraude más antiguo es la creencia de que la izquierda política es la ideología de los pobres y los oprimidos”.

- Thomas Sowell -

[Gabriel @Gbrlmb]

Remplace the calculators for AI

[Math Files @Math_files]

This 1986 protest literally happened. Teachers marched in the streets warning that calculators would ruin an entire generation’s ability to think.

[MajoMéndezCasariego ⚔️ @MJMendezCasarie]

“Ante todo, el gobierno no tiene dinero propio. Solo las personas tienen dinero. El gobierno únicamente obtiene dinero metiendo la mano en tu bolsillo y sacándolo de allí.”
— Milton Friedman

[The Rabbit Hole @TheRabbitHole]

The term far right has become so broadly applied that it includes normal people now.

[Gabriel @Gbrlmb]

Quiero una camiseta con este estampado

[Mathematica @mathemetica]

Quantum mechanics uses specialized symbols to describe the behavior of matter and energy at microscopic scales.

An alphabetical guide highlights core concepts like angular momentum, bound states, the fine structure constant 1/137, photon energy hν, the Schrödinger equation EΨ=ĤΨ, reduced mass mM/(M+m), Laplacian ∇², and uncertainty ΔxΔp.

These ideas are used to build the transistors in every computer chip and the lasers in fiber optic networks that connect the world.

[Made in Ancapia @MadeInAncapia]

✒️| "Yo quiero el estado más reducido posible (si puede ser, inexistente) y que me dejen vivir tranquilo."

– Miguel Anxo Bastos

[ScieVision @scievision369]

Quantum harmonic oscillator ✍️

It is a key system in quantum mechanics and is more than just a textbook example. It describes any situation where a particle feels a restoring force that pulls it back to an equilibrium position, similar to a mass on a spring. Its importance in modern physics lies in the fact that nearly every quantum system near its equilibrium can be seen as a collection of these oscillators. This model serves as the basis for quantum field theory, solid-state physics, molecular physics, and quantum optics all at once.

The large bowl-shaped curve in the image shows the potential energy of the oscillator related to its position. The lowest point of the bowl has the minimum potential energy, which is where a classical particle would rest. Moving away from the center, the potential energy increases sharply, creating walls that keep the particle inside. This bowl shape represents any harmonic oscillator and appears whenever a restoring force is proportional to displacement, whether around an atom in a crystal lattice, along a vibrating chemical bond, or within an electromagnetic field in a small cavity, among other systems in nature.

The horizontal lines inside the bowl, labeled from n equals zero at the bottom to n equals six, represent the quantized energy levels, which are the only energies allowed for the quantum oscillator. Unlike a classical spring that can have any energy in a continuous range, the quantum oscillator has a discrete set of possibilities. It cannot have an energy value between two adjacent levels. The remarkable aspect of this ladder is that its rungs are evenly spaced; the gap between any two adjacent levels is the same throughout. This equal spacing means the oscillator always absorbs or emits the same fixed amount of energy in any transition, forming the basis of the quantum description of light.

The lowest energy level does not rest at the bottom of the bowl. Instead, it hovers slightly above it, representing a small but non-zero energy. This is one of the most thought-provoking aspects of quantum mechanics. A particle at the center of the well would have both a perfectly defined position and momentum, something the Heisenberg uncertainty principle prohibits. Therefore, the particle must maintain a minimum level of motion known as zero-point motion, even at its lowest energy state. Nothing in the universe, not even things cooled to absolute zero, is ever completely at rest. This is not merely a theoretical point but a measurable fact with confirmed effects, like the Casimir effect, where two uncharged metal plates close together in a vacuum pull toward each other due to these quantum fluctuations in the electromagnetic field.

At each energy level, the wave-like curves inside the bowl represent the wave function of the particle. This wave function fully describes its quantum state, with the squared magnitude at any position giving the probability of finding the particle there. The ground state wave function at n equals zero is the simplest form, showing a single-humped bell curve centered in the bowl, with the particle most likely to be found near the center. As you ascend the energy ladder, the wave functions become increasingly complex, each step adding one more zero-crossing and another hump. By n equals six, the wave function has six crossing points and seven lobes. This rising complexity corresponds to increased energy; more oscillations lead to shorter wavelengths and higher momentum and kinetic energy.

[Physics In History @PhysInHistory]

Werner Heisenberg's canonical commutation rule for position and momentum variables of a particle, 1927.

[Math Files @Math_files]

In mathematics, silence can be the most beautiful proof.

[Math Files @Math_files]

For people who hate math

[Science Simplified @Scivf4]

Mt. Everest vs. the tallest mountain in the solar system.

Olympus Mons on Mars.

[Milton Friedman Quotes @MiltonFriedmanW]

“We must come to recognize that voluntary cooperation, through the market and in other ways, is a far better way to solve our problems than turning them over to the government.”

— Milton Friedman

[Philosophy Of Physics @PhilosophyOfPhy]

Quantum mechanics did not merely add new equations to physics. It changed what physicists believed reality could be.

Energy became quantized. Light and matter refused to remain only particles or only waves. Superposition replaced a single definite state with several possible outcomes. Heisenberg showed that uncertainty is not simply poor measurement, but part of the structure of nature itself.

Probability entered the foundations of physics. Entanglement revealed correlations with no classical equivalent. Quantum statistics explained why atoms have structure, why chemistry works, and why ordinary matter does not collapse.

These ideas are strange, but they are not philosophical decoration. Together, they form one of the most precisely tested frameworks in science.

Quantum mechanics taught us something deeply unsettling and beautiful.

"Nature is not required to behave according to the categories human intuition finds comfortable."

[ScieVision @scievision369]

Central Limit Theorem (CLT) ✍️

It often called the "CLT", is one of the most powerful and surprising ideas in statistics. In simple terms, it states that when we take many random samples from a population and calculate the average of each sample, those averages will form a bell-shaped curve, known as the normal distribution, as long as the samples are large enough. This is remarkable because it happens regardless of the original population's shape. Whether the population is skewed, lopsided, flat, or has unusual bumps, averaging smooths everything out and creates the familiar, symmetric bell shape. This is why the normal distribution often appears in nature and data, and why the CLT is seen as fundamental to modern statistical thought.

To grasp how it works, imagine a large population perhaps the heights of all people in a country, the incomes of all households, or the test scores of all students. This population has its own true average and spread, but its overall shape could be anything. Since studying every individual is not practical, you take a random sample of a certain size and calculate its average. You repeat this many times, taking sample after sample and recording each average. If you then examine all those averages and plot them, you will see something amazing: they arrange themselves into a beautiful, symmetric bell-shaped curve, centered around the true population average. This collection of averages is called the sampling distribution of the sample mean, and the CLT ensures it will be approximately normal whenever the sample size is large enough.

The theorem also provides three important insights about these sample averages. First, the averages tend to center around the true population mean, meaning if you keep sampling and averaging, your results will generally hit the correct target; the sample mean reflects reality fairly well. Second, the spread of these averages gets smaller as the sample size increases, so larger samples produce averages that cluster more closely around the true value. Third, the shape of the distribution of averages becomes approximately normal, no matter how strange the original population may look. Together, these insights explain why larger samples give more reliable estimates and why averages behave more predictably than individual data points.

The real beauty of the CLT lies in its universality. It does not matter if the population is symmetric, skewed, uniform, or has outliers as long as the samples are large enough and drawn randomly, the averages will behave in a predictable, bell-shaped way. A simple example illustrates this: imagine rolling a single die, where each outcome from one to six is equally likely, creating a flat and not bell-shaped distribution. But if you roll ten dice at once, calculate their average, and repeat this thousands of times, you will see a beautiful bell curve centered around 3.5. Even though individual rolls are unpredictable and evenly spread, the averaging process creates order, symmetry, and predictability. It’s akin to mixing many different colors of paint any single splash might be wild and unique, but when many colors combine, the result tends to be a smooth, predictable hue.

A common rule of thumb is that a sample size of about thirty or more is often enough for the CLT to work well, though this depends on the population's shape. If the population is roughly symmetric, even small samples can suffice; if it is highly skewed or has extreme values, you may need much larger samples sometimes fifty, one hundred, or more before the averages begin to resemble a proper bell curve. The more irregular the population, the larger the sample needed for the theorem's effects to be noticeable.

[Black Hole @konstructivizm]

A hidden side of our planet that most people never truly see: the vast, endless Pacific Ocean.Covering nearly one-third of Earth’s surface, the Pacific is so immense that it could swallow all the world’s continents and still have room to spare. It stretches from the Arctic to Antarctica and from Asia to the Americas — an endless blue frontier that remains one of the most mysterious and powerful places on our planet.Beneath its surface lies a world of hidden volcanoes, deep trenches plunging 11 kilometers into darkness, thriving coral kingdoms, and creatures found nowhere else on Earth. Yet for most of us, it’s just “that big ocean on the map.”Time to give this magnificent giant the attention it deserves.

[Philosophy Of Physics @PhilosophyOfPhy]

Paul Dirac in a letter to Erwin Schrödinger 1930s.

"The mathematics has a life of its own. When you write down the equations, you are not inventing a language; you are translating a message that is already there. There are times when the symbols on the page seem to guide my hand rather than the other way around. It is a frightening feeling, because you realize that the truth does not care if you understand it or not. It simply exists, cold and perfect, waiting for someone to look."

(📷 Erwin Schrödinger - R, Paul Dirac - L )