After all our goal is to explain physical phenomenon...what if we venture into this jungle of real functions and come up with a totally different theory which explains physical phenomenon.
Good luck to you.
First of all you are wrong that classical physics did not use imaginary functions. The solutions of Maxwell's equations expressed as imaginary functions are more general and universal than sines and cosines.
The simplest set of solutions to the wave equation result from assuming sinusoidal waveforms of a single frequency in separable form
$$\mathbf{E}(\mathbf{r}, t)=\mathrm{Re}\{\mathbf{E}(\mathbf{r} e^{i\omega t}\}$$
Imaginary functions are a useful tool in integrations and descriptions of real data.
((If you ask me to do research in theoretical physics, I'll throw all the QM books in garbage (no disregard though) and start thinking from this point of view...Thats my style of working!)
With such blind spots I am sure nobody will ask you to do research in theoretical physics.
The difference between the classical use of imaginary functions from the solutions of the wave equations and the quantum mechanical one is the postulate the posits that the square of the wavefunction is real and gives the probability of an elementary particle (or nuclear) interaction to be observed. When in the microcosm quantum mechanics reigns. There one cannot take a ruler, mark it and measure, it was found that the theories and data agreed when the probability postulate was imposed. One has to make many measurements and get the probability distribution for a particularly desired value.
The above link discusses the postulates of quantum mechanics which were not imposed out of a freak imagination, but were necessary to be able to calculate and fit known observations, like the hydrogen atom, and predict the outcome of experiments and observations.
EDIT to address the last part of the question:
((If you ask me to do research in theoretical physics, I'll throw all the QM books in garbage (no disregard though) and start thinking from this point of view...Thats my style of working!)
That works for art, art is much less dependent on data bases of observations and the tools that can be used.
The fact that for two thousand years people have been creating models of physical observations, and particularly the last 300 a data base of mathematical tools too, constrains creativity in science. The mathematical tools have been used to model all observations up to now. These models are in a way a shorthand description of nature that could be used in many ways instead of going back to the data itself. There exists a frontier of experimental research where the models have not been validated , and that is where new thinking can come in.
My expected answer is in this spirit, "Hey if you go in that direction, you are bound to end up in a quick sand, for so and so reason"
If you go into the direction of throwing everything away you will end up with vague models like the Democritus atomic model, or the phlogiston theory, in your own words. The mathematical models used now are validated, some of them to great accuracy. New mathematical tools to model the already modeled data would only be worth the attention if something new and unexpected is predicted and found in the experiments.
There are people working off the beaten track theories, trying to explain quantum mechanics by underlying deterministic theories. These people have a thorough knowledge of existing mathematical tools and the physics models that have been validated. They just want to work at the frontier by ignoring that mainstream physics considers their effort contradictory or impossible/prohibited by the postulates of quantum mechanics and special relativity. An example is the current research interests of G.'t Hooft who has also participated here a while ago .
So if you go in that direction you will end in quick sand surely if you do not have a thorough knowledge of the data and mathematical tools used by physics up to now. If you make the effort to acquire them, then of course you are free to prove mainstream physics "wrong" , as long as your new theory can accommodate the data shorthand of the models up to now . All new theories as they appeared in physics joined smoothly with the old ones, as limiting cases.
It doesn't really play a role (in a way), or at least not as far as physical results go. Whenever someone says
we consider a plane wave of the form $f(x) = Ae^{i(kx-\omega t)}$,
what they are really saying is something like
we consider an oscillatory function of the form $f_\mathrm{re}(x) = |A|\cos(kx-\omega t +\varphi)$, but:
- we can represent that in the form $f_\mathrm{re}(x) = \mathrm{Re}(A e^{i(kx-\omega t)})=\frac12(A e^{i(kx-\omega t)}+A^* e^{-i(kx-\omega t)})$, because of Euler's formula;
- everything that follows in our analysis works equally well for the two components $A e^{i(kx-\omega t)}$ and $A^* e^{-i(kx-\omega t)}$;
- everything in our analysis is linear, so it will automatically work for sums like the sum of $A e^{i(kx-\omega t)}$ and its conjugate in $f_\mathrm{re}(x)$;
- plus, everything is just really, really damn convenient if we use complex exponentials, compared to the trigonometric hoop-jumping we'd need to do if we kept the explicit cosines;
- so, in fact, we're just going to pretend that the real quantity of interest is $f(x) = Ae^{i(kx-\omega t)}$, in the understanding that you obtain the physical results by taking the real part (i.e. adding the conjugate and dividing by two) once everything is done;
- and, actually, we might even forget to take the real part at the end, because it's boring, but we'll trust you to keep it in the back of your mind that it's only the real part that physically matters.
This looks a bit like the authors are trying to cheat you, or at least like they are abusing the notation, but in practice it works really well, and using exponentials really does save you a lot of pain.
That said, if you are careful with your writing it's plenty possible to avoid implying that $f(x) = Ae^{i(kx-\omega t)}$ is a physical quantity, but many authors are pretty lazy and they are not as careful with those distinctions as they might.
(As an important caveat, though: this answer applies to quantities which must be real to make physical sense. It does not apply to quantum-mechanical wavefunctions, which must be complex-valued, and where saying $\Psi(x,t) = e^{i(kx-\omega t)}$ really does specify a complex-valued wavefuntion.)
Best Answer
The use of complex numbers is just a mathematical convenience. It makes calculation of derivatives especially easy, it has nice properties when you do Fourier transforms, etc. You're correct that you can do it all using real numbers, so that's not wrong. It's just - in most people's view - more cumbersome.
EDIT In light of the back and forth in the comments, let me provide more detail.
First, starting with classical mechanics: Let $f$ be a (potentially) complex solution to the wave equation. The physically relevant (i.e. measurable) quantity here is the amplitude as a function of space and time. Any complex function can be rewritten in terms of two real-valued functions $g$ and $h$ such that $$ f = g + ih $$ The amplitude of $f$ is $\| f \| = (g^2 + h^2)^{(1/2)}$. We basically have two free functions here where we only need one to meet this constraint, so we're free to choose $h=0$, which means that $f$ is actually real-valued. You could choose some other values for $g$ and $h$ that have the same amplitude, but you don't need the complex part. (Note that I'm not dealing with plane wave solutions here, although you could build up your solution from them. I'm dealing with general solutions to the wave equation.)
For quantum mechanics, we have the Schroedinger equation: $$ i\hbar \partial_t \Psi = -\frac{\hbar^2}{2m} \nabla^2 \Psi$$ (where I set $V=0$ because it's not going to figure in the rest of the point). This is typically written with complex numbers, as shown above, but this is again a short-hand only. We could instead write the solution in terms of two real-valued functions: $$ \Psi = f + ig $$ and then, doing a little simplification, get two, coupled, real-valued PDEs: $$ \hbar \partial_t f = -\frac{\hbar^2}{2m} \nabla^2 g $$ $$ \hbar \partial_t g = +\frac{\hbar^2}{2m} \nabla^2 f $$ So, again, we can avoid complex numbers in the formulation. The price here is that we now have coupled PDEs for real functions instead of a single PDE over complex values. It turns out for practical reasons, that working with the single, complex-valued formulation is easier.