Energy conservation and quantum measurement

For a high degree of accuracy you would have to probe the particle with a high energy (short wavelength) photon so there is plenty of energy that can go into vibrational excitation. After such hard hit the particle will be smeared across a wide range of states $$\Psi=a_0*\Psi_0+a_1*\Psi_1+...$$This is not an entanglement but a simple superposition of eigenstates. The expectation value of the particle's energy $\bar{E}=\sum_{i}a_i^2E_i$ should not be equal to the energy of any particular state and the second measurement will yield $E_i$ with $a_i^2$ probability.

So, the extra energy comes from an interaction with a probe particle and it doesn't have to be precisely equal to the energy of a certain vibrational state.

Dear Nigel, if you measure the particle's position and find it in a small region, you also change its state.

As you correctly wrote, a localized wave packet (I don't talk about a delta-function state vector whose average kinetic energy would be infinite) can be rewritten as a linear superposition of energy eigenstates.

It means that before you measured the energy, there was a nonzero probability for the energy to be $(21/2)\hbar\omega$, and this particular outcome was ultimately realized. There is no violation of the energy conservation law here.

What you may be annoyed by is that the final energy of the electron usually isn't equal to the expectation value of energy in the state before the measurement. It's surely true. But there is no reason why it should be. The expectation value isn't any "objective value" of energy. It's just a statistical average of many possibilities, and only some of them will be realized, as dictated by the probabilities predicted by quantum mechanics.

If you wish, the process of measurement violates the "conservation of the expectation value of energy".

Needless to say, this "violation" can't be used to obtain any sharp contradiction with the conservation law for the "actual" energy. You may only interpret the measured energy in a "classical way" after decoherence, which means after the measuring apparatus has interacted with the environmental degrees of freedom (that are needed for decoherence). If you prepare your harmonic oscillator plus the apparatus in a state whose total energy is sufficiently sharply well-defined to reveal the "violation of the conservation law for the expectation value", the environmental degrees of freedom will inevitably spoil this accuracy.