A variant of Schrödinger's cat experiment: Consider operators with eigenstates of the form |dead> + |alive>. If we could somehow measure such an operator we could kill a healthy person by just observing. So, what is going on here? Consider a simpler problem: A particle in symmetric two well potential separated by a potential barrier inbetween. If we put the particle in the left well and measure the energy, the wavefunction will collapse to an eigenstate of the Hamiltonian, which are symmetric or anti-symmetric linear combinations of the particle in the left and right well. So, if we measure the energy and then measure the position of the particle, the particle has 50% chance of moving from the left to the right well, regardless of how high the barrier in the middle is. So, the measurement must involve a strong interaction with the particle. The higher the barrier is the more difficult it is to measure the Hamiltonian because the symmetric and anti-symmetric wavefunctions become almost degenerate.