Let's say I have a time independent Hamiltonian so my system conserves energy. It's initially in an energy eigenstate with $E=1$ in whatever energy units you like. I measure a different observable that doesn't commute with $H$, then I measure $H$ again. I have some probability now of finding my system in an energy eigenstate with $E\neq1$.
What gives? If this was a harmonic oscillator for example, it could be the case that I end up in a state with hugely more energy. Where does this extra energy come from?
In this question, the accepted answer says that the probe particle imparts the energy: Energy conservation and quantum measurement
I get that idea, but after the probe interacts with the system and I measure the energy again, then my energy may go down, stay the same or go up. How does this relate to the energy of the probe particle? I feel I'm missing something fundamental here.