It's incredibly easy* to formulate quantum mechanics mathematically. The problem is, a lot of the time this is completely useless for the curious passer-by; or come to that the aspiring physicist. I thought I'd set down some of the principles, techniques, predictions and uses of quantum theory in a comparatively readable way, whilst giving hints as to how you'd go about making this into a complete mathematical theory (of which there is some more here).

* By which I mean briefly, rather than, say, comprehensibly...

What's Different and Why We Care

It's clichéd in introductory quantum mechanical texts to begin with the standard examples of why classical physics - the nice simple view of the world where you have lots of hard things bouncing off each other etc. governed by Newton's equation - is, in some situations, horribly wrong. It's clichéd, but it's a damn good idea.

Problems in classical physics

Aside: One thing that's worth briefly mentioning here is what's going on when we say a physical theory is "wrong", but still carry on teaching it and using it and saying it's "right". Usually, we mean something quite simple: the "wrong" theory is the correct approximation to the "right" theory - for example, suppose your "wrong" theory predicts a graph should be a straight line through the origin with slope 5, but your "right" theory says it's a straight line with slope 5 that doesn't quite go through the origin.

ctd. Then maybe in your day to day life, when you're looking at normal objects and so looking at the graph a long way from the origin, you don't notice it's just slightly off; maybe a 0.01% error. But then, one day, you get out the microscope and look at the start of the graph. Suddenly, the error looks bigger - and you get a bigger microscope - and then the error looks infinitely big as a percentage at the origin! Oops. But it's still an excellent approximation - and the "right" theory tells us it should be! So the "wrong" theory is still useful, provided it's - for example - easier to use than the "right" one.

ctd. We'll see below that quantum mechanics as formulated here is also "wrong" in this sense - quantum field theory is the next step on the way to truth (and it's not even right).

• Stability of atoms & the spectra of light - The start of the 20th century had a model of the atom as a positively charged nucleus surrounded by negatively charged particles. The problem is, classically, the only way you can have things that attract each other (nucleus and electrons) keeping their distance from each other means having the electrons orbiting the nucleus - moving around the nucleus in circles or ellipses or some such - but when electrically charged things change direction or speed (either way, they accelerate) they emit radiation. This is just how radio works - a current in the antenna makes electromagnetic waves. (Think of pulling your hand through water. The drag you feel is because you lose energy to the waves that go rolling away. To the electron, which is charged, the whole of space feels like it's filled with water!) But if electrons in atoms did this, they'd lose energy and spiral into the nucleus. Oops.
• The 'ultraviolet catastrophe'
• The photoelectric effect
• Wave-particle duality

Things that work because of quantum mechanics

• Modern electronics - transistors and diodes
• Chemistry
• Flash memory
• Lasers
• MRI (Magnetic Resonance Imaging)

Take-home Facts

[to come]

The Classical Limit

[to come]

Quantum Field Theory

[to come]

How To Build the Mathematics

[to come]

How To Think About Quantum Theory

A lay person's guide to the intuition behind quantum mechanics

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