Observing Quantum Mechanics

Observation implies exchange of energy. For a macroscopic body the energy exchanged for observation is relatively small and is not essential for the body dynamics. For a tiny object with quantum energy levels observation is not that innocent. Factually you observe an object thanks to its metamorphose. Before exchange you had one state, after you have another. So you have two states rather than one and the same object.

It's not possible to eavesdrop quantum information without modifying it. This is a central result in quantum information theory. The mere act of observation entangles the observed with the observer. This is a kinematic, and not dynamic restriction.

That being said though, if you unobserve and unmeasure, you can restore the process.

To approach this question it helps to have a mental model of how quantum experiments are carried out. In these, you don't generally get data from just one particle, but from what you might call a 'quantum ensemble'. That's effectively what the wave function describes: a special complex-valued distribution, which describes the statistics of particles that are all prepared identically in a coherent state.

Crucially you never measure each and every particle generated by the oven/filter combination, typically, you just sort of skim enough to get a trace of what each particle is doing in the ensemble.

So if the most you can do is measure an ensemble of particles, rather than track a particle individually, then how do you know that you are preparing them all in the same state? That question is difficult firstly because it requires acknowledging the limits of the scientific method, and secondly because it requires showing that whatever filters are used to prepare the wave function produce a perfectly coherent (unmixed) state, which requires making an almost impractical number of measurements to compute something known as the von Neumann entropy. But these are stories for another time.

The particles you do observe are generally scattered into a new state, and as others have mentioned, their dynamics cannot be reversed without you also 'undoing' your own observation (which is probably stronger than merely forgetting that the observation ever took place.) So it's fair to say that the part of dynamics involving your experience of observing and collecting data and learning and becoming yourself is essentially irreversible, for better or for worse. However, the dynamics of unobserved particles in the quantum ensemble is perfectly reversible (up until the moment a stray photon happens to reveal the state of one, at which point the 'coherent' ensemble shrinks further.)

Incidentally, whenever an electron is 'observed' it is never the electron itself that is seen. Instead, what you get is just a trace of the unscreened and therefore often long-ranged photon cloud that is stuck with it. With that in mind, it might be interesting to consider how the dots that your eyes observe on a photographic plate record a wave function collapse that is more subtle than localization of the electron's wavefunction alone, and that according to quantum mechanics, your eye's impression of the photographic plate doesn't fundamentally describe the plate itself, but rather is constructed only from the transitions between the states your eye is receptive to.

Here's a pretty terrible cartoon graphic summarizing what I just described:

(To clarify, the 'origins' of the lines are meant to represent interactions at a sort of matrix of detectors that interact weakly with the particle, but are capable of scattering it into a more localized semi-classical or classical path. Also, the 'filter' is somewhat unphysical, as it would need to prepare particles in states with definite orbital angular momentum. This could be done perhaps with a carefully tuned axial magnetic field, but likely not without significant contamination from nearby thermally excited states.)