# Problem with understanding Time Evolution of a Quantum State [closed]

I was given the following task and I'm having some troubles with understanding a few things about it:

There is given a system with Orthonormal basis $$|u_1 \rangle , |u_2 \rangle, |u_3 \rangle$$ and two operators: Hamiltonian $$H$$ and some operator $$A$$ given by:

\begin{align} H|u_1\rangle & = 2E_0|u_1\rangle − E_0|u_3\rangle, \\ H|u_2\rangle & = E_0|u_2\rangle, \\ H|u_3\rangle & = −E_0|u_1\rangle − 2E_0|u_3\rangle, \\ A & = a \left(\text{ } 4|u_1\rangle\langle u_1| + 2|u_2\rangle\langle u_2| - 2|u_3\rangle\langle u_3| \right ) \end{align}

In the moment $$t_0$$ there was a measurement of energy, that gave the highest possible value; in the moment $$t_1$$ there was a measurement of $$A$$, that gave the lowest possible value.

1. Give the evolution of state from $$t_0$$ to $$t_1$$
2. Give the state just after the measurement of A and the evolution of this state for $$t > t_1$$.

Now, as far as I understand correctly, the state in the moment $$t_0$$ had a form of the highest eigenvalue of energy and the eigenstate corresponding to that energy, e.g. $$E_1 |1 \rangle$$, and thus the time evolution of this state should look like this I guess: $$\exp \left(-iE_1 \left( \frac{t-t_0}{ħ} \right) \right) |1 \rangle .$$

If my reasoning is correct then I don't know what the measurement of $$A$$ actually tells us and what will be the state and the evolution of this this state after the measurement in $$t_1$$.

Any help would be greatly appreciated.

## closed as off-topic by Emilio Pisanty, ZeroTheHero, Cosmas Zachos, GiorgioP, YashasMay 2 at 7:04

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• Have you determined the explicit form of the states alluded to in your basis, as implicitly required to? – Cosmas Zachos May 1 at 14:04

## 1 Answer

Looks like you understand how the initial measurement of $$H$$ and time evolution from $$t_0$$ to $$t_1$$ works. Without giving away the details, the same reasoning applies to measurement of $$A$$: after measurement of $$A$$, given that the measurement returns the lowest eigenvalue of $$A$$, the state should be the corresponding eigenstate of $$A$$. The challenge will be finding this state and expressing it in an appropriate basis in order to find the time evolution.

• Thank you for your answer, however I still have some doubts: firstly, you state that after the measurement of A the state of the system will be given by eigenstate of A and not H (if I understand correctly). I thought that the whole system state should be given by eigenstates of energy (so eigenstates of H), but I am not certain about this, so correct me if I'm wrong. Secondly, by "appropriate basis", do you mean I have to express the state after measurement of A in basis of Hamiltonian eigenstates? I think this was suggested to me by my teacher, although once again - I'm not sure. – kbogucki May 1 at 14:50
• Since this appears to be a homework question, I'm trying not to give too much away. You are right that any state can be expressed as a superposition of energy eigenstates, but in general the system state need not be in a particular energy eigenstate. With regard to the choice of basis, I approve of your teacher's suggestion ;) You know how the energy eigenstates evolve in time, so writing your state of interest in terms of these will tell you how it evolves in time. – Will May 1 at 14:57
• I think this answers all my questions and I should be able to solve it now. Your insight on this have been a great help for me, so thank you! – kbogucki May 1 at 15:01
• No problem, glad it was helpful! – Will May 1 at 15:02