Intuition behind creation and annihilation operators? Here I am talking about Harmonic Oscillators with Hamiltonian
$$
H=\frac{1}{2m}(p^2+(m\omega x)^2),
$$
with eigenstates $|1\rangle,|2\rangle,\ldots$
Many textbooks define the annihilation operator to be
$$
A=\frac{m\omega x+ip}{\sqrt{2m\hbar\omega}},
$$
So $A|n\rangle=\sqrt{n}|n-1\rangle$.
The definition of $A$ look very complicated, so my question is, how can people possibly come up with it at the first place? Just imagine you are the first one who discover the operator $A$. How can you possible discover it?
More precisely, we want $A$ to have the property that, if $|\phi\rangle$ is an eigenstate, then so is $A|\phi\rangle$. Also, $A|\phi\rangle$ must not be a constant mutiple of $|\phi\rangle$. Can we obtain an expression of $A$ starting from this property?
 A: It's not so bad if you just set all the constants to one, $m = \omega = \hbar = 1$. You can always recover these later by dimensional analysis.
Classically, we have the Hamiltonian 
$$H = \frac{p^2 + x^2}{2}.$$
Even without knowing quantum mechanics, you can think of phase space as the complex plane, with $x$ as the real axis and $p$ as the imaginary axis. That motivates looking at quantities like $x + i p$. In fact, this is useful because then you can factor the Hamiltonian as a "difference of squares". Defining $a = (x + i p)/\sqrt{2}$, we have the simple form
$$H = \frac{(x + ip)(x - ip)}{2} = a^* a.$$
The variable $a$ is useful because it has a simple time evolution. We have
$$\frac{dx}{dt} = p, \quad \frac{dp}{dt} = -x$$
which means that
$$\frac{da}{dt} = \frac{p - i x}{\sqrt{2}} = - i a.$$
So the variable $a$ just rotates in a circle in the complex plane. It's all really nice, and doesn't require anything beyond high school algebra.
This all has nothing to do with quantum mechanics. Now, in the context of quantum mechanics, we find that $[a, a^\dagger] = 1$, which implies that $a$ and $a^\dagger$ behave like creation and annihilation operators. We also find that $H$ picks up an additional constant term. Finally, the fact that $a$ evolves by picking up a complex phase means that $[H, a] \propto a$, which implies $a$ shifts energy eigenvalues. However, just about all of these things are present in some modified form in the classical theory, where we're using nothing fancier than the complex plane.
