Why is Complex-Number Math Essential in Quantum Physics?

[Dense]

Is "i" Really Imaginary?

Complex-number math is very important in quantum physics. Is this because square roots of negative numbers are actual quantities being measured/calculated?

Or is it that imaginary numbers aren't occurring for real, but complex math nevertheless represents very well what's actually going on -- sort of like, "if you think of this as occurring in a complex plane, even though it isn't, it all works out."

 

[Galileo]

Well, the term 'imaginary' is ill-chosen, since all numbers (rather, all mathematical objects) are imaginary in the sense that they do not exist in a physical sense.

Complex numbers allows us to calculate stuff according to the laws of quantum mechanics, but in the theory all observable quantities are always real (as they should!). So in your latter remark you have the right idea.

Ofcourse we never really measure abstract numbers (expect for ratios). We don't measure '3', or '5', but rather '3 kilogram' or '5 meters'. Ratios wrt a chosen unit measure. So measuring imaginary numbers is nonsense.

 

[pr0pensity]

My calc 3 teacher showed us something that may or may not be trivial, but it made me think a lot. He showed the i axes as just being rotations around the x, y, or z axes. I guess these would be infinitely indistinguishable if the i axes modify points in space, but it opens my mind at least to the possibility of other dimensions.

 

[ZapperZ]

Complex-number math is very important in quantum physics. Is this because square roots of negative numbers are actual quantities being measured/calculated?

Or is it that imaginary numbers aren't occurring for real, but complex math nevertheless represents very well what's actually going on -- sort of like, "if you think of this as occurring in a complex plane, even though it isn't, it all works out."

Are you also aware that "complex numbers" are also extremely important in OTHER branches of physics AND engineering? Look in electrical engineering, for instance.

So is there a particular reason why you are asking this question only in reference to quantum physics?

Zz.

 

[berkeman]

One big motivation for using complex functions is that they are eigenfunctions for linear time-invariant systems. LTI systems are very common in engineering and physics, so that's why complex analysis is so prevalent.

 

[Meir Achuz]

Complex-number math is very important in quantum physics. Is this because square roots of negative numbers are actual quantities being measured/calculated?

Or is it that imaginary numbers aren't occurring for real, but complex math nevertheless represents very well what's actually going on -- sort of like, "if you think of this as occurring in a complex plane, even though it isn't, it all works out."

What you say in your second paragraph is a good interpretation.
Square roots of negative numbers are NOT actual quantities being measured.

All observable quantities in QM are real. Using complex numbers is a convenient mathematical way of doiong intermediate calculations.
It all could be done with trigonometry, but would just be more complicated.

 

[selfAdjoint]

Complex numbers are the largest algebraically complete field. This means they satisfy all the familiar laws of arithmetic plus, any polynomial with complex coefficients has all complex roots. You can't say that about "real" numbers.

Real numbers are embedded in the complex numbers as a special case; that's why you can get operators defined over the complex field that have real eigenvalues. And the rotation gimmick that teacher showed is crucial. ALL rotations in the plane can be represented by $$e^{i\theta} = cos \theta + i sin \theta$$.

 

[Integral]

The appropriate home for this disscussion is the math forum. I have moved it.

 

[Hurkyl]

We use a number system simply because it does what we want it to do.

Real numbers are ordered. The real number line has no holes. Thus, the real numbers tend to be good for describing ordered things like distance, or temperature.

Complex numbers have a phase and magnitude. The complex plane has no holes. Thus, the complex numbers tend to be good for describing things with phases and magnitudes, such as waves.

 

[Tide]

All of physics and engineering could be done without resorting to complex numbers. People choose to use complex numbers in these disciplines because it simplifies the analysis tremendously!

 

[Dense]

Thanks for all the responses, they have removed a bit of fuzz that was obstructing my understanding of some fairly basic concepts. (Well, all but one of the responses, anyway .)

 

[quantumdude]

Complex-number math is very important in quantum physics. Is this because square roots of negative numbers are actual quantities being measured/calculated?

No, it's because the objects of quantum theory cannot be accommodated with a mathematical structure 'smaller' than a complex vector space. In the first chapter of his book Modern Quantum Mechanics, Sakurai shows how the use of complex vector spaces is necessitated by the humble Stern-Gerlach experiment.

Or is it that imaginary numbers aren't occurring for real, but complex math nevertheless represents very well what's actually going on -- sort of like, "if you think of this as occurring in a complex plane, even though it isn't, it all works out."

I'll just echo what others have said: The imaginary numbers aren't any more or less real than the so-called real numbers. Further, I would not agree with the statement that observables must be real-valued. See the following threads for discussion on that.

Hermitean operators (Physics Forums)
http://www.scienceforums.net/forums/showthread.php?t=11193 (Science Forums Network)