Current quantum theory does not offer a dynamical mechanism to account for energy transfer in position-measurement, WHY? In Quantum Mechanics (vol. 1) by Tannoudji et al., there are two pages discussing "perturbation created by a position measurement" (p.278-279). They consider a particle in an infinitely deep potential well of width $a$ and prepare the particle in the ground state  $ ψ_1(x)$ =  $ \sqrt{\frac{2}{a} }\sin \frac{\pi x}{a} $.  It must be stressed that when the position is measured, the particle can never be an eigenstate of the position. Thus, once we say that the particle is found to be $x=\frac{a}{2}$, the position state can be chosen to be a long rectangle with small width $ \epsilon $, and the height is $\frac{1}{\sqrt{\epsilon}}$, which is determined by the resolution power of the apparatus of the position measurement. This step is exactly what measurement postulate indicates. 
Now we hope to know what the energy of the position state, i.e., the long rectangle, was transferred during the position measurement above. Results show that the mean value of the energy is divergent. It is absurd but it is can be easily amended with the long rectangle being replaced by a sharp Gaussian function. Surely, the mean value of the energy is greater than ground state over which the position measurement is performed. 
My question is: as far as I know, there is dynamical process within the current formalism of quantum mechanics which account for the energy transfer. WHY? does such a process would violate any principle of current quantum mechanics?
 A: I would say the answer is that any such measurement will necessarily involve some transfer of energy. For example, you might use a photon to measure the electron's position. But if that photon were smaller in energy than the gap between energy levels in the well, then it couldn't be absorbed, and it wouldn't measure anything. It needs instead to be a high energy photon, and thus carrying a lot of momentum. And in the process of scattering off the electron, it will transfer to it some of that momentum, leading to your localised high energy electron.
In other words, I think the problem is that you should describe the full quantum measurement process throughout suitable quantum equations, and you would find out that energy is overall conserved, and that the higher the precision in position you want, the higher the energy of the photon, the higher the energy transferred to the electron. The idea of a measurement and collapse are just approximations to these real physical processes that go all the way to the brain of the observer (and while there's still not a definite answer re: measurement problem, I am convinced that there is no true loss of unitarity involved).
A: In quantum theory, psi function gives probabilities and from those expected mean values (expected for an ensemble of experiments) can be calculated. Quantum theory does not predict, track or analyze particle position, momentum or energy in a single experiment.
If you would like to do just that, you need to search for ideas outside the standard quantum theory.
