Timeline for Energy measures and probability of measuring them
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 18, 2020 at 22:30 | vote | accept | schrodingal | ||
| Oct 18, 2020 at 20:20 | comment | added | Milarepa | Sorry maybe I just made things too complicated. A projective measurement is just a measurement in which by means of some experimental apparatus you project the state of your system onto one state of some chosen orthonormal basis. However, yes: if you are measuring the energy you do a measurement in the energy eigenstate basis. If instead you are measuring some other physical observable, say the momentum of a particle, the only allowed post-measurement states are eigenstates of the momentum operator. | |
| Oct 18, 2020 at 19:42 | comment | added | schrodingal | It's just that I don't understand what a projective measurement is. If we perform a projective measurement, the possible measures of the energy are still the eigenvalues of the Hamiltonian (in this case a and -a) right? | |
| Oct 18, 2020 at 17:52 | comment | added | Milarepa | You can write any state out of your Hilbert space as a linear combination of the two energy eigenstates, as these form an orthonormal basis. Given any state $|\psi_{1}\rangle$ you write as a superposition of the two, there is some other superposition of them $|\psi_{2}\rangle$ which is orthonormal to $|\psi_{1}\rangle$ (i.e. $\langle \psi_{1} | \psi_{2}\rangle=0$) such that they form another orthonormal basis together. If you perform a projective measurement with respect to this basis, the post-measurement state will be either $|\psi_{1}\rangle$ or $|\psi_{2}\rangle$. | |
| Oct 18, 2020 at 16:43 | comment | added | schrodingal | So $|u_{2}\rangle$ is an allowed post-measurement state because we can express it as a linear combination of $|\phi_{1}\rangle$ and $|\phi_{2}\rangle$, which are the eigenstates of the Hamiltonian. Did I understand right? | |
| Oct 18, 2020 at 16:36 | comment | added | Milarepa | This does not contradict any of our statements above: given any two orthonormal states defining a basis which spans the Hilbert space you can perform a projective measurement in this basis. These two states are related to some observable represented by an hermitian operator, whose eigenvalues are the allowed measurement results for the projective measurement. | |
| Oct 18, 2020 at 16:25 | comment | added | schrodingal | That bothers me because in an exercise I did, there was this question: what is the probability of finding the system at the moment t in the state $|u_{2}\rangle$? But the state $|u_{2}\rangle$ is not an eigenstate. | |
| Oct 18, 2020 at 15:36 | comment | added | Milarepa | No. Well it depends. As long as you measure the energy, the only allowed post-measurement states are energy eigenstates. However, if you measure some other observable, the post-measurement state is one of the eigenstates of the given observable | |
| Oct 18, 2020 at 15:30 | comment | added | schrodingal | Okay, thank you very much. So the only values we can measure for energy are the eigenvalues of the Hamiltonian. However, the system can be found in a state that does not necessarily have to be an eigenvector of the Hamiltonian, right? | |
| Oct 18, 2020 at 14:29 | history | answered | Milarepa | CC BY-SA 4.0 |