Energy conservation in QM [duplicate]

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.

• Performing a measure means that you interact with the system, for example by sending photons and measure the emitted ones... Interacting with the system means that you perturb it – Kevin De Notariis Oct 11 '18 at 12:51
• Yes I understand that, but looking specifically at the energy difference between the initial and final state, how can a simple measurement of (e.g.) position account for a potentially huge energy change? Where does that energy come from / go to? – bRost03 Oct 11 '18 at 12:54
• My guess is that in reality when we measure, for example, position, we don't actually confine the system to a position eigenstate because our measurement is not infinitely precise. So the resulting linear combination of energy eigenstates may not necessarily contain the highest energy states. To truly confine the system to an eigenstate would require an infinitely precise measurement which would require infinite energy? But if I make a very precise measurement (lots of energy) then measure the energy again, I may end up in a high energy state but I may also end up in a lower energy state. – bRost03 Oct 11 '18 at 13:14
• Possible duplicate of Energy conservation and quantum measurement – Hugo V Oct 11 '18 at 13:38
• @bRost03 In order to measure again, you must have another probe interact with the system. – probably_someone Oct 11 '18 at 15:53