"If you think you understand Quantum Mechanics, then you don't understand Quantum Mechanics" -Richard Feynman
(You are not mostly empty space & Quantum Rule That Makes Our Existence Possible)
If you were to look at what your body is made of, at smaller and more fundamental levels, you'd find a whole miniature Universe of structure inside you. Your body is made of organs, which are in turn made of cells, which contain organelles, which are composed of molecules, which themselves are linked-up chains of individual atoms. Atoms exist on extremely tiny scales, just 1 angstrom across, but they're made of even smaller constituents: protons, neutrons, and electrons. The tiny sizes of the protons and neutrons making up each atom's nucleus are known: just one femtometer apiece, 100,000 times smaller than an angstrom. But the electron itself is indistinguishable from point-like, no more than 1/10,000th the size of a proton or neutron. Does this mean that atoms and by extension, everything made of atoms are mostly empty space? Not at all. In our normal experience, if you want to know how big something is, you just go ahead and measure it. For non-quantum objects, this isn't a problem, as different methods of measuring an object all give you the same answer. Whether you use a measuring stick (like a ruler), high-definition imaging, or a physics-reliant technique like Brownian motion or gravitational settling, you'll arrive at identical solutions.
But for the smallest objects of all, like single atoms, these techniques are no longer effective. The first attempt to probe the interior of atoms came shortly after the discovery of radioactivity, and was actually ingenious. By firing the particles emitted by radioactive material at a thin sheet of atoms, Ernest Rutherford attempted to determine what happened when you examined an atom's interiors. What he found shocked the world.These fast-moving particles were fired at a very thin sheet of gold foil, hammered so thin that it would fall apart if touched by bare human hands. While most of the particles went straight through, a small but substantial fraction were deflected, with some even returning back i the reverse of their original direction. This type of technique for measuring the sizes of particles is known as deep inelastic scattering, and is used today to constrain the sizes and measure the properties of fundamental particles inside protons and neutrons. For more than 100 years, from Rutherford to the Large Hadron Collider, this is an important way to measure the sizes of fundamental particles.But these high-energy conditions, where conventional atoms and atomic nuclei are bombarded with particles moving close to the speed of light, are not the conditions that the atoms in our everyday lives typically experience.
We live in a low-energy Universe, where the atoms in our bodies and the collisions that take place between various particles are less than one-billionth the energy of what the Large Hadron Collider reaches. In our quantum Universe, we frequently talk about wave-particle duality, or the idea that the fundamental quanta that make up the Universe exhibit both wave-like and particle-like properties, depending on what conditions they're exposed to. If we go to higher and higher energies, the quanta we're examining act more like particles, while at lower energies, they act more like waves.We can illustrate why by examining the photon: the quantum of energy associated with light. Light comes in a variety of energies, from the ultra-high energy gamma rays down through the ultra-low energy radio waves. But light's energy is closely related to its wavelength: the higher the energy, the shorter the wavelength.The lowest energy radio waves we know about are many meters or even kilometers long, with their oscillating electric and magnetic fields being useful in causing the electrons inside antennae to move back-and-forth, creating a signal that we can use and extract. Gamma rays, on the other hand, can be so high in energy that it takes tens of thousands of wavelengths to fit across even a single proton. If the size of your particle is larger than your wavelength of light, the light can measure its size.
But if your particle is smaller than the light's wavelength, the light won't be able to interact with that particle very well, and will behave like a wave. This is why low-energy photons, like visible light photons, will create an interference pattern when they're passed through a double slit. So long as the slits are large enough that the light's wavelength can get through them, you'll get an interference pattern on the other side, demonstrating this wave-like behavior.This is true even if you send the photons through one-at-a-time, indicating that this wave-like nature isn't occurring between different photons, but that each individual photon is interfering with itself somehow. This remains true even if you replace the photons with electrons, as even massive particles can act like waves under low-energy conditions. Even low-energy electrons sent one-at-a-time through a double slit can add up to produce that interference pattern, demonstrating their wave-like behavior.When we picture an atom, most of us instinctively revert to that first model we all learned: of a point-like electron orbiting a small, dense nucleus. This "planetary model" of the atom first came about due to Rutherford, and was later refined by Niels Bohr and Arnold Sommerfeld, who recognized the need for discrete energy levels.
But for the better part of the past century, we've recognized that these models are too particle-like to describe what's actually occurring. Electrons do occupy discrete energy levels, but that doesn't translate into planetary-like orbits. Instead, the electrons in an atom behave more like a cloud: a diffuse fog that's spread out over a particular volume of space. When you see illustrations of atomic orbitals, they're basically showing you the wave-like shape of the individual electrons. If you were to send a high-energy photon or particle in there to interact with an electron, sure, you could pin down its position precisely. But and here's where quantum mechanics trips most of us up the act of sending that high-energy particle in there fundamentally changes what's going on inside the atom itself. It causes the electron to behave like a particle, at least for the moment of that one interaction, instead of like a wave.But until such an interaction occurs, the electron has been acting like a wave all along. When you have an isolated, room temperature atom, or a chain of atoms linked up in a molecule or even in an entire human body, they're not acting like these individual particles with well-defined points. Instead, they're acting like waves, and the electron is actually located all throughout this —1 angstrom volume, rather than in one particular point-like location. The better way to think about an electron is like a "fog" or a "cloud': spread throughout the space around an atomic nucleus.
When two or more atoms are bound together into a molecule, their electron clouds overlap, and the electron's extent in space gets even more diffuse. When you press your hand up against another surface, the electromagnetic forces from the electrons on that surface push against the electrons in your hands, causing the electron clouds to distort and deform in their shapes. This is counterintuitive, of course, because we're so used to thinking of the fundamental constituents of matter in terms of particles. But it's better to think of them as quanta instead: behaving like particles under high-energy conditions but behaving like waves under low-energy conditions. When we're dealing with atoms under normal terrestrial conditions, they're wave-like, with individual quanta occupying large volumes of space all on their own.There's a big problem whenever we rely on our intuition to make sense of the Universe: intuition is borne from experience, and our own personal experience of the Universe is entirely classical. Our Universe is made up of particles at a fundamental phenomena, and collections of particles can compress, rarify, and oscillate in ways that appear wave-like.
If it weren't for the Pauli Exclusion Principle, the matter we have in our Universe would behave in an extraordinarily different fashion. The electrons, you see, are examples of fermions. Every electron is fundamentally identical every other electron in the Universe, with the same to charge, mass, lepton number, lepton family number, and spin. If there were no Pauli Exclusion Principle, there would be no limit to the number of electrons that could fill the ground (lowest-energy) state of an atom. Over time, and at cool enough temperatures, that's the state that every single electron in the Universe would eventually sink to. The lowest energy orbital the 1s orbital in each atom would be the only orbital to contain electrons, and it would contain the electrons inherent to every atom. Of course, this is not the way our Universe works, and that's an extremely good thing.The Pauli Exclusion Principle is exactly what prevents this from occurring by that simple rule: you cannot put more than one identical fermion in the same quantum state.Sure, the first electron can slide into the lowest-energy state: the 1s orbital. If you take a second electron and try to put it in there, however, it cannot have the same quantum numbers as the previous electron. Electrons, in addition to the quantum properties inherent to themselves (like mass, charge, lepton number, etc.) also have quantum properties that are specific to the bound state they're in. When they're bound to an atomic nucleus, that includes energy level, angular momentum, magnetic quantum number, and spin quantum number. The lowest-energy electron in an atom will occupy the lowest (n = 1) energy level, and will have no angular momentum (I = 0) and therefore a magnetic quantum number of 0 as well. The electron's spin, though, offers a second possibility. Every electron has a spin of 1/2, and so will the electron in the lowest-energy (15) state in an atom. When you add a second electron, it can have the same spin but be oriented in the opposite direction, for an effective spin of -1/2. This way, you can fit two electrons into the 'I s orbital. After that, it's full, and you have to go to the next energy level (n = 2) to start adding a third electron.
The 2s orbital (where I = 0, also) can hold an additional two electrons, and then you have to go to the 2p orbital, where I = 1 and you can have three magnetic quantum numbers: -1, 0, or +1, and each of those can hold electrons with spin of +1/2 or -1/2. The Pauli Exclusion Principle and the fact that we have the quantum numbers that we do in the Universe is what gives each individual atom their own unique structure. As we add greater numbers of electrons to our atoms, we have to go to higher energy levels, greater angular momenta, and increasingly more complex orbitals to find homes for all of them. The energy levels work as follows: The lowest (n = 1) energy level has an s-orbital only, as it has no angular momentum (I = 0) and can hold just two (spin +1/2 and -1/2) electrons.The second (n = 2) energy level has s-orbitals and p-orbitals, as it can have an angular momentum of 0 (I = 0) or 1 (I = 1), which means you can have the 2s orbital (where you have spin +1/2 and -1/2 electrons) holding two electrons and the 2p orbital (with magnetic numbers -1, 0, and +1, each of which holds spin +1/2 and -1/2 electrons) holding six electrons. The third (n the 3) energy level has s, p, and d-orbitals, where d-orbital has an angular momentum of 2 (I = 2), and therefore can have five possibilities for magnetic numbers (-2, -1, 0, +1, +2), and can therefore hold a total of ten electrons, in addition to the 3s (which holds two electrons) and 3p (which holds six electrons) orbitals.
Each individual atom on the periodic table, under this vital quantum rule, will have a different electron configuration than every other element. Because it's the properties of the electrons in the outermost shells that determine the physical and chemical properties of the element it's a part of, each individual atom has its own unique sets of atomic, ionic, and molecular bonds that it's capable of forming. No two elements, no matter how similar, will be the same in terms of the structures they form. This is the root of why we have so many possibilities for how many different types of molecules and complex structures that we can form with just a few simple raw ingredients. Each new electron that we add has to have different quantum numbers than all the electrons before it, which alters how that atom will interact with everything else.There is no limit to the possible combinations that atoms can come together in; while certain configurations are certainly more energetically favorable than others, a variety of energy conditions exist in nature, paving the way to form compounds that even the cleverest of humans would have difficulty imagining. But the only reason that atoms behave this way, and that there are so many wondrous compounds that we can form by combining them, is that we cannot put an arbitrary number of electrons into the same quantum state. Electrons are fermions, and Pauli's underappreciated quantum rule prevents any two identical fermions from having the same exact quantum numbers. If we didn't have the Pauli Exclusion Principle to prevent multiple fermions from having the same quantum state, our Universe would be extremely different. Every atom would have almost identical properties to hydrogen, making the possible structures we could form extremely simplistic.
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