Complex numbers-roles in phyiscs

[neginf]

How are complex numbers used in physics ?
Do they have any physical meaning ?

 

[LostConjugate]

The square root of a negative number is a negative number when squared and a positive number when cubed. This is convenient as it has the properties of being oscillatory.

It also happens that $$e^{ix}$$ = cos(x) + (i)sin(x)

It also provides a solution to the equation $$x^2 + 1 = 0$$

 

[Dyson]

You can find complex numbers in many famous equations such as Schroedinger equation, Dirac equation...
It seems we often use the form of e(ix). X often represents the phase factor or a certain type of operator.
The frequent Fourier transformation is also relative to the complex numbers. We can not go further without the complex numbers.
The introduction of complex numbers also bring us much convenience. For instance, the calculation of total amplitude in single slit diffraction.
There are also many applications...

 

[neginf]

Are there any physical quantities that are complex ?
Do physical quantities need to be ordered ?

 

[LostConjugate]

Are there any physical quantities that are complex ?
Do physical quantities need to be ordered ?

There are no physical quantities that are complex. Don't let this fool you into thinking that it is not a valid mathematical tool though. Just as you can't have an irrational number of apples, or a fraction of people.

I am not sure what your second question is asking.

 

[neginf]

Thank you for helping, and I won't be fooled into thinking the complex numbers aren't valid tools.
I want to just forget about the second question.
Thanks again.

 

[Studiot]

The square root of a negative number is a negative number when squared and a positive number when cubed.

Excuse me?

Are you asserting the i is negative, since i x i x i = -i ?

 

[Studiot]

neginf,

You are right is spotting that complex numbers do not possesses the ordering property, like real ones do.

If you are familiar with the Argand Diagram you will understand the idea that ' i 'can introduce something at right angles.
Not only at right angles to a line as in Argand, but at right angles to a plane. This has implications in fluid mechanics where we can introduce (or take away) fluid at right angles to the plane of a flow pattern.
The mathematics of this is called conformal mapping and is part of complex analysis. Complex analysis is (like) Real analysis, only using complex numbers, instead of real ones.

 

[K^2]

The square root of a negative number is a negative number when squared and a positive number when cubed. This is convenient as it has the properties of being oscillatory.

I'm guessing it's just a typo, but i³ = -i = 1/i. You need 4th power to get positive.

Also, in terms of actual algebra, it is incorrect to say that i = √(-1). It can be shown that this leads to contradictions. It's is better to define i² = -1. It might seem like the same thing until you consider the fact that (-i)² = -1 as well, which can lead to ambiguities with the square root definition. To avoid such ambiguities, square root function is defined only on [0, ∞).

Strictly speaking, you can never take a square root of a negative quantity. It can sometimes be taken as a shortcut, but you have to always re-check that the answer you get by taking such a shortcut makes sense. Fortunately, most formulae where you can run into trouble already have the ± sign built in. Think of quadratic formula for example.

 

[LostConjugate]

Excuse me?

Are you asserting the i is negative, since i x i x i = -i ?

I am confused, what I said was correct, and your statement is correct as well. I don't see how that shows that i is a negative number though, i is the square root of a negative number.

Edit. Oh I see the typo now. I did not mean to the power of 3 when I said cubed, meant the power of 4.

 

[Studiot]

Whilst i raised to any even power is definitely positive, i itself is neither positive nor negative.

 

[K^2]

Now you're making mistakes. Imaginary unit to even power is real, but it can be positive or negative.

 

[Cleonis]

How are complex numbers used in physics ?
Do they have any physical meaning ?

As to the second question:

Compare negative numbers. Do negative numbers have physical meaning? For instance, is there such a thing as negative temperature? Well, as we know: if you use the Kelvin scale to represent temperature, then there is no negative temperature, but if you use the Celsius scale there is.

More generally, whether there is an application for a particular class of numbers depends on how the physics is represented.

The set of positive integer numbers has the widest range of applications. The more specialized the class of numbers, the smaller the range of applications.

There are quite a lot of situations where complex numbers offer the most efficient mathematical representation. Anything that can be represented with complex numbers can be represented with other mathematical means too, (such as vectors) but not with equal efficiency.

An even more specialized class of numbers is quaternions.

 

[Jsanders]

There was also some speculation a while ago, about the equation M=M(observed) / (Lambda) in special relativity, and say the speed was 2c (speed of light), this would lead to a mass of about -i M/1.7 .

M/sqrt(1-V^2/c^2) = M/sqrt(-3).

Some theorized that this was resembling the tachyon, a particle that travels faster than the speed of light. No evidence has been found for this existence, but interesting nonetheless. It also comes up in relativity when relating to spatial distance as ids and time as ds.

 

[Studiot]

Now you're making mistakes. Imaginary unit to even power is real, but it can be positive or negative.

Yes you are right, perhaps I should retire?

 

[LostConjugate]

Whilst i raised to any even power is definitely positive, i itself is neither positive nor negative.

i to the power i is also a positive number. Which is kind of cool.