This is my attempt.
We start with the following tricks:
We assume $M$ is a large number. This enables us to say the distribution observed $\{ M \}$ is the most probable distribution.
We count the frequency of the of a particular eigenvalues with
$\tilde \lambda_k$ with $m_i$. Hence, for a par...
This is my attempt.
We start with the following tricks:
We assume $M$ is a large number. This enables us to say the distribution observed $\{ M \}$ is the most probable distribution.
We count the frequency of the of a particular eigenvalues with
$\tilde \lambda_k$ with $m_i$. Hence, for a par...
@MoreAnonymous "We assume M is a large number. This enables us to say the distribution observed {M} is the most probable distribution." I guess $M$ is the number of measurements, that is, the number of collected samples. What do you mean "the most probable distribution?" the distribution is uniquely determined by state being measured and observable
@MoreAnonymous but what I mean is that there isn't "a most probable" distribution here. As far as I understand, you have a state $|\psi\rangle$ that you are measuring multiple times in some basis. The distribution is simply givens by the squared overlaps of $|\psi\rangle$ in the measurement basis. It's unique, so why talk of "the most probable one"?
@glS How would I other wise reach the maximum (most probable) distribution point (6). Also I think there are 2 equals sign in point 3
Note the 2nd one is gained by empirical probability
And must match after the maximization
@glS I think I'm saying using the distribution of eigenvalues it should be possible to construct the eigenbasis and thus the Hamiltonian . Also slight edit done.
ok so I still don't get what state is being measured here. You have some $|\psi\rangle$ and you measure it in the eigenbasis of some Hamiltonian, correct? Then you collect many samples and thus find the overlaps of $|\psi\rangle$ in this basis. If you change the measurement basis you are not measuring the Hamiltonian anymore
@MoreAnonymous I don't know what "use units and dimensions and see it's energy" means, but that's beside the point. If there is an intermediate operator than my sentence is not correct
@MoreAnonymous that is not clear at all from the question. It should be in the background section
so, you have a state $|\psi\rangle$, and you measure this state sometimes in the energy eigenbasis and sometimes in some other random measurement bases?
@MoreAnonymous no, you would get one of the possible outcomes, we already discussed this. Then if you measure immediately after the first measurement yes you do get the same outcome
@glS Sorry I personally find it easier to think like this
@glS You do not know the states (you only have the eigenenergies)
So basically the variables are the possible Hamiltonians and the eigenvectors
/energy -eigenfunctions
and you are given . only the distribution of eigenvalues
If you think this is impossible to solve (even non-probabilistically) I think you can use an argument of counting number of variables and number of equations? (I think it is solvable and hence the solution)
if you are measuring the same state $|\psi\rangle$ multiple times, regardless your knowing the state or not, doing measurements and discarding the results is pointless
the state is reset before every measurement, so it's not like the post-measurement states matter in this case
I don't get the point of these "intermediary measurements" that you do and discard. If you measure $|\psi\rangle$ in a basis $|u_k\rangle$ and then measure $|\psi\rangle$ in a basis $|v_k\rangle$, the results in the different bases do not affect one another
@MoreAnonymous that's just words, I don't know what you mean by that. The only interpretation I can think of is the one we discussed a few days ago, when I told you that no I don't see how that makes sense.
@glS Does it work if you read the answer with this perspective (provided by words)? Also can you find the permalink or something for that version (we've done this too many times :( ) So I'm not sure which version
Honestly I'd like to include diagrams,etc maybe I should do that in the answer?
This is my attempt.
We start with the following tricks:
We assume $M$ is a large number. This enables us to say the distribution observed $\{ M \}$ is the most probable distribution.
We count the frequency of the of a particular eigenvalues with
$\tilde \lambda_k$ with $m_i$. Hence, for a par...
