Imaginary numbers and Quantum mechanics

[Arp220]

Ok, so this has always puzzled me a bit. Tell me what you think.

Consider the following:

Over a century ago, some geeky mathematician types were sitting around, and idly thinking about number theory. Or something. And they got to chatting about square roots and cube roots and so on. And realised that mathematics as they knew it had no way to deal with the root of a negative number. So they decided address this by calling the square root of '1 'i'. And then had a grand old time messing about with this utterly arbitrary, completely theoretical mathematical construction that had no practical application and no basis in reality whatsoever.

And then, a few decades later, a whole bunch of scientific thingies came along, most notably quantum mechanics, but also things like signal processing, where imaginary numbers were essential to their success.

Now, we can all see examples where whole numbers, fractions, percentages, negative numbers, etc etc are 'real' - for example credits and debts require negative numbers, computing ares requires fractions (usually), and so on.

But then a whole bunch of sciency stuff comes along that absolutely requires imaginary numbers for their theoretical underpinnings, despite the fact that imaginary numbers were devised as little more than an intellectual masturbation exercise several decades earlier, with no though that they represented, or indeed could represent, anything 'real'.

So what puzzles me is this - by this logic, imaginary numbers are also real. But in what sense are they real? What aspect of reality does the square root of minus one describe?

[knowledgeORbust]

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In what sense are they real?

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This seems to be summed up in Plato's conception of a mathematical world, along with the mental and physical worlds. The math is created, for us, in the mental, and its laws and predictions are observable in the physical. So we can't empirically *sense* math, and it appears separate from our mental plane; But for the mathematicians who create math, and for the students who recreate it, the experience is certainly real. The laws are real. Math has real uses.


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What aspect of reality does the square root of minus one describe.

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Maybe none; maybe we'll find one; maybe it's useless in our physical world. It doesn't really need to describe anything to exist.

[gumpzilla]

Well, in quantum mechanics, you do use imaginary numbers in calculating wavefunctions. But, as you might also know, one of the axioms of QM is that any physically observable quantity is represented by a Hermitian operator. Those operators have the special property that their eigenvalues are purely real, which also means expected values of those operators will be real. So, to a certain extent, you dodge the issue that way.

There are some sort of tricky formal exceptions to this. Eventually you learn to deal with states that have finite lifetimes - this is how you explain something like stimulated emission. Formally, one way to handle this is to allow the energy eigenvalues of those states to have an imaginary component. Then when you put the time dependence in, in addition to the usual oscillatory factor you also get an exponential decay in the probability of measuring your particle in that state.

Given all of this, it's obvious the complex numbers are super convenient. They're also convenient for classical oscillatory problems, but in those situations you can also do all of the same tricks with purely real functions, provided you also account explicitly for phase differences. (That is, you still need two numbers to address the solution.) I suspect, but am by no means sure, that you could conceivably do the same thing in QM. But it would probably suck.

Complex numbers (and complex valued functions) are very, very special. I've been slowly reading through Penrose's The Road to Reality, and he has a bunch of chapters on there about some of the special properties of complex functions that might be approachable if you want to learn more.

[Siegmund]

I have to point out that imaginary numbers weren't invented by people idly sitting around thinking about number theory; they were used, over 450 years ago, in the intermediate steps of the practical problem of solving cubic equations. It was quite a long time later that they were accepted as interesting theoretical constructions in their own right, and it ceased to be necessary to "keep them hidden under the hood".

In that light it's somewhat less amazing that science has found subsequent uses for them.

[thylacine]

Don't confuse the technical and colloquial uses of the words `real' and `imaginary'.

i is more real than most real numbers.

Most real numbers are more imaginary than i.

If you realize how the real numbers are constructed mathematically, you'll realize there's no way you can realistically model reality using the real numbers. For real!

[Metric]

I knew a guy that was fascinated by this kind of question, and it motivated him to work out the theory of quantum mechanics over the quaternions. Nothing really earth-shaking came out of it, though, as I recall.

Basically, complex numbers are a "quantum leap" more useful than the reals because they're closed under algebraic operations (unlike the reals). If you write down a polynomial equation using only complex numbers, you can rest easy that its solutions will still be complex numbers. The reals don't have this feature, e.g. x^2 = -1

http://en.wikipedia.org/wiki/Fundame...rem_of_algebra

[PairTheBoard]

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for example credits and debts require negative numbers ...

What aspect of reality does the square root of minus one describe?

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If you can accept magnitudes with the additional feature of a + or - sign attached to them, then complex numbers can be thought of as magnitudes with the additional feature of "phase" attached to them. Just as two negatives become a positive under multiplication, so the Phases add under multiplication of complex numbers.

So complex numbers are made to order for dealing with wave functions where the phases of the waves act to interfere with each other. Furthermore, ordinary functions can be decomposed into wave functions so that we could think of wave functions as being the fundamental ones if we happened to think that way. Thus making complex numbers most natural for descibing wave functions the more fundamental type numbers for describing reality.

PairTheBoard

[m_the0ry]

Couldn't agree more, PTB. Waves are so ubiquitous in nature, and complex numbers explain them better than 'real' numbers could ever hope to. Phase not only correlates to observable phenomenon but also establishes the idea of orthogonality between dimensions.

[pzhon]

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But then a whole bunch of sciency stuff comes along that absolutely requires imaginary numbers for their theoretical underpinnings,

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You don't have to deal with imaginary numbers if they bother you. You can deal with real 2x2 matrices of the form

a b
-b a

Such matrices add and multiply just like complex numbers do.

Complex numbers don't have to be used in many of the disciplines where they are used. They are just much simpler than the alternatives, because they allow you to deal with things such as exponentials instead of sines and cosines, since sin x = (e^ix - e^-ix)/2, and to deal with one number instead of a vector or matrix.

Similarly, if you are discussing rotations in 3-space, you don't need to use quaternions. Most people just use matrices. However, for some purposes, a quaternion is more convenient.

[DarkMagus]

Here's a way I read to think about it:

Think of the natural numbers. They're used to count things... 1 person, 2 apples, whatever.

Now think of the integers, which introduces negative numbers. You can't have -1 people or -2 apples. But you can have something like a bank balance of -$100, meaning you're overdrawn, or a velocity of -10m/s, meaning you're going in reverse. It doesn't make sense to use negative integers to count discrete objects, but there are other contexts for which it does make sense.

Now the next step is real numbers. They're used to measure indiscrete things. You can't have 4.35354 people or own sqrt(2) cars, but you can measure your height to be 1.78 m, and whatnot. Again, they make sense in some contexts, but not in others.

Now step up to complex numbers. They make perfect sense in quantum mechanics, signal processing, and other mathematical applications; but in everyday life, there just aren't any contexts in which they make sense. Complex numbers are very real and are used to describe real things, it's just a lot more abstract than the real numbers we're used to (not to mention that their name is misleading!).

Hopefully this way of thinking might help to clear the issue up in your mind. It does for me.