# Questions tagged [time-evolution]

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 Schrödinger/Heisenberg/Dirac pictures. For time-independent Hamiltonians, the time evolution operator is simply exp(-iHt).

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### Plausibility of Heisenberg equation for the canonical momentum:

In this question, I want to to know wether my reasoning on the plausibility of the Heisenberg equation is flawed: Let's say I want to describe my system in the quantum-mechanics framework: ...
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### Schrödinger equation for time dependent Hamiltonian and conjugation

The Schrödinger equation for the evolution operator reads: $$\frac{\partial U}{\partial t} = -\frac{i}{\hbar}HU$$ where for a time dependent Hamiltonian which need not commute with itself at ...
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### Time-evolution of localised particle [duplicate]

I am interested in the question of, if a particle is initially localised at some position $x_0$ what it will evolve to at a later time assuming a free Hamiltonian $H = p^2 /2m$. Long story short, I ...
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### Why is time-evolution operator unitary?

When we shift the system's time from $t=0$ to $t = t$, we can define the following operator $\hat{U}$. $$\hat{U} = e^{- i \hat{H} t / \hbar} \, .\tag{1}$$ So many (as far as I read, almost all of) ...
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### Exactly solvable time-dependent Schroedinger equation [closed]

We know there are some (many actually) exactly solvable models, like the Hydrogen atom, the harmonic oscillator, etc. But these models are solvable often only in the sense that the eigenstates or ...
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### Convert time operator from momentum space to position space

I'm trying to transform the time evolution operator from momentum space to position space. I know that $$U(t) = e^{-iHt/h} = \int_{-\infty}^\infty e^{-ip^2t/2uh} | p \rangle \langle p | dp$$ and ...
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