Okay, so when we talk about classical systems, our description of the configuration of the different states is given by a probability. So I might say my bit is a mixture of 50% 0 and 50% 1. Now when we move to quantum theory we no longer have probabilities but instead have complex numbers. But what symbol am I supposed to use for this? My state is a mixture of 1/sqrt(2) q% 0 and 1/sqrt{2} q% 1? Mabye we should invent a new symbol which is the % sign but with the slash the other direction? Or turn those 0s in the % sign into “q”s?
O brave new quantum world!
maybe some notation with a sqrt sign and/or a partial circle (phase) around the percent sign.
I really like the idea of q/q, and I bet with some clever fussing with baselines and spacings one could make it look nice in LaTeX.
Basing it on the % sign seems like it would only be a good idea if you intend to multiply by 100. I like incorporating a square root, since that reflects what you’re doing as well.
Hmmm. I’m not sure I agree with the premise. Perhaps you shouldn’t try to generalize the percent sign in the first place.
Why? Because, in a nutshell, it promotes a deceptive intuition, which is that superpositions really are, in some sense, mixtures of a particular basis.
Of course, we all know and abhor the preferred ensemble fallacy. We still teach it that way, though: “Here’s the |+> state; it’s a superposition of |0> and |1>.” We do that because this stuff makes no sense at all to students at first, and a gentle seduction is needed. It’s wrong, though — |+> isn’t a superposition of |0> and |1> in any fundamental sense, any more than it’s a superposition of a bajillion other orthogonal pairs. It’s a state all of its own, and it’s got rights, dammit! Every state is precious!
Tuning down the facetious rhetoric, my point is that “per cent” is an ancient term that implies probabilistic, convex combination. Mud is 50% dirt and 50% water; take 100 parts of mud, and it can be separated into 50 of dirt and 50 of water. We have a concept like that in quantum mechanics, convex combination, and it works just like the classical case. Students often absorb the misconception that somehow classical mixture gets replaced with quantum superposition — which, of course, isn’t true at all. Superposition is new, different, and freakin’ weird!
I’d rather see somebody try to teach quantum mechanics _without_ using superposition notation (at least for a while), than see a snazzy notation for it. For one thing, more students might say, “Wait, how the heck do we pick a basis?” which is a darn good question. Superposition notation can imply a preferred basis, which in turn induces an unjustified comfortability with something that’s really a fairly major conceptual issue.
*quietly breaks down soapbox, exits stage left*
Robin, didn’t you get the memo establishing a canonical basis standard?
What? Oh, you mean that? I musta missed it…
…I used to get all this annoying junk mail from The Environment, trying to tell me what to measure…
…I finally got an unlisted Hamiltonian.
I’m not sure if the 1/\sqrt{2} q% notation is a good choice. The “percent” of the % symbol means parts of hundred and q% implies something similar: “parts of a whole”. But amplitudes of different states summed don’t give the same value. Example: 1/\sqrt{2} + 1/\sqrt{2} != \sqrt{3}/2 + 1/2
(This is similar to what Robin argued.)
What’s wrong with the term “amplitude”?
How does the q% symbol relate to probabilities involved when measuring pure quantum states or mixed quantum states?
You all take my posts way to seriously 🙂 😉 Trudging onwards in spite of all the criticism, how about [tex]$\sqrt{\%}$[/tex]?
I think mathematical notation is starting to approach Chinese ideograph-like levels of complexity; it seems to be driven by the theorist’s desire to have one symbol for each concept.
Computer languages evolved from 1-character variable names a long time ago, maybe math can, too.
Yes, I worry about these things.
What about manipulating the percentage sign so that it is actualy a q. The circle at the top of the q becomes the top circle in the percentage sign, and the tail of the q is curved to look like a partial circle (the bottom circle in %). These two characters would then look similar, but still be easily distinguishable.