Significance of complex exponential representation of waves

Question

I am starting to learn about the complex exponential form of waves and SHM and was wondering what the significance of the imaginary part is. I know this question has been asked many times in different ways, but I think my question here will be slightly different.

From what I have seen, we use the complex representation because of its convenience as many calculations are simplified (like integration, differentiation, wave superposition and enveloping etc) but we have to extract the real part to get the physical reporesentation. The complex part is useful for this also because it consists of the real and imaginary part which are orthogogonal and so when we apply linear operators such as derivatives we can take the real part before or after applying these. This is not so for non[linear operators, such as getting the power/energy/intensity of a wvae which all depend on the square of the amplitude; the real part must be taken before squaring.

I also know that there is some significance of the imaginary part in quantum mechanics as probabilities are found by multiplying the wavefunction with its complex conjugate, which does not give the same result as just squaring the real part.

So it seems to me that the complex representation can just encode extra information in a convenient way. From what I have read I have not seen there being any extra phsyical interpretation of the imaginary part. I was wondering if someone could clarify or add to this? Parhaps I am also missing something- some physical phenomena where the complex nature of waves becomes apparent and physical results cannot be explained simply from a real representation of waves (whether in quantum mechanics or not)?

Answer 1

Here's one convenience of the complex solution. Imagine observing an object on an (invisible) bicycle wheel from the side. It is oscillating only in one dimension.

$$x=x_0\sin(\omega t)$$

will describe that. However, let's bring in conservation of energy and break that model.

Consider the energy of the system at time t, first we need velocity. $$\dot x=x_0\omega \cos(\omega t)$$ then kinetic energy at time T is $$\frac{m\dot x^2}{2}=\frac{mx_0^2\omega^2 \cos^2(\omega t)}{2}$$ Notice that when $\omega t=\pi /2$ then this is 0. Energy appears not to be conserved, it oscillates over time. This can't be a full description of our system.

To cause oscillatory motion physically (and generate waves) there has to be a restorative force, and that doing work stored as energy. In this case of the bicycle wheel it is stored in the hidden dimension of motion of the object as it moves away from and back towards the observer.

so we could imagine such motion exists even though we cannot observe it (in this case such conjecture would be correct). All we really need to assume is that energy must be conserved "somehow".

$$x=x_0(\sin(\omega t)+i\cos(\omega t))$$

We have imagined where the energy went, and the energy expression comes out

$$\dot x=x_0\omega (\cos(\omega t)-i\sin(\omega t))$$ $$\frac{m\dot x^2}{2}=x_0^2\omega^2 \bigg(\cos^2(\omega t)+\sin^2(\omega t)-2i\sin(\omega t)\cos(\omega t)\bigg)$$ We know that $\sin^2+cos^2=1$ We know that the imagined motion is in an orthogonal/perpendicular direction to the real - that no imaginary motion can be interpreted as real motion. So we can discard terms that are zero in this case (the imaginary terms). gives a nice constant expression as desired $$\frac{m\dot x^2}{2}=x_0^2\omega^2$$

Energy is conserved. We haven't only assumed that energy is conserved and imagined that something like equivalent to circular motion is going on.

In practice the cause of the restorative force and therefore the hidden energy storage mechanism behind harmonic motion, including that which emits waves, can be something else entirely from that imagined. Extension of a spring, gravitational potential, electric potential etc.