Why conserved quantities in quantum mechanics are experimentally interesting? Part of the QM lecture notes of Prof. Biswas, available online, on the harmonic oscillator reads:
"Now it is easy to see from theorem 4.2 that neither P nor X is conserved. The only
conserved quantity is H. Direct position measurements, like in scattering experiments, are
not possible as that would mean directly measuring interatomic distances in molecules.
This makes the measurement of P or X experimentally uninteresting. Hence, we shall
discuss the measurement of H alone. These measurements are actually made indirectly
in molecular spectra."
1-I do not see why it is impossible to measure interactomic distances based on the fact that x does not commute with the hamiltonian (i.e being not a conserved quantity)
2-The text gives the feeling also that only conserved quantities are experimentally interesting, why is that?
Could someone elaborate on 1 and 2 please?
 A: Re question 2 :
A conserved quantity is that which does not change with time. If some operator commutes with Hamiltonian operator then it will be conserved (the reason is simply because of the way time evolution of operators is defined, i.e. via Heisenberg equation). In quantum mechanics (and more generally in any science) those characteristics of a physical system which are conserved in time (or which at least do not change very rapidly or randomly) can be used as a natural name of that system. Non-conserved quantities are not good for this purpose; e.g. elementary particles are classified by their qualities like charge, spin etc, rather than by their position in space time. As another example, people usually remember each other by their face rather than hairstyle because the latter may keep changing.  
A: The only point of this statement is that in a molecule, there is a superposition of different interatomic distances in the ground state (or in any excited states). The atoms are not definitely at a single point, but spread out over a little region.
This means that if you measure the nuclear positions very accurately, you will necessarily collapse the wavefunction of the nuclei you are measuring, and then the molecule will not be in the ground state anymore. It will either break apart, or go into an excited state of vibration or rotation, depending on the accuracy of your measurement. This is because an accurate measurement of position makes the momentum uncertain by a large amount, leading to a large uncertainty in the energy state of the molecule.
This is an overstated worry. The fact that the nuclear position X is uncertain in the energy eigenstates doesn't mean it's all that uncertain. The nuclei are more massive than the electrons by several tens of thousands of times in normal molecules, which means that their ground state is much much more localized than the electrons. You can measure inter-atomic separation in principle by neutron diffraction, and you get a sensible answer for the positions of the nuclei because they aren't too spread out. The majority of chemistry is done in the Born-Oppenheimer approximation, which is simply the idea that the nuclei are classical objects and only the electrons are quantum mechanically spread out. In this approximation, every question about the nuclei, including their instantaneous position and velocity, is allowed (although of course this is only an approximation, and in the full quantum mechanics, these cannot be known with an accuracy that violates the uncertainty principle)
The idea that only measurements of conserved quantities is interesting is silly. It's not true. Even the energy of an excited state is not strictly conserved when you consider the electromagnetic field (the molecule spontaneously decays). The mass of an unstable particle is not any kind of conserved quantity. The measurement of bond-angles and mean internuclear distances is an important and interesting prediction of quantum chemistry.
I would just ignore this statement, as it is bad motivation for labelling quantum states with classically conserved quantities, like angular momentum, energy, etc. The reason one does this is because in systems with a classical limit, you can identify the quantum numbers with the classical conserved action variables of the system (see this answer for a too-brief intro: How does one quantize the phase-space semiclassically? ). This motivation is outdated, because people nowadays would rather forget the period 1913-1926, and just deal with modern quantum mechanics.
It is possible that the rest of the information in the course is equally flawed, but this is highly unlikely. The best thing to do is to just ignore this statement as just one wrong thing the guy says. People say wrong things all the time, that doesn't mean that the other things they say are wrong.
