New answers tagged time-evolution
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Connection between Real time evolution and Imaginary time evolution
The eigenvalues of the Pauli matrix $$\sigma_y= \left( \begin{array}{cc} 0 & -i \\ i & 0 \end{array} \right)$$ are $+1$ and $-1$ with associated (normalized) eigenvectors $$\chi_+= \frac{1}{\...
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Time-reversal without time translation symmetry
Time reversal operator is defined through a sequence of postulates, that generally includes
Its action on position operator $T\mathbf xT^\dagger = \mathbf x$
Its action on momentum $T\mathbf p T^\...
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2
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Accepted
Is Kubo formula the integrated version of Ehrenfest?
TL;DR Wikipedia article on Kubo formula is wrong. See article on Green-Kubo relations.
The Heisenberg equation of motion for an arbitrary operator is
$$
\frac{d}{dt}A = \frac{1}{i\hbar}[A,H] + \frac{\...
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Time evolution of a tripartite quantum state
If you have a tripartite system and you do want to compute it's evolution, you need to write the global Unitarty Operator, evolve the global system and get the informations of some subsystem, if you ...
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Accepted
Why are the energy eigenvalues not also dependent in time?
First, switch off the time-dependent perturbation. We know that the set of all eigenstates $\lbrace \phi_a \rbrace$ of the unperturbed $H_0$ forms a complete basis of the Hilbert space. Therefore, ...
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Accepted
Confusion about integral form of interaction potential in Dyson series
In short, it is important (for us all) to distinguish between operators defined in various pictures, and how these definitions relate to each other. P&S define the interaction in equation 4.12, ...
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Why are the energy eigenvalues not also dependent in time?
But if the Hamiltonian changes, why should it not be the case that at a later time $t$, that the energy eigenvalues also change?
Eigenvalues of a time-dependent Hamiltonian indeed do change in time. ...
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Why are the energy eigenvalues not also dependent in time?
In general, if the Hamiltonian is time-dependent, its eigenvalues and eigenvectors also will. A silly example could be the spin one-half Hamiltonian
$$
H = E_0\cos(\omega t) \sigma_z + J\sin(\omega t) ...
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Time evolution of energy eigenstates in the Heisenberg picture
For time-independent Hamiltonians, the Heisenberg picture transfers time-dependence from wave functions to operators almost by definition; explicitly, Heisenberg-type states need to be constant in ...
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Time evolution of energy eigenstates in the Heisenberg picture
Change from the Schroedinger picture to the Heisenberg is not like a coordinate transformation, which leaves vector the same, and only changes its coordinates (components wrt some basis vectors). ...
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Accepted
Time evolution of energy eigenstates in the Heisenberg picture
It is a tricky subject. Three things are however to note:
Finding the eigenvectors of an operator is ambiguous. If $|i\rangle$ is an eigenvector of an operator $\hat{o}$, then $e^{i f(t)} |i \rangle$ ...
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QM: General Solution of TDSE
Bransden & Joachain is a great book. You have uncovered a part that is somewhat cryptic.
If you trace the argument from Equation (3.19) to (3.21) and then scrutinise the crucial transition from ...
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Accepted
QM: General Solution of TDSE
But, how can the energy eigenfunctions at all be derived, if $V$ is time-dependent? That doesn't make sense compared to the paragraphs before, where $V=V(t)$ was explicitly assumed for separation of ...
- 41.4k
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QM: General Solution of TDSE
In typical (non-relativistic) quantum systems, there is a time-independent "background" Hamiltonian $H_0$ that describes the system in equilibrium conditions, and an array of time dependent ...
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QM: General Solution of TDSE
It is said that the generic solution of the TDSE (they explicitly refer to the most general form) $\Psi(r,t)$ can be written as $$\Psi(r,t)=\sum_EC_E(t)\psi_E(r)\tag{3.145},$$ with $\psi_E(r)$ the ...
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QM: General Solution of TDSE
Given that they don't explicitly state that the solution is for $V=V(r,t)$, and given that the solution they propose is separable, I believe that they mean the general solution for a time independent ...
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Why do people say the dynamics of quantum mechanics is always linear?
People(even high credentialed ones) say false things all the time. Gross-Pitaevskii is an approximate replacement of the Schr. equation.The statement you ask about is implicitly about QT in general (...
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