Let's say I have a simple isolated quantum system of one particle, 1-D, without spin or relativistic effects. The system was prepared in a way that it begins in an eigentstate of the Hamiltonian, with Total Energy = E1.
Then, I measure the position of the particle. The wavefunction collapses, and since the Hamiltonian and position operators don't commute (in most cases at least), that means the system is no longer in an eigenstate of Total Energy = E1, but in a linear combination of various eigenstates, E1, E2, E3, etc. So the Total Energy changed.
For example, a finite potential well. Let's say the potential outside the well is 100 J, inside it's 0 J, and that the initial Total Energy is 20 J. Upon measuring, I find that the particle is outside the well. Its potential energy is now 100 J. The kinetic energy is not -80 J (that is impossible, right?), rather, there is now a superposition of kinetic energy eigenstates with various values.
At first I would think that Energy Conservation was violated (was it?).
But perhaps what happened is that the interaction with the measuring device meant the system wasn't truly isolated. The device 'gave' or 'took' the extra energy. Then that makes me wonder:
Do the classicaly forbidden sections of the wavefunction (like outside the well in the example above) 'rely' on someone measuring the particle in order to be probable? Could you propose that, as long as nobody measures it, the particle will never actually be in that section (since it would have to have negative kinetic energy)?