Infinite Square Well (Conceptual)

[Nicolaus]

Homework Statement

Say, for example, a wave function is defined as 1/sqrt(2)[ψ(1)+ψ(2)] where ψ are the normalized stationary state energy eigenfunctions of the ISQ.
Now, say I make a measurement of position. What becomes of the wavefunction at a time t>0 after the position measurement (i.e. after it has had time to evolve)?

The Attempt at a Solution

I would assume that the wave function would become a superposition of inifinitely many stationary states, but such that the expectation value of the Hamiltonian (energy) is the same so as to stay in line with the conservation of energy.

 

[blue_leaf77]

I would assume that the wave function would become a superposition of inifinitely many stationary states, but such that the expectation value of the Hamiltonian (energy) is the same so as to stay in line with the conservation of energy.

That's why the time evolution operator commutes with Hamiltonian.

What becomes of the wavefunction at a time t>0 after the position measurement (i.e. after it has had time to evolve)?

Let's say at t=0, you found your particle at $x=x'$ with probability $|\langle x' | \Psi \rangle|^2$, then right after this point in time, the initial state before measurement collapses to $|x' \rangle$. If you want to know how it evolves with time, then just apply the time evolution operator. If you want it to be expressed in the known entities such as the eigenstates of the ISW, insert the completeness operator in the space spanned by the ISW eigenstates in between the time evolution operator and $|x' \rangle$.

 

[vela]

I would assume that the wave function would become a superposition of inifinitely many stationary states, but such that the expectation value of the Hamiltonian (energy) is the same so as to stay in line with the conservation of energy.

Do you mean that the expectation value of energy would be the same before and after the measurement? If so, that's not the case.