What does it mean for a wave to have an imaginary part?

[IronHamster]

This post is in General Math because it is focuses on the complex plane and justifications for using it.

I do not understand what it means for a wave to have an imaginary part.

I can understand expressing a wave as e^(iθ) and then extracting the information you want since complex exponentials are easier to work with than adding sines and cosines with different arguments. But according to the http://en.wikipedia.org/wiki/Schrödinger_equation#Derivation", we do not use this form for convenience but because the wave really has imaginary components.

What does this mean, that the wave has imaginary components?

 

[Hurkyl]

This post is in General Math because it is focuses on the complex plane and justifications for using it.

The justifications for using it are physics, not math. And the most basic and most important justification is simple -- QM agrees with reality, and kets in a (complex) Hilbert space (e.g. the Hilbert space of square-integrable complex-valued functions) have proven themselves to be a useful way to express states.

What does this mean, that the wave has imaginary components?

Just what it says -- the wave-function is complex-valued, not real-valued.

For what it's worth, components of the wave-function at a point are not physical observables -- the wave-function is not waving through "stuff" whose position and displacement-from-equilibrium we can measure, for example.

And you could split the wave-function into real and imaginary parts, but I know of no useful purpose for doing such a thing. (beyond appeasing people who are ideologically opposed to complex numbers)



Other ways to represent states can be useful. For example, density matrices are commonly useful. Also, the (pure) state space of a qubit can be represented as the Bloch sphere, giving a geometric picture of the state space of a qubit in three-dimensional Euclidean space.

 

[Petr Mugver]

I do not understand what it means for a wave to have an imaginary part.

Without necessairly going to QM waves, but considering all kinds of waves one encounters (EM waves, air waves, water waves etc) the wave function f(x,t) is conveniently expressed as a Fourier integral

$f(x,t)=\int g(k,\omega)e^{i(kx-\omega t)}\frac{dk d\omega}{4\pi^2}$

wich is essentially a sum of exponentials with a certain weight g(k,w). This is useful for a number of reasons. For example, certain integro-differential equations involving f become algebraic equations involving g. The cool thing that usually happens is that most physical operators are linear, so you can just see what theese operators do to a single exponential, and then sum up with g.

About reality, you can close an eye to the fact that the exponential is complex, and be sure that, after the sum is done, if the result must be real, it will be. (one often sees conditions like g*(-k,-w)=g(k,w)...)

 

[IronHamster]

The justifications for using it are physics, not math. And the most basic and most important justification is simple -- QM agrees with reality...

Just what it says -- the wave-function is complex-valued, not real-valued.

For what it's worth, components of the wave-function at a point are not physical observables -- the wave-function is not waving through "stuff" whose position and displacement-from-equilibrium we can measure, for example.

And you could split the wave-function into real and imaginary parts, but I know of no useful purpose for doing such a thing. (beyond appeasing people who are ideologically opposed to complex numbers)

I feel you are contradicting yourself here. You say that the wave function really is complex-valued in a physical way, but the imaginary part is neither physically observable nor useful to think of on its own, which suggests it is not physical but a mathematical convenience, which I suggested in the OP.

So I'll restate my question: is using the complex plane in QM a necessity (that is, matter waves must be expressed with the complex plane because they really do have complex parts) or is it a convenience due to techniques like the one suggested by Mr. Mugver (and you could theoretically express matter waves in real terms)? Is this even a valid question?

 

[disregardthat]

Using the complex plane is never necessary, as you will find many other algebraic structures satisfying the same properties. But why are one not inclined to make the same objections towards an algebraic structure isomorphic to the complex field if it is used in its stead? I think the core of the problem is naively psychological. It lies in the name, the complex numbers have "imaginary parts".

A physical model won't have any fundamental preference when it comes to mathematical notation.

 

[Hurkyl]

I feel you are contradicting yourself here. You say that the wave function really is complex-valued in a physical way, but the imaginary part is neither physically observable nor useful to think of on its own, which suggests it is not physical but a mathematical convenience, which I suggested in the OP.

The real part isn't a physical observable either. The overall phase of a wave-function has no physical meaning -- if $\psi$ is a wave-function, then both $\psi$ and $e^{i \theta} \psi$ denote the exact same quantum state. (for $\theta$ real)

Complex numbers aren't an intermediary as you might see when analyzing a real matrix (with complex eigenvalues) or summing a real sequence using techniques of complex analysis. The differential-operators-and-wave-function formalism for quantum mechanics really is set up for wave-functions to be complex-valued.


Technically, the use of any mathematical object to represent a physical state is a "mathematical convenience". But you have to have at least one mathematical convenience or you can't do anything.


As I mentioned, you can always split a complex-valued function into a pair of real-valued functions -- take the real part and the imaginary part. So if you wanted a purely real version of QM, you could reformulate everything in those terms. But it's a superficial way of getting rid of the complex numbers -- complex functional analysis really is still being used in the theory whether you express it in those terms or not.


I don't know if it is a useful example, but consider the pure state space of a qubit. It is spanned by two states: "spin up along the z axis" and "spin down along the z axis", which I will write as $|z+\rangle$ and $|z-\rangle$.

As mentioned $i |z+\rangle$ is also a ket describing "spin up along the z axis".

Some other spin states are:

• Spin up along the x axis: $(1/sqrt{2}) (|z+\rangle + |z-\rangle)$
• Spin down along the x axis: $(1/sqrt{2}) (|z+\rangle - |z-\rangle)$
• Spin up along the y axis: $(1/sqrt{2}) (|z+\rangle + i|z-\rangle)$
• Spin down along the y axis: $(1/sqrt{2}) (|z+\rangle - i|z-\rangle)$

Note the y-axis spins cannot be represented as a real linear combination of the z-axis spin states...

 

[octopus]

I think imaginary parts suggest some extra degrees of freedom for the wave function other than the three physical dimensions.
However, we can always combine wave functions in such a way that the projections on these degrees of freedom cancel out and we get a real wave function- that exists in the real physical 3D world.

 

[GustavoMerchan]

Please correct me if I'm wrong, but the question is not whether the imaginary part of the wave function represents any imaginary components of it, and whether these components have any physical meaning. Just to clear away any basic confusion, Quantum Mechanics, like all physics, deals with physical reality, so an "imaginary physical reality", such as an imaginary wave component, is a contradiction in itself. I agree that this should be treated in the General Math section because the question is a purely mathematical one. And it is why do we represent a wave function through complex exponentials. The answer is because not only it makes working with it easier (due to Euler's formula), but fundamentally because a complex representation of a wave (any wave, not only in QM) gives a fuller description of it at any given point. The real part describes the displacement vector form equilibrium at any point (x,t) of a traveling wave; and the imaginary part describes whether this displacement is increasing or decreasing (but not its rate - here I'm not too clear and it's why I got into reading the thread), without the need for a phase term. Simple. Since the wave has the same displacement for exactly two points of any of its cycles, the imaginary component carries the additional information needed to disambiguate and tell you "whatever real" displacement from equilibrium "and rising", or "and falling". It should be remembered that the complex plane had a real horizontal axis and an imaginary vertical axis. According to this representation we can define a unit circle centered in the origin, circulating through which we can obtain a simple harmonic oscillator, and since this is a two dimensional shape, well then we need two pieces of information to uniquely define any of its given points.