Quantum Physics

J.B.S. Haldane: Now my own suspicion is that the Universe is not only queerer than we suppose, but queerer than we can suppose.

Richard Feynman: I think that nature's imagination is so much greater than man's; she's never gonna let us relax.

Reporter: Is it true that only three people in the world understand Quantum Theory?
Sir Arthur Eddington: [beat]
Reporter: Sir?
Eddington: I'm trying to think who the third might be.

By the close of the nineteenth century, classical physics was beginning to reach the limits of what it could describe. It had mastered what Richard Dawkins calls Middle Earth, the realm of the every day, but at the boundaries of the very large, the very small, and the very fast, it was breaking down and reaching insoluble dilemmas. A new model of the universe was needed, and a number of brilliant men and women were waiting to step into the gap with Quantum.

In the first half of the twentieth century, it almost seemed that scientists were gleefully overturning what we thought we knew to be true, discovering a wildly strange and wholly counter-intuitive new physics, based on integers and probabilities, a physics that thew out the notion of the universe as an infinitely large and complex Swiss watch and replaced it with a frightful smear of strange mathematics, abstruse notations, and huge swaths of Uncertainty.

The was the era of the wunderkinden, the wonder children, in which brilliant young men of twenty-five to thirty-five years of age took the stage. The names of Heisenberg, Schrödinger, Feynman, Dirac, Freeman, and many others impressed themselves on the popular consciousness, and particularly on the scientific establishment. Einstein began the era with a bang in 1905, his annus mirabilis, his Miracle Year, and the world never looked back.

Quantum Physics and Quantum Mechanics and Quantum Chromodynamics are famously difficult to understand. In truth, so long as you go into them with the assumption that everything will be weird and that that's okay, it's really not to difficult to get a handle on. Also, you have to spend half a decade studying math, but that's not a problem, right?

The Fundamental Concepts of Quantum Physics

Wave–particle duality: Classical physics treated energy as a wave. We still do, but not the same way. In classical physics, the wave was understood as an ocean wave. Waves had frequency, speed, and amplitude. Speed and frequency depended on the medium and amplitude told you what the energy of the wave was. To put it in terms of the ocean, taller wave means more energy (this is correct). To put it in terms of light, brighter light means more amplitude (correct) means more energy (only half right). A brighter light does deliver more energy, but not in the way physicists thought. Their error was brought to light with the Ultraviolet Catastrophe.

Turn on your stove. It glows dull red. As it gets hotter it turns cherry red, then very bright red. If you could get it hot enough (don't), it would turn yellow, then white, then blue. This is known as black body radiation and it's the means by which every object in the universe, including you, cools off. Physicists attempting to find a mathematical model for this phenomenon using the classical understanding of energy failed spectacularly. Their models predicted that the intensity of the radiation would go as the cube of the frequency. That is, starting at zero frequency you have zero amplitude, then it goes upward like x³. In reality, this does not happen. As ultraviolet light has a higher frequency than visible light, this was known as the ultraviolet catastrophe. Physicists wrote this off as a failure of mathematics and assumed a bright light would one day solve the problem.

Enter young physicist Max Planck. He saw that their math was correct and their conclusion incorrect. He therefor realized that the problem lay with their assumptions. He took the bold step of assuming an entirely different kind of electromagnetic radiation, one which could only exist in quanta, that is to say in discreet packets of energy rather than continuous waves. For quite some time, Planck believed that the quantization of energy was purely a formal approach, much like the electric field, rather than an actual physical phenomenon. It was only later that the world saw this as the first step in the development of Quantum Physics.

To solve the mathematics, Planck was forced to introduce what is now known as the Planck constant, 'h', with units of joule*seconds. This was one of the tools used by Einstein in his annus mirabilis of 1905. The problem Einstein tackled was what is known as the "photoelectric effect"; when you shine light (photus) on a piece of metal, it will occasionally fire off an electron (electric). According to classical theory, the energy of the light is measured by the amplitude and the frequency has nothing to do with it. Therefor increasing the amplitude (brightness) of the light should increase the energy of the electrons leaving the metal. Instead, increasing the amplitude of the light increased the number of electrons leaving the metal and didn't change their energy at all. Rather, changing the frequency of the light changed the energy of the electrons, and below a certain frequency, no electrons escaped at all.

Einstein saw the Planck constant and realized that Planck's understanding went beyond mere mathematical formality, but that the quantization of E=hv held a physical reality. The energy of each photon, or particle, of light was directly equal to the product of its frequency (v, actually the Greek letter 'nu') and Planck's constant. Thus the light striking the metal consists of many many photons, each with some chance to be absorbed by the electrons of the metal. These electrons are held by the metal with some energy Phi, and the photon has to have more energy than that to liberate it. Thus the minimum frequency is related to Planck's constant. Further, the amplitude of light is related to the energy because more amplitude means more photons to carry energy, which explains why more electrons are kicked off by a brighter light. There's a reason Einstein is famous across the globe.

If waves can be particles, why not the other way around? Can a particle also be a wave? In fact, yes. A grad student by the name of de Broglie proposed as his thesis that just as waves are particles, so might particles be waves. The smaller something is and the slower it is, the longer is its wavelength. To put it another way, a thrown baseball will have a wavelength, but because it's so massive it's wavelength is ridiculously small compared to the size of the ball itself, so in practice the wave doesn't matter. A moving electron also has a wavelength, but because an electron has such a tiny mass its wavelength can be much larger than the electron. This was the genesis of wave-particle duality that forms the core of quantum physics.

Quantum and the Quantum Leap: 'Quantum' comes from the latin quantus for 'how much'. In physics, it means a unit, almost always of energy. For electromagnetic radiation, light is quantized into photons, each of which is a discrete packet of energy. In quantum physics, all systems are quantized, and only discrete units of energy of the right amount can enter a system. In practice, there are enough different systems in play that energy can flow freely into or out of almost any system. The rotation of a molecule is quantized: it will only spin at certain energy levels. The vibration of a molecule is quantized, and it can only vibrate in certain ways. How electrons can arrange themselves in an empty box is quantized.

The Quantum Leap is a popular and popularly misunderstood phrase. In popular parlance, it means a huge leap forward. "Physics took a quantum leap forward with the discovery of X which I, a guy with a journalism degree who took a biology class in college for my science requirement, will now completely misreport and make it sound like something out of Star Wars, which I think we can all agree was a pretty cool movie." The phrase has its genesis in the behavior of electrons. Recall that electrons can only exist in certain orbits about an atom. An electron at a lower energy level can absorb a photon of light (a quantum of energy) and leap to a higher energy level. Thus the quantum leap, wherein an electron moves from one discrete energy level to another, without crossing the intervening levels of energy. As such, a quantum leap is, by definition, one of the smallest possible leaps that could be made.

Uncertainty Principle: If you want to measure the particle's position accurately, you must sacrifice the accuracy of its velocity, and vice versa. On a practical level, this can be understood as trying to find the position of a baseball by hitting it with another baseball. You cannot see a subatomic particle in any real sense; we see by interacting with light that bounces off of something, and light has roughly the same size and shape as a subatomic particle, definitely about the same wavelength. Therefor the interaction of a particle and a photon of light is like two baseballs colliding. You'll learn where it was, but not where it was going, or vice versa.

However, this isn't just a matter of pragmatism. It's a fundamental truth of reality. Thanks to wave-particle duality, a particle doesn't exist at a single defined location. Rather, they're smeared across a volume, existing more at some places than at others. The electron of a hydrogen atom is mostly inside the nucleus, but somewhat outside it, with the amount of the electron existing at any location decreasing with distance from the nucleus. This is sometimes called the 'electron cloud'.

For quantum particles, position and momentum are thus very different concepts than we're used to seeing on a daily basis. The end result of this is that a particle's location and momentum are not independent of one another and our knowledge of either limits and is limited by our knowledge of the other.

Superposition: in which subatomic particle can exist in two or more places at once. Well, states. And it's not just a particle that does this, it's a system.

Say you enter a room and see five chairs, all exactly the same, and are told to sit down. You sit in one of the chairs. End of story. An electron enters a room with five identical chairs and sits down in all five of them at once. What happens is that the electron is one-fifth in chair A, one-fifth in chair B, etc. It sits in all five chairs identically and at the same time, but only one-fifth of itself. If you take the integral over all five chairs, you find exactly one electron, which is what you expect, but when you examine any given chair, you find 20% of the electron.

On the other hand, say four electrons enter the room and are told to sit in five identical chairs, they do. Take an integral over all five chairs and you find four electrons, but each chair only has 80% of an electron. The four electrons are sharing all five chairs. If two different ways of arranging a system are identical, then the system won't favor one over the other and will arrange itself both ways at the same time. If three ways, all three. But what if they're not identical? Then it will still occupy both positions, but not equally. It will exist mostly in the lower energy/higher entropy state, but also a little bit in the higher energy/lower entropy state. Give an electron an entire lecture hall of chairs and it will mostly sit down low at the front, but also a little but up high at the back.

This is why a single particle can interact with itself. In the famous double-slit experiment, electrons fired one at a time through a piece of card with two slits produce an interference pattern just like light, even though there's no other electrons to interfere with it. The electron travels through both slits at the same time and interferes with itself.

The Wave Function: Einstein famously said that if you couldn't describe relativity or quantum theory without resorting to mathematics then you didn't understand either. When a layman asked him to explain relativity without math one summer, Einstein repeatedly found himself stymied; his own understanding of the science was so rooted in his ability to think and communicate through mathematics that he couldn't communicate it otherwise. Although it's possible to relate the concepts of modern physics without math, to truly understand and work with them, a powerful understanding of mathematics is required. For quantum physics, this means working with the wave functions developed by Erwin Shcrödinger.

The wave function is a probability amplitude describing the states of a system as a function of space and time. That is, if a system is known perfectly, then you can map the system at all points in space and then follow it in time. Or vice versa. First, a probability amplitude isn't the same as a probability. A wave function has both real and imaginary parts. What happens is that you take the function multiplied by its complex conjugate to find the probability of finding a particle somewhere (for its position-space wave function) or of its momentum (for its momentum-space wave function)^[1]. If a wave function were (a+bi), then its complex conjugate would be (a-bi), and their product would be (a+bi)*(a-bi)= a²+b².

Why not just stick with that and ignore the imaginary stuff? For several reasons, not least of which that the imaginary stuff, though we can't observe it, is still vitally important to understanding the system. Second, wave functions are far more complex than just (a+bi). Third, a whole lot more really complex stuff. Finally, because a wave function, in describing a system, allows you to find out more information about the system using other mathematical operators.

Certain operators (mathematical functions) when applied to a wave function will produce the original function multiplied by a scalar quantity. For example, if you take the first derivative of e^3x, it returns 3e^3x. The scalar quantity is the number three, the function is e^3x and the operator is d/dx, the first derivative. In quantum mechanics, operators give us information about a system and the information is the scalar quantity that results is the value of the information. That is to say, if you were to set up that system and measure it, that scalar would be the measured quantity.

Wave functions are undeniably useful, but very difficult to work with, because they require taking multiple derivatives across multiple dimensions and finding complex conjugates and, worse, taking integrals across multiple dimensions. For a simple system, like a hydrogen atom, this is manageable, but the functions quickly degenerate into painful headaches. For this reason, Schrödinger's equations have been largely replaced by Heisenberg's matrices. Werner von Heisenberg developed a different mathematical formulation for quantum mechanics roughly contemporaneously with Schrödinger using matrix mathematics rather than functions and integrals. At first, everyone understood Schrödinger's math, as difficult as it was to work with, and no one understood Heisenberg's matrices. Over time, though, Heisenberg's formulation has become the standard for multiple reasons, not least of which that they're much easier to work with and, once you can read a matrix, give information in a much more straightforward fashion.

One of the problems in working with quantum physics is the lack of commutation. Commutation is the rule of arithmetic that tells us that A x B = B x A. For quantum physics, this isn't always the case. For some properties, such as momentum and position, A x B <> B x A. This is the basis/result of uncertainty. AB - BA <> 0. For Schrödinger's wave equations, this lack of commutation was dealt with using complicated operators involving derivatives and the like, which took time to work through and, particularly for complicated systems, left many opportunities for human error. For Heisenberg's matrices, non-commutation is built right in. Matrix math is a class unto itself, but suffice it to say that the matrix equivalent of multiplication is not a commutative property and is much simpler than working in multi-variable calculus. Where the wave equations require you to do multivariable calculus, the matrix requires you to add, subtract, multiply, and divide. You'll still fill up a few pages with funny looking math, but it's much easier math.

Wave function collapse: Superposition tells us that a particle or system can exist in an indefinite (or definite) number of states. Until we look at a system, it exists in all of those states simultaneously, and to the degree determined by the amount of energy in the system and the energy of the states and the number of states and other factors. This condition is described by the wave function, which defines the states and their energies. Once we look at the system, however, the wave function collapses and the superposition of states is replaced by the system achieving a single defined state. This is known as collapsing the wave function and is part of the explanation behind the double-slit experiment (below).

Quantum Tunelling, in which a particle can tunnel through a barrier. In classical physics, if you roll a ball up a hill without enough energy to get to the top, it won't get to the top, and thus won't go over the hill. In quantum physics, if you fire a particle at a barrier without enough energy to get through the barrier it can still get through the barrier. This is related to uncertainty; there's no such thing as a zero percent chance of something (and the corollary, there's no such thing as a 100 percent chance, after all a 0% chance of something happening, is a 100% chance of it not happening). You can't say something will never happen. The odds might be enormously small, but they're there. Put away the voodoo doll, this is for real. This is how a number of technologies function, such as Scanning Tunneling Microscopes (STM); a probe is brought very close to the surface of something and a voltage applied that allows some electrons to tunnel through the space between the surfae and the probe, which allows the microscope to get a very fine record of the surface. How fine? Individual atoms. This is also something that has to be overcome, as well. Computer engineers have to design their hardware's architecture to minimize electron tunneling and prevent glitching.

As with most other quantum phenomena, this is really only true of small particles; you should still expect classical mechanics to apply at the level of every day life. There is a non-zero probability that, for example, that remote control just out of reach might spontaneously fly to your hand without you having to get up. That non-zero probability is so small that you'll spend hundreds of billions of years waiting for it to happen.

Quantum Entanglement: when two particles interact, they can become linked such that the alteration of one causes a mirrored response in the other, despite a separation by arbitrarily large distances. This is a sub-case of quantum superposition; the two particles continue to form a system even after they've been separated and continue to share a single quantum state until a measurement has been made. Until that measurement has been made, each particle holds either (or multiple) positions simultaneously. Measuring/determining the state of one causes the other to take a different or opposite state.

Say Alice and Bob have a pair of particles known to possess the quantum property of spin, and can take a spin up orientation or spin down. Once the two particles are entangled and separated, if Alice measures her and finds it to be spin up, Bob's will be found to be spin down. Prior to that measurement, both particles are indeterminate and will exist in a superposition state of both spin up and spin down. Uncertainty once again rears its head because there are other properties that can be measured and measuring a property in one particle brings about uncertainty not just in that particle but also in its entangled twin. Spooky, which was how Einstein derided entanglement. He didn't like it and thought that the math justifying entanglement meant the theory was incomplete rather than just weird. One of the problems he had with it is that entanglement has been experimentally demonstrated to transmit information faster than the speed of light. Another experiment occurred at relativistic speeds such that each measurement occurred before the other (something possible in relativity) and they still confirmed one another.

An obvious conclusion might be that each particle in fact had a state prior to the measurement, but this isn't the case. In quantum theory, and this has been born out by experiment, a particle that can occupy multiple states does so until a measurement forces it to only occupy one. This is the explanation behind the double slit experiment (see below). The particles of an entangled pair don't have a particular value until that value is measured and that information is transmitted, perhaps instantaneously, to the entangled partner.

Quantum Foam: At its finest structural detail, the universe is a seething mass of instantaneously appearing and disappearing particles, antiparticles, black holes, and wormholes. Maybe. Essentially, quantum theory predicts that there's no reason a particle/antiparticle pair can't appear from nothing very briefly, before annihilating back into nothingness, with the energy necessary for their creation resulting from the annihilation. Wait, what? Yes, the universe borrows energy from itself at zero percent interest and pays it back in full.

At very very very very very tiny scales of time and space, the smallest energy fluctuations are relatively enormous and allow for these sorts of shenanigans. Because the travel of forces across these times and distances are very strange and relativistic, space no longer follows a smooth curve, but is instead rough and foamy. Our understanding of quantum foam is still fairly rough, because we're talking about scales that make the interior of an atom seem enormous. However, many predictions have been born out; those particle/antiparticle pairs are responsible for the evaporation of black holes (thank you, Hawking), and the constant flux of appearing and disappearing particles is responsible for much of the mass of a proton, the elucidation of which recently won someone a Nobel prize. We're on the cutting edge, here.

The Little Guys that inhabit the Quantum World

All matter is composed of atoms, which can be broken into:
- Electrons, which can't be broken down into smaller particles.
- Protons and neutrons, which form the nucleus of the atom. But even protons and neutrons can be broken down into even smaller particles: quarks. The quarks are fundamental, meaning that (like electrons) they can't be broken down into smaller particles.
Of course, there are other particles, which are:
- The six leptons which are: electrons, muons, taus,' and the three types of neutrinos. Leptons are also fundamental.
- The Antimatter particles, which are the opposite of the normal matter particles.
- The six quarks which are: Up, down, charm, strange, top, and bottom. Except for exotic conditions such as quark plasmas, quarks are never found on their own. They are normally only found combined with other quarks in either:
  - Baryons, which are composed of three quarks or three antiquarks. (The proton is composed of two up quarks and one down quark; the neutron is composed of one up quark and two down quarks.) Or
  - Mesons, unstable particles which are composed of one quark and one antiquark.
- The force carrier particles, also known as "vector bosons", which carry the four forces between subatomic particles. The force carrier particles are: Photons (the basic units of light), which carry the electromagnetic force; W & Z bosons, which carry the weak force; and gluons, which carry the strong force. As for gravity, the weakest of the four forces, it is said to be carried by the hypothesized gravitons, but so far, no gravitons have been found.
- There is also the hypothesized Higgs Boson, which, if found to exist, provides an explanation why sub-atomic particles have masses while being so small.
- Virtual particles, particles that only exist for a short amount of time. Force-carrier particles are a kind of virtual particle, but Quantum Electrodynamics established that any particle (or pair of particles) can come into existence ex nihilo if (A) electric charge is conserved, and (B) the length of time for which the particle(s) exist is less than Planck's constant divided by the particles' mass-energy.

Experiments

The Double Slit Experiment
This experiment is used to describe and explain the dual nature of quantum particles, that they are both particle and wave. It also explains the effect of observation and the collapse of the wave function.

Take a barrier and cut a slit in it, then shine a light on it. On a wall behind the barrier you'll see a single bar of light corresponding with the single slit. If you cut a second slit and shine a light on the two, you won't see two bars of light. Rather you'll see a bar of light bookended by two dark bars, bookended by two dimmer bars of light, bookended by darkness, bookended by two bars that are dimmer still, and so on. This is called an interference pattern, and is caused by the waves of light reinforcing one another (bars of light) or canceling one another (bars of dark) when their peaks and valleys align (light) or not (dark).

Take the same barrier and now, rather than light, you fire bullets at it randomly. With a single slit you get a single bar where bullets strike the wall behind it. With two slights you get two bars. This is because bullets don't interfere with one another. They are singular and particulate in nature.

Now take the barrier and fire electrons at it. Are electrons particles or waves? With two slits, you get an interference pattern! Of course, electrons are negatively charged and propagate an electric field, thus electrons can interfere with one another. What happens if you slow down the rate of fire so that only one electron is fired at a time? Without other electrons to interfere, shouldn't you get the same result as the bullets? The savvy consumer knows that the answer is "No.". The superposition of states means that the electron goes through both slits simultaneously, it is smeared across the entire region before it strikes the barrier (collapsing the wave) and interferes with itself.

Now imagine you set up a detector just before the barrier so you can determine which of the two slits the electron enters through. What happens then? Without your observation, the electron would continue unimpeded and the superposition would remain in place until it struck the barrier, collapsing the wave function. However, by setting up the observation, by measuring the electron, you collapse the wave function prematurely and remove the superposition. The electron is no longer a probabilistic smear, but has become a particle again. You now see a particulate profile emerge, looking like bullets from a gun. No interference.

This isn't just a mental exercise. The experiment has been performed numerous times under the conditions described and these are the results observed. They match the predictions of quantum mechanics, much to the consternation of stodgy 19th century types.

Schrödinger's Cat
Unlike the double slit experiment, Schrödinger's Cat isn't an actual experiment. Rather, it's satirical derision. However, many people don't realize this and assume it's some sort of actual experiment or clever notion. Schrödinger, like others, wasn't fond of the probabilistic theories being developed by Heisenberg and Bohr, outlined in the EPR paper of 1935. This was the Copenhagen Interpretation of Quantum theory (below).

The basic thought experiment is that within a box is a cat. There is also an atom of a radioactive isotope that may or may not decay. Upon decay, it will trigger a Geiger counter, that will then trigger a vial of poison that will kill the cat. According to Schrödinger, the superposition of states means that until the box is open and the system observed, the cat is simultaneously alive and dead. Schrödinger never intended this as an actual result of quantum theory, but as a reductio ad absurdum to demonstrate what was to him the ridiculousness of the Copenhagen interpretation.

These days the thought experiment is used to analyze and interpret the various schools of though, showing their strengths, weaknesses, and potential implications. No one ever actually tries to poison a cat. Scientists aren't dicks.

Interpretations of Quantum Physics

There have been many intepretations for how Quantum Physics exactly work. Listed here are some of those many interpretations:

Copenhagen Interpretation
There's nothing in the math to say that a particle should exist in any given state rather than another. Indeed, the wave functions and matrices tell us that it has a certain probability of existing in one state, a differing probability of existing in another state. This is the interpretation of quantum theory most widely held to be true. The Copenhagen Interpretation states:

A system is completely described by a wave function ψ, representing an observer's subjective knowledge of the system.
The description of nature is essentially probabilistic, with the probability of an event related to the square of the amplitude of the wave function related to it.
It is not possible to know the value of all the properties of the system at the same time; those properties that are not known with precision must be described by probabilities.
Matter exhibits a wave–particle duality. An experiment can show the particle-like properties of matter, or the wave-like properties; in some experiments both of these complementary viewpoints must be invoked to explain the results, according to the complementarity principle of Niels Bohr.
Measuring devices are essentially classical devices, and measure only classical properties such as position and momentum.
The quantum mechanical description of large systems will closely approximate the classical description.

The Cat Question: Schrödinger's paradox states that the cat is both alive and dead. The Copenhagen interpretation is that the Geiger counter, not the scientist, acts as the observer and collapses the wave function until such time as the isotope decays. Douglas Adams later noted that the cat also acts as an observer. Terry Pratchett noted that there are three states: Alive, Dead, and Bloody Furious.

Many Worlds Interpretation
This interpretation is more popular with storytellers and certainly catches hold of the imagination, but is less popular with scientists due to its inherent untestability. Essentially, this interpretation asserts the reality of the wave function but denies the collapse. At the moment of apparent collapse, the universe splits in two and two possible futures emerge. Whereas observation is a fundamental part of the Copenhagen Interpretation, in this interpretation, observation is no longer special. What would have been collapse of the wave function at observation is instead recognition of a branching that occurred sometime at or before the moment of observation. In this interpretation, the superposition of states is in fact a superposition of multiple universes.

The Cat Question: Depending on which universe you're in, the cat is either alive or dead, regardless of observation.

The Ensemble or Statistical Interpretation
This states that the superposition of states is just a substate of a larger system. Rather than take the view of individual particles, this interpretation views systems as large collections, or ensembles, of particles and states and the wave functions apply to these rather than to individual particles and states. In this view, the wave function need never collapse because the probabilities tell you what fraction of the ensemble exists in a certain state rather than the probability a single particle exists in a certain state. One of the principle criticisms of the Ensemble Interpretation is that it ignores real experimental evidence of single particle behavior in an attempt to adhere to classical principles.

The Cat Question: The experiment deals not with a single cat in a single box but many cats in many boxes, a fraction of which are alive and the rest of which are dead. The question is trivial.

↑ Remember uncertainty, you can measure either momentum or position, but not both.

[1] Remember uncertainty, you can measure either momentum or position, but not both.

[1]