# Are wave functions really complex?

#### kiwiheretic

Why is the wave function complex containing both a real and imaginary part? The wave function I am talking about is the one which when you take the square of it it gives the probability density distribution of where the particle might be found.

Are we saying that the real world has an imaginary component? Or is the use of imaginary numbers simply a mathematical technique to avoid more complex mathematical machinery and there is a more complex mathematical description that does not employ these imaginary numbers?

#### studiot

What do you have against the complex representation, it is used everyday by electrical engineers supplying your power.

It is also used when we have radial symmetry and can reduce the number of dimensions by one from 3 to 2.

#### kiwiheretic

I have nothing against complex numbers when used as a mathematical convenience for simplifying otherwise complex calculations. (That would be how they are used in electrical engineering. Also in EE the use of complex numbers does not preclude an alternative, albeit less simple, mathematical model.) However, I dislike the propensity of making up stuff as we go along and calling it fact (like string theory).

#### studiot

One of the beauties of mathemtics is that there are often more ways than one to skin a cat.

But the current in an inductor or capacitor is purely imaginary when complexors are invoked.

The exponential form often give algebraic and/or calculational advantages over the trigonometric forms.

#### kiwiheretic

One of the beauties of mathemtics is that there are often more ways than one to skin a cat.

But the current in an inductor or capacitor is purely imaginary when complexors are invoked.

The exponential form often give algebraic and/or calculational advantages over the trigonometric forms.

I have no idea what a complexor is. Please post link illustrating what and how they are used in EE.

#### studiot

I thought you understood the use of complex variables in electrical engineering from your post 3.

A complexor is a special form of vector that represents the current or voltage or impedance in an electric circuit and the equations that govern these circuits.

For example the writing in red here

http://www.regentsprep.org/regents/math/algtrig/ato6/electricalresouce.htm

#### kiwiheretic

I thought you understood the use of complex variables in electrical engineering from your post 3.

A complexor is a special form of vector that represents the current or voltage or impedance in an electric circuit and the equations that govern these circuits.

For example the writing in red here

http://www.regentsprep.org/regents/math/algtrig/ato6/electricalresouce.htm
Yes, I understand that complex numbers *can* be used in EE. I am not getting that complex numbers *must* be used and that no other formulation is possible. My understanding of phase and frequency is that both these quantities can be represented by real scalar quantities. I am not following your assertion that they *must* be used in this instance.

#### studiot

After review of my posts, I honestly can't see what gives the idea the I am suggesting complex numbers must be used. Indeed I though I had commented how fortunate it is that there are usually several ways to solve a given problem.

A further question.
Do you understand the complex conjugate and what happens when you multiply a complex number by its conjugate?

This is important because, in your initial post, you mention the square of the wave function.
In fact if you just square it you will still have a complex number, but if you multiply by the conjugate you will not.
This is part of a larger area of mathematics called self-adjoint problems.

#### kiwiheretic

Yes, I understand what the complex conjugate does. Yields a "length like" property akin to vectors.

As I am not a professional physicist I am not always able to come up with the correct terminology for phrasing my question so I see how the purpose of my question was misunderstood. Maybe a better way would be to ask if the imaginary component of aa wave function is "ontological". (Still not supremely confident this will cause my question to be understood but I'll try that word and hope for the best.)

I understand the notion of impedance in EE and how its made up of resistance and reactance with reactance being the imaginary component. However even this notion is based on pure sine wave signals and doesn't work so well for square waves, etc. You can say you have complex voltages and currents on paper but you can't measure those with a volt meter or amp meter or an oscilloscope and as far as I know there are no imaginary volt meters, etc, for measuring such things.

The difference with EE and QM is that with EE most people admit a non imaginary formulation is possible (although not as simple). I dont think the same claim is made for QM as the QM crowd seem to believe "thats how the world really is".

#### topsquark

Forum Staff
For the wavefunction comment consider that the wavefunction of an electron does not actually vibrate in the "real world." (Most wavefunctions don't.) For that reason alone we need something like a complex number. Spin is another example...One of the spin 1/2 operators needs to be non-real, otherwise the spin measurements make no sense. etc. etc.

Other cases where we can get away with not using an imaginary component but where it is used anyway is typically to turn two equations into a single equation that explains two related things. For waves (optical and otherwise) we can use complex numbers in the index of refraction not only to get intensity but also absorption information. In EE it is common to use the strategy to give information not only about transmission properties but also resistance. (Generically when we use a complex exponent we expect to see losses in the property we are measuring.)

Sometimes complex numbers are required and other times they are simply useful. And I know of no instance where complex numbers have been introduced for no reason...in fact in most calculations we avoid them where possible. For example there is no Quantum treatment of the infinite square well potential where we assume that the normalization coefficient is complex because we can achieve the same results by assuming it is real.

Other kinds of complex numbers show up all the time but they are buried in the Math terminology we use. For example the spin 1/2 operators actually fall into a representation of the "quaternion group" where we have not one kind of imaginary unit, but three. i^2 = j^2 = k^2 = -1 where none of the i, j, k are equal.

On the one hand the imaginary unit is, well, imaginary. But it's so d*mned useful that I see no reason to not admit that such a thing really exists.

-Dan

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