Inverse of Wave Reduction Let's consider a system of three apparatuses. Sequentially these act as


*

*A device that measures momentum of an electron

*Parallel plates where electric field between them changes randomly.

*Same device as in 1


A electron with $\int{\psi_p dp}$ enters and after we measure with device 1 wave reduction occurs $\int{\psi_p dp} \rightarrow \psi_p$. It passes through device 2 
and we measure momentum again by device 3, say $\psi_{p'}$. Now before measurement 3 we don't know in which momentum state the electron is. 
I think we should consider a superposition of momentum states before a measurement by device 3. 
Now before device 2 momentum was definite but after device 2 it is indefinite. So how come device 2 cause an "inverse" of reduction $\psi_p\rightarrow \int\psi_pdp$
This is as mysterious as wave reduction. How come possible values of momenta come into play? What is the mechanism?
I don't adhere to any interpretation of QM whether be it Copenhagen or many worlds, etc. I would like to know how any interpretation can deal with this.    
Note: By the way is the system I've described realizable? I'm not sure.
 A: The two physical mechanisms are entirely different. To formalize a little bit what you wrote, and if I understood it correctly, you start with an initial state:
\begin{equation}
\lvert \psi_{0} \rangle =\int dp \psi_p \lvert p\rangle
\end{equation}
which means you start in a state that is a superposition of different momenta. Once you measure it, the wave function will collapse in an given momentum state. This will correspond to:
\begin{equation}
\lvert \psi_{1} \rangle = \lvert p\rangle.
\end{equation}
Now it will evolve in time according to the Schrodinger equation. This time evolution depends on the Hamiltonian, or, if you want, in the "environment" where the electron is put in. By interacting with the environment, the electron might evolve into a state where, once again, it is in a superposition of momentum states. An electric field which changes randomly, as you propose, will do the job. But even if you have a constant harmonic potential, it will still evolve into a superposition of states. You can compute the state after a time $t$, by applying the time evolution operator $U\left(t\right)$:
\begin{equation}
\lvert \psi_{2} \rangle = U\left(t\right) \lvert p\rangle.
\end{equation}
Unless $\lvert p\rangle$ is an eigenstate of $U\left(t\right)$, you will find it in a superposition of momentum states, once again, after some time $t>0$.
I am sure it would not be difficult to realize such experiment. A conceptually even simpler setting would be


*

*A device measures the position of a particle;

*You let the particle "be", without interacting with anything;

*Same as in 1.


A similar thing will also happen in this case. You measure the position of the particle and it will collapse in a position state. Because this is not an eigenstate of the free particle Hamiltonian, it will evolve in time to a state which is a superposition of position states. Once you measure it again, you may find it in another position.
If you want, you can think of this in terms of the uncertainty principle. When you put the particle in a well defined position state, you don't know its momentum. When you measure it again you can find it in another position, even if the particle did not interact with anything. This means that, when you measured it again, the particle was, once again, in a superposition of position states. There is nothing fundamentally different here than in the setting that you propose.
