# Tagged Questions

The quantum mechanical time evolution operator governs how observables and/or states evolve during finite time steps, and is always unitary. Use this tag for questions about the time evolution operator, or the different equations of motion in the SchrÃ¶dinger/Heisenberg/Dirac pictures. For time-...

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### Using Dyson formula in Schrodinger picture

From Time-ordering and Dyson series and what I learnt, Dyson formula is used in the situation of interaction picture: $$i\frac{dU_I}{dt} = H_{I}(t)U_I$$ where $H_I(t)$ is interaction Hamiltonian ...
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### Extending the ergodic theorem to non-equilibrium systems

I try to make this as short and concise as possible. For equilibrium systems in statistical mechanics, we have the Liouville's theorem which says that the volume in phase space is conserved when the ...
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### How long does it take to a local perturbation to propagate along a quantum system?

Imagine to have a one-dimensional system in its ground state, and to apply a local perturbation at one edge of the system. How does the system evolve after being perturbed? More specifically, how ...
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### Time dependent and time independent Schrödinger equations

I'm trying to understand the relation between the time dependent and time dependent SchrÃ¶dinger equations. In particular, we know that the TDSE is $$H\Psi=i\hbar \frac{\partial \Psi}{\partial t}$$ ...
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### Picture-independence of quantum mechanics

I've been thinking about the equivalence of the Heisenberg and SchrÃ¶dinger pictures of quantum mechanics in the following terms lately: a quantum system is a Hilbert space $\mathcal{H}$ equipped with ...
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In M. Nakahara's book Geometry, Topology and Physics on page 19, the probability amplitude for a particle to move from $x_i$ at time $t_i$ to $x_f$ at time $t_f$ is given as $$\tag{1} \langle x_f, ... 2answers 139 views ### Time evolution in Quantum Mechanics abstract state space As I've learned the first postulate from Quantum Mechanics can be stated as follows: The states of a quantum system are described by vectors in a complex Hilbert space \mathcal{H}. The book ... 1answer 151 views ### Completeness relations of eigenstates in the Heisenberg picture I've been reading Srednicki's introduction to path integrals and I'm slightly unsure of the notation that he uses for the completeness relation of position eigenstates in the Heisenberg picture. In ... 1answer 66 views ### Time evolution of expectation value of an operator I'm studying QM from Sakurai, and I have a doubt regarding the proof given that in the case of time independent Hamiltonian the expectation value of an observable doesn't change with time. The ... 2answers 454 views ### The curious case of the time derivative of the expectation value of the position Having defined the expectation value of position as follows$$ \langle x \rangle = \int x {\lvert\Psi(x,t)\rvert}^2dx $$The time derivative of the expectation value is derived in my literature in ... 1answer 122 views ### What is time evolution operator? Could you explain to me (level 1 years undergrade) what is a time evolution operator? I read on Wikipedia, and it confuses me. 1answer 67 views ### Effective theories and unbounded operators If you have two operators, one the true Hamiltonian H and one we call an effective Hamiltonian H_{eff} and say they agree on every eigenvector with eigenvalue up to E_{eff}. Above that, they can ... 2answers 333 views ### Time-ordering and Dyson series In Dyson series we use a time-ordered exponential by arguing that a Hamiltonian at two different instants of time does not commute. Why is it that so? Can anyone explain with example why should the ... 0answers 75 views ### Evolution of a 'state' in the Heisenberg picture Suppose that we have a Hamiltonian, \hat{H}, and an operator \hat{A} which satisfies the Heisenberg equation^{[a]}$$i \frac{d}{dt} \hat{A} = [\hat{A},\hat{H}].$$Can we create a 'state' by ... 2answers 92 views ### How can a solution of the time-independent Schrödinger equation evolve in space? I understand that if the Hamiltonian does not depend on the time, the SchrÃ¶dinger Equation becomes separable, so you get$$ H \psi(x) = E \psi(x) $$and$$ \Psi(x,t) = \psi(x)\exp\left(-\frac{\...
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I have been trying to solve a Lindblad Equation and then thought about whether there is a closed form Lindblad Equation solution for most types. Googling hasn't lead me to anything useful. So, is ...
I was reading Feynman's Lectures III's Hamiltonian Matrix. There I found this property of Hamiltonian Matrix: The Hamiltonian has one property that can be deduced right away, namely, that $$H^*_{... 6answers 516 views ### Why does time evolution operator have the form U(t) = e^{-itH}? Let's denote by |\psi(t)\rangle some wavefunction at time t. Then let's define the time evolution operator U(t_1,t_2) through$$ U(t_2,t_1) |\psi(t_1)\rangle = |\psi(t_2)\rangle \tag{1}$$and ... 1answer 147 views ### How to get Heisenberg equation of motion? [closed] A system Hamiltonian is given by$$ H=\hbar\omega_{1}\hat{a}^{\dagger}\hat{a}+\Sigma_{i=1}^{N}\left(\hbar\omega_{se}\hat{\sigma}_{ss}^{i}+\hbar\omega_{ge}\hat{\sigma}_{ee}^{i}\right)-\hbar\Sigma_{i=1}...
Consider the quantized real scalar field acting on the vacuum state $\vert 0 \rangle$. We can interpret the state $\phi(\textbf{x})\vert 0 \rangle$ (defined in the Schrodinger picture at $t=0$) as a ...