Timeline for Understanding how to determine the time evolved state vector for a unitary operator constructed from non-commutating operators
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 6, 2021 at 21:43 | vote | accept | DJA | ||
| Apr 30, 2021 at 9:56 | vote | accept | DJA | ||
| Apr 30, 2021 at 9:56 | |||||
| Apr 13, 2021 at 2:49 | vote | accept | DJA | ||
| Apr 30, 2021 at 9:53 | |||||
| Apr 13, 2021 at 2:48 | vote | accept | DJA | ||
| Apr 13, 2021 at 2:49 | |||||
| Apr 13, 2021 at 1:09 | answer | added | QuantumEngineer | timeline score: 1 | |
| Apr 12, 2021 at 22:27 | comment | added | DJA | Still dont see how to solve the problem above though. Sorry that I cant quite see what to do! | |
| Apr 12, 2021 at 22:02 | comment | added | user87745 | Yes, but the point of my comment is that you only need to solve the eigenvalue problem for the Hamiltonian. The $H\vert\psi\rangle = E\vert\psi\rangle$ that you mentioned. Once that is done, it is trivial. You don't need to work with the exponentiated operators because it takes the form of exponentiated scalars in the eigenbasis of the Hamiltonian. | |
| Apr 12, 2021 at 21:59 | comment | added | DJA | Thats very true, but normally I am used to working out stuff such as $H |\psi \rangle = E|\psi \rangle$ solving things that are of the form $exp(-iHt /\hbar)|\psi \rangle$ is a bit more difficult for me at the present moment. | |
| Apr 12, 2021 at 21:54 | comment | added | user87745 | I would also mention that the Hamiltonian being given as a summation of operators that don't commute isn't anything exotic. Almost all our examples in QM courses involve Hamiltonians of the type $\hat{p}^2 + V(\hat{x})$ and we know that $[\hat{p},\hat{x}]\neq 0$ ;-) | |
| Apr 12, 2021 at 21:48 | comment | added | user87745 | Did you try finding the eigenbasis of the Hamiltonian first? Once you know the eigenbasis of the Hamiltonian, you can express the propagator simply as $\sum_n e^{-iE_nt}\vert n\rangle\langle n\vert$ and the state at time $t$ will be simply $\sum_n e^{-iE_nt}\vert n\rangle\langle n\vert\psi(0)\rangle$. | |
| Apr 12, 2021 at 21:43 | history | asked | DJA | CC BY-SA 4.0 |