Regarding Energy and Momentum in QM I've been trying to learn Quantum Mechanics for a few months now, and there's a pretty fundamental thing I never quite understand: What is energy and momentum in quantum mechanics?
I've been following MIT's QM course, and the instructor talked about the energy and momentum of light/wave-like objects, and said that De Broglie proposed that $E=\ \hslash \omega$ and "it turned to be true". 
My question is, what is/was the QM definition of energy before de Broglie, and how did $E=\hslash \omega$ "turn out to be true"? Same question goes for momentum (the only definition of momentum I know is $\mathbf p = m\mathbf{\dot x}$).
 A: There wasn't much of a quantum definition of energy/momentum before Einstein, Bohr, and Planck.
Before de Broglie, most physicists considered energy to be a continuous quantity in the mathematical sense. That is, if you had a "bucket" of energy, you could divide it up into arbitrarily small pieces without end. Einstein and Planck were actually among the first few physicists to suggest discretized energy transfers, with the theoretical discussion of the photoelectric effect in 1905. In this view, there is a limit to how small the pieces of energy may be partitioned from some larger "source". 
One of the greatest theoretical triumphs of quantum mechanics in its early days was the resolution of the so called "ultraviolet catastrophe." Classical derivation of the total energy emitted by a blackbody involved mathematically integrating a function that diverges, resulting in an infinite total energy (a ridiculous result!). Of course, this function, without going into too much detail (if you are interested, read any book on quantum statistics - see the so-called Rayleigh-Jeans formula), was derived on the classical assumption that energy is continuous. 
It was actually Planck who first showed that if one instead assumes a discrete energy exchange for photons of a blackbody (in discrete packets of $\hbar\omega$), and changes the classical integral over a continuous energy to a discrete sum over discretized energy states, the ultraviolet catastrophe is avoided theoretically and a finite total energy can be recovered from the blackbody spectrum. Furthermore, the theory matched closely with experiments. This, combined with the results of the photoelectric effect, gave strong evidence that energy comes in quantized packets, which we associate with photons. 
It's actually pretty easy to perform the photoelectric effect experiment and be convinced that the expression is numerically indeed $\hbar\omega$. Most university physics labs for first or second year students do just this. If you are savvy with basic electronics, its not too hard to do on your own (at least think it through).
As for momentum (a quantity intimately related to energy), a similar result of overturning a nonsensical classical problem came from the postulation of the "Bohr atom", where electrons orbiting atoms are restricted to orbits with fixed angular momentum (and corresponding energy as well). Classically, accelerating charges should radiate and loose energy, and so electrons would ultimately spiral into atomic nuclei according to such a notion. There is clearly an abundance of stable matter that exists in our universe, so the classical description is obviously not quite correct. 
However, most practicing physicists (particularly following after Schrödinger and Heisenberg) usually see energy and momentum as "quantum observables." These are represented by mathematical operators that have matrix/tensor structure. The mathematical framework of linear algebra suits most elementary quantum mechanics problems (as one of the other commenters mentioned, Griffiths is an excellent resource for this), so matrix operations arise naturally as an organized description (if you know much about differential equations, then this checks out because the Schrödinger equation is a linear partial differential equation). More complicated problems involving angular momentum coupling and/or multiple electrons may use tensor operators and tensor product spaces. A good read on this lies in Sakurai's "Modern Quantum Mechanics."
A: This is actually more of a comment than an answer but I don't have enough reputation to post a comment yet. Anywho, it's important to distinguish between classical dynamical definitions of dynamical properties and quantum mechanical definitions. First of all the definition you gave for momentum is that of linear momentum $\mathbf{p}=m\mathbf{\dot{x}}$. Furthermore, what you are discussing are the deBroglie relations for matter waves. For e.g. the energy of a photon would be given by $E=hf$ where $h$ is plancks constant and $f$ is the frequency so the momentum can be obtained by the deBroglie relation $\lambda = h/p$ where $\lambda$ is the wavelength. The description of energy (and the beginning of the quantum era) pre-deBroglie is given by Einstein and Planck. Perhaps reading the wiki article will help, given here. then of course, we have the classical $E=T+V$ relation in classical conserved systems. 
P.S. with regards to the course you are taking, I think you will find that in terms of mathematics, the book by Griffiths is good and very doable the whole way through. Good luck!
