↑ comment by pragmatist ·
2014-04-11T21:52:45.129Z · LW(p) · GW(p)
I don't recommend Eliezer's sequence as a first introduction to QM, at least not if you're interested in developing a reasonably deep understanding of the theory. If you want a minimal-math introduction, I think a better bet would be to check out the first few chapters of David Albert's Quantum Mechanics and Experience, illegally available in it's entirety here. I don't think Albert's book is an ideal introduction either, but I do think it does a better job than the sequence at getting the salient points across in simple language. Also, since you're interested in ontology, the latter half of Albert's book contains a pretty incisive analysis of various interpretations of QM (not just MWI).
And here's an attempt at explaining quantum amplitudes:
Let's start with classical configuration space, since you understand that. The possible states of a classical system are represented by individual points in its configuration space (well, technically, the configuration space doesn't give you the complete state of the system, because it leaves out information about velocities, but let's ignore that for now).
In quantum mechanics, the configuration space looks just like classical configuration space, but its interpretation is very different. It's no longer true that the state of a quantum system is represented by an individual point in configuration space. The state of a quantum system is represented by a function on configuration space (the function has to satisfy certain other requirements in order to qualify as a bona fide representation of a quantum state, but ignore that detail also).
So imagine your configuration space is only two-dimensional for now, and again imagine that we are only considering real-valued functions on configuration space. In that case, you could construct 3-D plots of the various functions that correspond to quantum states, with the x-y plane representing the configuration space and the z axis representing the value of the function. Here's an example of one such function (this function actually doesn't satisfy those extra requirements I mentioned above, but we're ignoring those). Looks kind of like a wave, doesn't it?
So that function represents one possible state of the quantum system. Other possible states are represented by other functions. A quantum state is simply a function mapping each point on configuration space to a number; in other words, it associates a single number with every individual classical configuration. This number is what we call the amplitude of that particular configuration. Go back to our 3-D plot from above. The amplitude of any particular configuration (any point on configuration space) is just the height of our wave-like function at that point. And the quantum state is just the collection of all these amplitudes.
Now things are a little more complicated in quantum mechanics because the functions we consider aren't necessarily real-valued. They don't have to associate a real number with each point on configuration space. They are actually complex-valued functions, which means that each point in configuration space is associated with a complex number. That's no longer easy to plot in three dimensions unfortunately, so you'll have to use your imagination a little bit. And of course, the configuration spaces for actual systems we are interested have many many more dimensions than just two.
Now what the amplitude does is give you the probability of observing some particular configuration when you decide to look at the system.In order to figure out the probability of seeing a certain configuration, take the complex number associated with that configuration by the quantum state, and then calculate its squared magnitude. That will give you the probability.
Hope that helped a little.
Replies from: jd_k
↑ comment by jd_k ·
2014-04-16T23:18:26.057Z · LW(p) · GW(p)
(Sorry for the delay in response.)
That is extremely helpful; it is just the kind of explanation I was looking for. I have begun working through some of the materials linked here, as well. Many thanks. Now that I am starting to piece the picture together, I need some time to mull over it and let my intuitions adjust to the ideas, but I may send you a message when I next get hung up on it.