Doesn't the non-commutivity of $x$ and $p$ complicate the measurement of a quantum system's (e.g.) Energy? A classical system is parameterized by $x$ and $p$ coordinates, and so any other observable -- such as energy -- is some function $E(x,p)$ of them.  I assume, then, that to measure the energy one must measure $x$ as well as $p$ and then place them into the formula.
Nothing analogous can hold for a quantum system's energy.  As is well known, I can measure $x$ and then I can measure $p$, but I cannot construe those values as having jointly obtained at any common time.  It would be nonsense to put them both into any formula.
Quantum "measurement" is hopelessly idealized in the standard undergrad treatment, and so I try to imagine the actual procedure someone would make use of to learn the energy.  Yet I cannot.  The argument above would show (if it succeeds) that a measurement of energy would have to access it "directly" (whatever that means), instead of as a function of other non-commuting quantities.
Does this make energy a more fundamental thing in QM than it was in CM?  In CM you can just say "Oh energy is some conserved function, and obviously it's handy to know conserved functions.  That's all it is."
 A: In QM, performing a measurement perturbs the system with the probe you're measuring with. The state of the system before and after the measurement are different.
For example, you want to measure the energy of the system using a photon. Before the measurement the system was in the ground state. You probe the system with a single photon, say from a laser source, which is absorbed by the system. The system is now in an excited state. After some time, the system returns to the ground state, releasing a photon, which is then detected by your detector.
Your measurement data is then whether a photon is detected at the detector (which confirms that the excited state was available to the system), the energy of the photon that was detected (which can be different from your probe, in the case of phosphorescence and fluorescence), and the time elapsed between the photon you sent out and the photon you detect. All three measurements are limited by your experimental setup (for instance, sensitivity of the detector).
Say, instead of measuring the energy, you want to measure the momentum using electron scattering. You shoot an electron beam to the system (because using a single electron might not be enough due to the finite cross-section). The electron beam is then scattered by both the ions and electrons in the system, which at the same time also perturbs the electronic density of the system. You then use an analyzer to separate the scattered electrons with different momenta. Your measurement data is then the number of electrons that are scattered at a specific solid angle (where you place your analyzer), and the magnitude of the momentum of the analyzed electron (higher momentum = higher deflection in the analyzer). From there, you can analyze the composition of the system. This measurement technique is in common use in experimental condensed matter to investigate atomic structures, known as low-energy electron diffraction (LEED).
Furthermore, if the system was in an entangled state before the measurement, the measurement collapses the system into a single eigenstate. The fact that measurement is a non-reversible process is one of the basic tenets of quantum information.
