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# 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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### Integral of Schrodinger equation for a time-dependent Hamiltonian

I am given the following Hamiltonian, $H=H_1=\frac{p^2}{2m}+\frac{1}{2}m\omega_1^2x^2$ for $t<0$ and $H=H_2=\frac{p^2}{2m}+\frac{1}{2}m\omega_2^2x^2$ for $t\geq 0$. Now I want to integrate my ...
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### The formal solution of the time-dependent Schrödinger equation

Consider the time-dependent Schrödinger equation (or some equation in Schrödinger form) written down as $$\tag 1 i \partial_{0} \Psi ~=~ \hat{ H}~ \Psi .$$ Usually, one likes to write that it has a ...
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### State of a system at previous time

If I am given the state of a quantum system at $t=0$ as $| \psi \rangle$ and I know the Hamiltonian $H$ of the system for time $t<0$, how can I write the state of the system at some time $t<0$?
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### Dyson series for Hamiltonian with $c$-number commutator

I am trying to derive the evolution operator for a time dependent Hamiltonian which satisfies the commutator $$[H(t_1), H(t_2)]=I f(t_1,t_2)$$ Where $I$ is the identity operator, and $f(t_1,t_2)$ is ...
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### Are there 'closed' solution of Schroedinger equation?

Are there closed solution for the wave function of schroedinger equation? i mean solutions in the form $\Psi (x,t)= f(x-t,y,z,t)$ that are not given by infinite series. For example for the 1+1D ...
317 views

### Why does time evolution preserve the norm of a wavefunction?

I saw an awesome derivation of Schrodinger's equation on Wikipedia. Part of it relies on: Since time-evolution must preserve the norm of the wave-function, it follows that $U(t', t)$ must be a ...
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### How to numerically implement a Wick rotation?

I'm solving a Schroedinger-type differential equation using numerical methods (RK4 for precision, explicit Euler to get a rough idea). I have an initial condition to start. I understand that replacing ...
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### Visualising traveling waves as solutions of $\partial_t ^{2} u - \partial_x ^{2} u=0$

I have seen that there are particular solutions for the wave equation called traveling waves. I have also seen stationary waves, but I would like to understand the physical meaning of the former. ...
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### Why is the Hamiltonian in QFT the generator of time evolution?

In non-relativistic Quantum Mechanics one can derive that the time translation operator that acts on quantum states is given (in natural units) by $$e^{-iHt},$$ where $H$ ...
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### Why do excited states decay if they are eigenstates of Hamiltonian and should not change in time?

Quantum mechanics says that if a system is in an eigenstate of the Hamiltonian, then the state ket representing the system will not evolve with time. So if the electron is in, say, the first excited ...
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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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### Another way to calculate the time constant of a system approaching thermal equilibrium

I derived a formula for the time constant $\tau$ by which a toy-system of identical particles having two energy levels $E_1$ and $E_2$ approaches equilibrium. I'd like to ask if this derivation may be ...