This is one of several shortened indices into the Quantum Physics Sequence. It is intended for students who are having trouble grasping the meaning of quantum math; or for people who want to learn the simple math of everything and are getting around to quantum mechanics.

There's a widespread belief that quantum mechanics is supposed to be confusing. This is not a good frame of mind for either a teacher or a student. Complicated math can be difficult but it is never, ever allowed to be confusing.

And I find that legendarily "confusing" subjects often are not really all that complicated as math, particularly if you just want a very basic - but still mathematical - grasp on what goes on down there.

This series takes you as far into quantum mechanics as you can go with only algebra. Any further and you should get a real physics textbook - once you know what all the math means.

Quantum Explanations: Quantum mechanics doesn't deserve its fearsome reputation. If you tell people something is supposed to be mysterious, they won't understand it. It's human intuitions that are "strange" or "weird"; physics itself is perfectly normal. Talking about historical erroneous concepts like "particles" or "waves" is just asking to confuse people; present the real, unified quantum physics straight out. The series will take a strictly realist perspective - quantum equations describe something that is real and out there.
Configurations and Amplitude: A preliminary glimpse at the stuff reality is made of. The classic split-photon experiment with half-silvered mirrors. Alternative pathways the photon can take, can cancel each other out. The mysterious measuring tool that tells us the relative squared moduli.
Joint Configurations: The laws of physics are inherently over mathematical entities, configurations, that involve multiple particles. A basic, ontologically existent entity, according to our current understanding of quantum mechanics, does not look like a photon - it looks like a configuration of the universe with "A photon here, a photon there." Amplitude flows between these configurations can cancel or add; this gives us a way to detect which configurations are distinct.
Distinct Configurations: Since configurations are over the combined state of all the elements in a system, adding a sensor that detects whether a particle went one way or the other, becomes a new element of the system that can make configurations "distinct" instead of "identical". This confused the living daylights out of early quantum experimenters, because it meant that things behaved differently when they tried to "measure" them. But it's not only measuring instruments that do the trick - any sensitive physical element will do. There is no need to suppose that the universe cares what we think.
Can You Prove Two Particles Are Identical?: You wouldn't think that it would be possible to do an experiment that told you that two particles are completely identical - not just to the limit of experimental precision, but perfectly. But quantum mechanics makes it easy.
Classical Configuration Spaces: How to visualize the state of a system of two 1-dimensional particles, as a single point in 2-dimensional space. A preliminary step before moving into...
The Quantum Arena: Instead of a system state being associated with a single point in a classical configuration space, the instantaneous real state of a quantum system is a complex amplitude distribution over a quantum configuration space. What creates the illusion of "individual particles", like an electron caught in a trap, is a plaid distribution - one that happens to factor into the product of two parts. It is the whole distribution that evolves when a quantum system evolves. Individual configurations don't have physics; amplitude distributions have physics.
Feynman Paths: Instead of thinking that a photon takes a single straight path through space, we can regard it as taking all possible paths through space, and adding the amplitudes for every possible path. Nearly all the paths cancel out - unless we do clever quantum things, so that some paths add instead of canceling out. Then we can make light do funny tricks for us, like reflecting off a mirror in such a way that the angle of incidence doesn't equal the angle of reflection. But ordinarily, nearly all the paths except an extremely narrow band, cancel out - this is one of the keys to recovering the hallucination of classical physics.
No Individual Particles: One of the chief ways to confuse yourself while thinking about quantum mechanics, is to think as if photons were little billiard balls bouncing around.
Decoherence: A quantum system that factorizes can evolve into a system that doesn't factorize, destroying the illusion of independence. But entangling a quantum system with its environment, can appear to destroy entanglements that are already present. Entanglement with the environment can separate out the pieces of an amplitude distribution, preventing them from interacting with each other.
The So-Called Heisenberg Uncertainty Principle: Unlike classical physics, in quantum physics it is not possible to separate out a particle's "position" from its "momentum". The evolution of the amplitude distribution over time, involves things like taking the second derivative in space and multiplying by i to get the first derivative in time. The end result of this time evolution rule is that blobs of particle-presence appear to race around in physical space. The notion of "an exact particular momentum" or "an exact particular position" is not something that can physically happen, it is a tool for analyzing amplitude distributions by taking them apart into a sum of simpler waves. This uses the assumption and fact of linearity: the evolution of the whole wavefunction seems to always be the additive sum of the evolution of its pieces. Using this tool, we can see that if you take apart the same distribution into a sum of positions and a sum of momenta, they cannot both be sharply concentrated at the same time. When you "observe" a particle's position, that is, decohere its positional distribution by making it interact with a sensor, you take its wave packet apart into two pieces; then the two pieces evolve differently. The Heisenberg Principle definitely does not say that knowing about the particle, or consciously seeing it, will make the universe behave differently.
On Being Decoherent: When a sensor measures a particle whose amplitude distribution stretches over space - perhaps seeing if the particle is to the left or right of some dividing line - then the standard laws of quantum mechanics call for the sensor+particle system to evolve into a state of (particle left, sensor measures LEFT) + (particle right, sensor measures RIGHT). But when we humans look at the sensor, it only seems to say "LEFT" or "RIGHT", never a mixture like "LIGFT". This, of course, is because we ourselves are made of particles, and subject to the standard quantum laws that imply decoherence. Under standard quantum laws, the final state is (particle left, sensor measures LEFT, human sees "LEFT") + (particle right, sensor measures RIGHT, human sees "RIGHT").
Decoherece is Pointless: There is no exact point at which decoherence suddenly happens. All of quantum mechanics is continuous and differentiable, and decoherent processes are no exception to this.
The Born Probabilities: The last serious mysterious question left in quantum physics: When a quantum world splits in two, why do we seem to have a greater probability of ending up in the larger blob, exactly proportional to the integral of the squared modulus? It's an open problem, but non-mysterious answers have been proposed. Try not to go funny in the head about it.
Decoherence as Projection: Since quantum evolution is linear and unitary, decoherence can be seen as projecting a wavefunction onto orthogonal subspaces. This can be neatly illustrated using polarized photons and the angle of the polarized sheet that will absorb or transmit them.
Entangled Photons: Using our newly acquired understanding of photon polarizations, we see how to construct a quantum state of two photons in which, when you measure one of them, the person in the same world as you, will always find that the opposite photon has opposite quantum state. This is not because any influence is transmitted; it is just decoherence that takes place in a very symmetrical way, as can readily be observed in our calculations.
Bell's Theorem: No EPR "Reality": Originally, the discoverers of quantum physics thought they had discovered an incomplete description of reality - that there was some deeper physical process they were missing, and this was why they couldn't predict exactly the results of quantum experiments. But Bell's Theorem rules out any local deterministic physics with single unique outcomes. The math of Bell's Theorem is surprisingly simple, and we walk through it.