Let $\Omega$ be an observable and let $\Psi = \sum a_i\phi_i$ be a decomposition of a state of a system (satisfying the Schrödinger equation) in eigenfunctions of $\Omega$ (assume for simplicity it is a finite linear combination and there is no degeneracy). Measuring the observable $\Omega$ will give the value $\lambda_i$, the eigenvalue of state $\phi_i$, with probability $|a_i|^2$, in which case the wavefunction will collapse to $\phi_i$.
However, there is no reason why $\phi_i$ should be a state of the system. How is that possible? A few ways out occur to me:
- the state instantanously "evolves" to a new, valid state
- such a measurement is somehow not possible
Concretely, consider a particle in a 1-D box, and let $\Omega$ be the momentum operator. Then the state can be written as an "linear combination" (in this case an integral combination) but no state of definite momentum is a solution.
If we imagine the walls of the box, the infinite potential barriers, to be finitely wide, then if 1) is the case, it looks like the measurement of the momentum could cause the particle to tunnel through an infinite barrier (it would collapse to a state with uniform position probability, and evolve to a state that is nonzero on both sides of the barrier).
What is actually going on? Or did I just totally misunderstand something?
EDIT: Strictly speaking such a $\phi_i$ always is a state of the system (only the time-evolution is governed by the Schrödinger equation). To be more precise would be to remark that it seems that measurement could make it go into a state of (extremely) much higher potential without any other effort than making that measurement.