Mathematica Physics and the 'i' (mathematical term)

[Pengwuino]

Do imaginary numbers ever manifest themselves in the physical world as we know it? And when I say that, i mean do they ever appear in solutions to problems that we know have physical meaning and aren't in the solutions or derivations purely as mathematical shortcuts?

Hopefully this question makes sense as I am not well versed on that advanced of mathematics.

 

[franznietzsche]

Do imaginary numbers ever manifest themselves in the physical world as we know it? And when I say that, i mean do they ever appear in solutions to problems that we know have physical meaning and aren't in the solutions or derivations purely as mathematical shortcuts?

Hopefully this question makes sense as I am not well versed on that advanced of mathematics.

In quantum mechanics and in Electronics. They are used to separate orthogonal quantities.

 

[topsquark]

Do imaginary numbers ever manifest themselves in the physical world as we know it? And when I say that, i mean do they ever appear in solutions to problems that we know have physical meaning and aren't in the solutions or derivations purely as mathematical shortcuts?

Hopefully this question makes sense as I am not well versed on that advanced of mathematics.

As far as the appearance of anything physical, usually an "i" will show up in something, such as an exponential term, that represents a quantity leaving a system. In particular it often shows up in a damping term in oscillatory motion or as the source term of an energy "bleed" where energy is leaving a system.

-Dan

 

[Pengwuino]

topsquark,

Are imaginary numbers the only way to correctly show the system or is it just a mathematical shortcut?

I got to stop asking questions where i don't expect to understand half of the responses...

 

[topsquark]

topsquark,

Are imaginary numbers the only way to correctly show the system or is it just a mathematical shortcut?

I got to stop asking questions where i don't expect to understand half of the responses...

I suppose if we knew everything we would be able to have a theory to track, say, what happens to the energy of an electric field as it penetrates a nonconductor and attribute the energy loss to the molecules in the non-conductor. That's a perfect world and as there are rather many molecules in even a small sample of matter, I doubt anyone is going to work on the problem in the near future!

Still and all, there are many places in which "i" is useful...so many that I don't know where Physics would be without it. One of my favorite spots in Physics is when we try to describe a Quantum Field Theory for spin 1/2 particles. There is something called a "Grassman Algebra." A Grassman algebra consists of a set of "anti-commuting complex numbers." That is to say all of the elements of the set may be represented by a+ib. But in this algebra (a+ib)(c+id) is not the same as (c+id)(a+ib), in fact one is the negative of the other! The use of this algebra is not a mere mathematical trick: in order for the field theory of spin 1/2 particles to give any sort of realistic results we MUST have such a set of numbers. In this part of Physics, at least, "i" is a very real number. (Sorry, I couldn't help the pun! )

-Dan

 

[Pengwuino]

So the answers yes and I've learned to stop asking questions where the answers will leave me speechless adn confused haha

 

[MiGUi]

Complex functions are sometimes useful, but physics quantities must be real. In QM, observables are hermitian operators, so its eigenvalues are real. We use, for example, imaginary exponentials to make it easier cause it is hard to work with trigonometric functions.

 

[Meir Achuz]

topsquark,

Are imaginary numbers the only way to correctly show the system or is it just a mathematical shortcut?

They are just a math shortcut. It could all be done with trigonometry, but physicists hate trig for good reason.

 

[Hurkyl]

Of course, real numbers aren't the only way to represent things either. You can express everything in terms of the number "0" and the "increment" operation... and arranging them in clever ways.

Of course, it would be very cumbersome to do that -- so we use real numbers for things that are well described by real numbers, and complex numbers for things that are well described by complex numbers.

The predjudice against complex numbers is just a historical thing from back in the day when numbers really meant lengths of curves, areas of shapes, and other similar things.

 

[HallsofIvy]

All numbers are "mathematical shortcuts"!

 

[Integral]

One could say that any oscillation is a manifestation of imaginary numbers. We use Trig functions to hide the fact that imaginary exponentials are the more fundamental mathematical construct.

 

[chroot]

Actual physical measurements are always real, as has been said.

Complex numbers make life easier for many fields, like quantum mechanics, but one is not required to use them. You can always just cast your system in terms of a two-dimensional variable (that's all a complex number really is -- a two-dimensional vector) and proceed as usual.

- Warren

 

[Hurkyl]

Of course, a real number is nothing more than a special kind of complex number, so you're working in the complexes anyways.