The fundamental basis for Quantum Mechanics is the idea that all energy (and, therefore, all mass) is only available in discrete multiples of a single unit called a quantum. Quantum Mechanics is, therefore, the study of how physical systems like atoms interact given that rather bizarre and unexpected constraint.

One of the beautiful things about Quantum Mechanics is that idea of the valence of electrons which is the basis for all of Chemistry is derivable directly from quantum mechanical theory.

An important philosophical consequence of Quantum Mechanics is that the quantizing of energy necessitates that all particles are described by a waveform which is something like the complex square-root of a probability distribution. That is, the fundamental description of all matter is probabilistic and not deterministic. The Strong Law of Large Numbers, of course, applies to the probabilistic events which take place at the atomic level, and so our macroscopic experience of the universe is fairly deterministic. Nevertheless, the Newtonian ideal that universe is deterministic and predictable was destroyed by Quantum Mechanics.

Also, several creatures from Nethack whose attack can effect a character's location and the consumption of whose corpses can effect a character's speed.

Quantum mechanics can be summarized by the following:
Each particle or combination of particles (including the universe as a whole!) has a complex-valued function, usually denoted Ψ (Psi), which contains all of its state information. This function is also called the "Wavefunction" because the constraint on its time-evolution is a Wave Equation - Schroedinger's Equation (or, for taking into account special relativity, various field equations such as the Dirac Equation)

These functions can be added to each other. That is, it is possible to add one state to another state; the result is a different state which is in a very specific sense part way between the two you combined. Moreover, if you take one state and let time progress in it, and you take a second state and let time progress in it, then you add the states together, you get the same result as adding them together in the beginning and then let time progress.

Importantly, you can also split a state up into different parts. Slicing the states in different ways yields all of the interesting part of the theory. In particular, one often chooses a way of slicing the states up so that some quantity - position, momentum, energy, angular momentum, etc. - has a specific, different value for each slice.

The measure of a state is its squared amplitude. So, the expected value of some physical quantity can be found by slicing the state up so each slice has a specific value of that quantity, then getting the measure of each slice, and multiplying it by the value for that slice, and adding it all up. Alternately, you can stop short of adding it all up to get the distribution over those values for the full state.

One thing that's important is, you can't keep slicing to narrow things down - at least, not without losing the meaning of the old slices. A slice that has perfectly defined position, say, does not as a matter of mathematical identity have a perfectly defined momentum, even if you sliced it to perfectly define momentum first. Indeed, such a slice is a pure mix of all possible momenta. If you try to slice this one point function slice down to get specific momentum values, then the resulting slices are no longer confined to single points in space. This is what is known as the Heisenberg Uncertainty Principle.

There are other confusing results. The most illuminating is the two-slit experiment. But even short of that, you run into troubles of interpretation. If you shoot an electron at a barrier, part of the wave is reflected and part of it continues onward. What do you say happened? The electron was split in two parts? Sure, of course. You have a slice of it that was blocked and a slice that was permitted. But, well, no -- if you actually look, then you will find it entirely reflected or entirely permitted. However, if you repeat the process many times, the number of outcomes will be proportional to the two corresponding slices of the wave. This is known as the Born Probability Rule. What can we make of that?

It seems that we only get to perceive one slice. Yet, what sort of slice it is varies from case to case in a way that is on the surface baffling (it takes a very complicated theory to make sense of this).

Interpreting Quantum Mechanics is something even a lot of people who use QM in their research simply don't have the time to deal with, since it can become somewhat of a philosophical obsession. The Copenhagen interpretation is very unsatisfying, but most physicists have aesthetic and/or comprehensional problems with the main alternative, the Many-Worlds Interpretation. While nonlocal-variable deterministic theories work and are not particularly mind bending at first glance, they are inelegant and the least popular of all.

Local Variable deterministic theories were ruled out by experiments in the 80's which took advantage of the Bell inequality, which showed that they were not equivalent to the official formulation of QM and pointed out a difference.

One thing that can clear up a little of the mess is this second example: Instead of shooting an electron at a barrier, we shoot a cannon ball at a barrier. Using a quantum calculation, we would either determine that the vast bulk of the wavefunction of the cannonball would bounce off the barrier, or conversely determine that the vast bulk of the wavefunction of the cannonball would penetrate the barrier. While in a technical sense the wavefunction is still split between permission and reflection, due to the tremendous mass of the object in question, the non-favored outcome is extremely strongly suppressed -- not a factor of a ten or a thousand in magnitude, but a factor of a ten to the thousandth power (to give a conservatively SMALL estimate). So it seems we could just say, "Oh, phew! All the quantum effects go away on their own for heavy objects!" But the problem is, they don't ALL go away. See Schroedinger's Cat.

One common misperception of quantum mechanics is that it always limits energies to particular discrete levels. It does do this whenever particles are trapped in a potential well. When particles are free, their energies may vary continuously. However, even in this case energy exchange is mediated in quanta, which is to say in discrete chunks. The size of these chunks can be any value, but still, the rules prohibit continuous energy transfer.

Though Quantum Mechanics, via the Schroedinger equation, reproduces Newtonian mechanics for reasonably heavy objects, and in the formulation of Quantum Field Theory has incorporated the predictions of Special Relativity, QM has so far resisted all attempts to reconcile it with General Relativity. Though it is possible to define wave functions in curved space-times, these efforts have not yet succeeded completely. A complete solution would have several notable difficulties - for example, making the definition of space-time itself a function of the wavefunctions. This is especially tricky since superposition of wavefunctions would cause different mass distributions, which would in turn cause different definitions of space-time, so you end up in the sticky position of adding functions which are not defined on the same domain. So long as the mass distributions only stretch space, it is possible to resolve this with greater abstraction in the mathematics -- but if mass distributions cause space's topology to change (e.g. wormholes), that is extremely awkward. Also, since gravitation is so weak compared to the other forces (only being significant for large masses), it has so far resisted attempts to measure it on the scale of individual particles. So, even if we could rule out topology changes, it would be very difficult to verify any theory connecting the two.

In the discussion above, I may have inadvertently given the impression that QM sprung fully-formed from the forehead of Schroedinger. This is utterly false, and the history writeup of bigmouth_strikes, below, lists a few of the milestones.

Quantum Mechanics celebrates 100 years
The development of Quantum Mechanics - Important Contributions

What the next 100 years of quantum mechanics will bring us ? Well...

Finally, a quote from the physics teacher of Max Planck, the man who started it all, in 1876:

Don't spend your time on physics; there's hardly anything left to discover

###### Source: ne.se, britannica.com for English terminology
Mauler's Layman's Guide to Quantum Mechanics

1. Very Large Particles = Newtonian Mechanics

Imagine several bullets are shot one at a time through two slits in a wall and hit another wall several feet behind the first wall. According to classical physics, any particle such as a bullet is subject to Newton’s laws of motion and therefore has mass, inertia, momentum, etc. Thus, each bullet could only travel in a straight line and would hit the second wall only in one of two very limited regions directly in line with the slits in the first wall.

2. Very Small Particles = Quantum Mechanics

When several photons are shot one at a time at a plate with two slits and hit a photosensitive screen behind the plate, the photons would land on all different parts of the screen, even those not directly in line with the slits. Measured over time, a pattern would emerge in which photons hit certain parts of the plate very often and other parts not at all, in a frequency pattern that could only be explained if the photons were acting as a wave. Somehow, even though they are shot one at a time, the photons are interfering with themselves like waves do, increasing the frequency in certain areas of the screen, and canceling it out in others. If light were a particle it would theoretically be limited by Newton’s laws and therefore produce a pattern similar to the one created by the bullets. According to classical physics, particles are incapable of diffracting and interfering to form a wave-like pattern and there is no way particles could curve around the solid sections of the slit plate to land in areas not in straight line paths from the "gun" unless an outside force acted upon them (which is not the case here). Thus, light must have wave properties.

3. Where Does This Lead Us?

If photons were true particles, no matter how many are released they should exhibit a distribution pattern similar to that of the bullets. Experiments show they do not. Rather, when many photons have been released, even just one at a time, the frequency distribution pattern measured over time is identical to the interference wave pattern. The only explanation for this phenomenon is that light somehow simultaneously acts as both a wave and a particle. Although Newton had once suggested that light was a hail of tiny particles, classical physicists became convinced (after much experimentation) that light is a wave. However, in 1905 Albert Einstein revived the particle theory of light because it was the only explanation for the photoelectric effect - the ejection of electrons from certain metals subjected to high-frequency light (even from a dim source). If light were a wave, low frequency light should also produce this same effect although it might take longer. However, only high-frequency light can cause the photoelectric effect. Einstein reasoned that light must be made be up of packets of energy (photons). Obviously, low-frequency light does not have enough energy in each packet to eject electrons, no mater how many packets there are. If this reasoning were correct, said Einstein, light must be made up of particles. This jived with Max Planck's quantum theory (1900) that radiation consists of "quanta," or individual packets of energy.

The next step in the puzzle came in 1924 when Louis de Broglie wondered, "If light can have both wave and particle properties, cannot all particles have wave properties?" De Broglie proved that all particles, even those things as large as a baseball, exhibit wave properties. His equation:

wavelength = Planck's constant / momentum
The effect of something as large as a baseball’s wave properties is negligible, however, because its mass is so great (momentum = mv), and therefore wavelength is effectively zero. Thus, thanks to De Broglie's equation, we can see that the examples of the bullets and the photons that at first appeared so different are both part of the same phenomenon!

In 1925, Erwin Schrodinger came up with an equation whose solution is known as the wave function. Schrodinger applied his wave function to the atom, theorizing that it represented the physical orbit of an electron around the nucleus. Thus, Schrodinger accounted for the energy levels of electrons in atoms (circumferences must be whole number multiples of the wavelength).

To account for the collapse of the wave function when only a single photon is launched in the two slit experiment, Max Born postulated in 1926 that the wave function was not physical as Schrodinger had contended, but rather, was a probability wave akin to a "crime wave" or mortality table that simply shows the probability of a particle being in a given location. Thus, the wave function collapses when only a single photon is launched because once the photon is detected the probability of it being located anywhere else becomes zero and the wave ceases to exist.

4. The Uncertainty Principle

Werner Heisenberg’s "Uncertainty Principle" is the idea that one can never be certain of both the momentum and the position of a particle or both the energy of a particle and the time it has that energy. The more one is certain of one element, the less one is certain of the other. This uncertainty increases the smaller the units one is trying to observe. With very large objects, uncertainty becomes negligible. The Uncertainty Principle is based upon the fact that we cannot observe anything without our observation effecting it in some way.

"Position" and "momentum" are not properties of particles. Rather, they are artificial concepts we have created to help us describe particles. Too often, we confuse words and language - abstract human creations - with actual reality. Language is not reality but only a crude (though the best we have) method for describing reality. Niels Bohr recognized this distinction. Bohr realized that physics cannot discover reality. Physics can only attempt to describe reality as closely as possible. What Bohr is saying is that one must give up the idea of an objective reality outside human experience in order to predict the outcomes of experiments. This loss of objective truth is the heavy price paid in exchange for the power of prediction.

5. Some Theories to Explain Quantum Mechanics

In Niels Bohr’s "Copenhagen Interpretation," Heisenberg's Uncertainty Principle reigns supreme. The collapse of the wave function is completely random and entirely uncertain. When the wave function collapses, the fuzzy, nebulous, ghostly world of the atom materializes into a random but concrete reality. The cause of the collapse is human observation.

Albert Einstein’s "Hidden Variables Theory" is the belief that "God does not play dice." For Einstein, concrete reality always exists, but quantum mechanics is not a good enough system to describe it outside of human observation. There are hidden factors behind the scenes: all events happen for a reason. For Einstein, the word "random" was just an easy way to explain away our own ignorance. Einstein dreamed up an experiment to support his view. Suppose, he said, a particle explodes into two fragments which are allowed to travel a long distance. Each fragment should demonstrate evidence of the other through action/reaction, Einstein believed. In the 1960s, John Bell proved the existence of a limit to the degree of cooperation between the two fragments if Einstein was correct. In experiments conducted by Alain Aspect in 1981, this limit was exceeded, proving Einstein wrong.

Hugh Everett’s "Many Worlds Theory" is the belief that for every possible outcome of an uncertain event a new universe in which that outcome occurs is created in some parallel yet utterly unreachable dimension. Thus the paradox of the ghostly simultaneous existence of contradictory outcomes until an observation is made is avoided. A good example is the paradox of Schrodinger’s Cat, in which a cat has a 50% chance of dying of cyanide poisoning. The Many Worlds Theory bypasses the unfathomable concept that the cat is in a limbo-like state of both life and death until human observation by stating that the cat either dies or does not die in our reality. If the cat dies in our reality, it lives in another reality. If it lives in our universe, it dies in another, parallel universe (Now you know where Star Trek got all its plots).

The "Mind Over Matter Theory" is the belief that the human mind is fundamental in the creation of reality: human consciousness alone is responsible for the reality we perceive around us. In 1979, John Wheeler demonstrated that a simple modification of the two slit apparatus could delay the decision of whether to measure location or momentum until after the photon has passed through the screen. Thus, humans can affect reality in the past, after it has occurred! The Mind Over Matter concept is further supported by the experiments of John von Neumann and the hypothesis of Eugene Wigner, which seem to indicate that quantum theory breaks down when it comes into contact with the human mind. If a human replaces Schrodinger’s cat, does he or she experience both life and death? No, says Wigner. However, many physicists reject Mind Over Matter because it can be used to support paranormal phenomena.

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