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.

  • $\begingroup$ Have you determined the explicit form of the states alluded to in your basis, as implicitly required to? $\endgroup$ – Cosmas Zachos May 1 '19 at 14:04

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.

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  • $\begingroup$ 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. $\endgroup$ – kbogucki May 1 '19 at 14:50
  • $\begingroup$ 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. $\endgroup$ – Will May 1 '19 at 14:57
  • $\begingroup$ 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! $\endgroup$ – kbogucki May 1 '19 at 15:01
  • $\begingroup$ No problem, glad it was helpful! $\endgroup$ – Will May 1 '19 at 15:02

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