Here you will find a list of the postulates of quantum mechanics.
To have a quantum mechanical theory one needs a wave equation whose solutions are consistent with the postulates of quantum mechanics.
It is the wavefunction postulate that separates quantum mechanical models from classical physics models.
The solution of a wave equation $Ψ(x,t)$ , in space and time is interpreted differently in quantum mechanics, as the probability of finding a particle at a particular time and space.
It is not the energy that is "waving" in a quantum mechanical model, but the probability of finding a particle at a specific space time point is calculated by use of the wavefunction. Same equations and wave functions in classical and quantum mechanics, but different quantities are represented by the solutions. In quantum probabilities, in classical energy densities .
See my answer here for an experimental illustration.
Also, why are we seeking a quantum theory of gravity
For aesthetic reasons, the hypothesis that a mathematical unified theory of all four fundamental forces must exist, and since the three , electromagnetic, weak and strong are quantum theories, gravitation should follow suit. There are various unifying theories for the first three, and the hypothesis is that gravity should also be quantized.
An additional reason is that quantum theories avoid singularities, due to the probabilistic nature. For example the 1/r of the coulomb potential is no problem in quantum mechanics: the electron cannot fall on the proton in the hydrogen atom.
Already in the Big Bang cosmological model effective quantization is used for the beginning of the BB instead of the classical singularity.