# 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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### Time-dependent perturbation theory in a degenerate system

In the derivation of probability transition of time-dependent perturbation theory (see for example these notes, from Ben Simons from Cambridge University), I have only encountered treatments of non-...
232 views

### Does the Schrodinger wave function associated with a non-moving free particle change in time?

I'm a bit confused by an answer given on this question. In the answer with the animation of a moving free (chargeless) particle and a non-moving free particle (or a free particle with a non-zero ...
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### Phase in time evolution operator for time-dependent Hamiltonian [duplicate]

In Quantum Mechanics, a state vector $|\psi\rangle$ will evolve in time according to $$|\psi(t)\rangle=e^{-\frac{i}{\hbar}\hat H t}|\psi(0)\rangle$$ Imagine we have a system such that, for a short ...
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### Timescales of plasma recombination and fluorescence?

I am currently working on a very simple model for the radiation from electric arcs. As both fluorescence (internal electronic transition) and plasma recombination occur, I would like to compare the ...
228 views

### What does Ehrenfest's theorem actually mean?

I am told that Ehrenfest's theorem, applied to a physical observable $\hat A$, is: $$\frac{d\langle\hat A\rangle}{dt}= \frac{i}{\bar h}\langle[\hat H,\hat A]\rangle$$ I don't understand how to use ...
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### Time evolution of stationary states [closed]

Let's say we have a state $\phi=\sum_i c_i \phi_i$ where the $\phi_i$ denote energy eigen vectors with non degenerate eigen values. Now if a measurement of the energy is done this state ...
328 views

### Does Haags Theorem forbid Time-Evolution?

I didn't quite grasp the essence of Haags Theorem in the the way it is presented (for example on wikipedia), but the issue seems to be that if one wants to represent infinitely degrees of freedom ...
223 views

### What's the time derivative of the Annihilation operator?

I've been dealing with annihilation operator recently where you can see related information Time derivative of the state vector as expressed in abstract Hilbert space vs. as a wavefunction How to ...
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### Does it make sense to ask: what is the probability of a particle being found in a certain state at time $t>0$?

I am dealing with a problem which involves a quantum system of orthonormal two states, $\left|\nu_1\right>$ and $\left|\nu_2\right>$, which are eigenstates of a time-independent Hamiltonian, ...
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### What is the state of particle at time $t$ if at $t=0$ it is in an eigenstate of $\hat{A}$, and $\hat{A}$ commutes with $\hat{H}$?

EDIT: added (assuming $\lambda$ to be non-degenerate). Based on the specifics of the question, we don't in fact know whether this is the case, so it may be that $\left|\lambda\right>$ is not an ...
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### Can I use time evolving block decimation (TEBD) to simulate the dynamics for many body localized systems?

In the many-body localized phase, the system is described by quasi-local integrals of motion ("l-bits"). The entanglement does grow logarithmically with time. So if I use TEBD to get the real-time ...
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### What is meant by unitary time evolution?

According to the time evolution the system changes its state the with the passage of time. Is there any difference between time evolution and unitary time evolution?
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### Time evolution operator in QM

I am reading a introduction to quantum mechanics right now. There is a part about the time evolution operator: \begin{align*} i\hbar \partial_t \,\psi(\vec r, t) = \hat H(t)\, \psi(\vec r,t) \end{...
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### Heisenberg and Schrödinger pictures - clarification

Question related to The equivalence between Heisenberg and Schroedinger pictures. I understand what's explained in the link provided above. My textbook (Breuer and Petruccione's Theory of Open ...
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### Solving the Schrodinger equation with a time-dependent Hamiltonian

I am trying to find the general solution to the Schrodinger equation with a time-dependent Hamiltonian: $$i \frac{\partial}{\partial t}| \psi(t) \rangle = H(t) | \psi(t) \rangle.$$ My Hamiltonian ...
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### How do base kets satisfy Schrödinger's equation in Schrödinger picture and why don't they evolve with time?

According to Sakurai, eigenvalue equation for an operator $A$, $A|a'\rangle=a'|a'\rangle$. In the Schrödinger picture, $A$ does not change, so the base kets, obtained as the solutions to this ...
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### Are the fundamental concepts in Heisenberg Picture and Schrodinger Picture different?

In Heisenberg Picture, for a free particle, $[x_i(t),x_i(0)]=\frac{-i\hbar t}{m}$. This relation implies that even if the particle is well localized at t=0, its position becomes more and more ...
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### Infinite square well and Heisenberg picture

The infinite square well is often a mainstay of introductory quantum physics courses. Its boundary conditions at the well-walls are easily solved to the find the Hamiltonian's eigenfunctions in the ...
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### How do I add decoherence to an oscillating system

If a have an initial (two-qubit) system in the state $\rho_i= \begin{pmatrix} 0&0&0&0\\ 0&1&0&0\\ 0&0&0&0\\ 0&0&0&0 \end{pmatrix}$ and this state ...
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### Time evolution of eigenstates superposition

If a system is in a state $\psi$ which is superposition of, let's say two, energy eigenfunction, namely $\psi_1$ and $\psi_2$, so that $$\psi(t)=\psi_1e^{-i\omega_1t}+\psi_2e^{-i\omega_2t}$$ (I am ...
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### Is there a unitary transformation such that the Hamiltonian in the time-dependent Schrödinger equation becomes real symmetric?

The time-dependent Schroödinger equation is given as (with $\hbar=1$): $$i\dfrac{d}{dt}\psi(t)=H(t)\psi(t)\ ,$$ where $\psi$ is some normalized column vector and $H(t)$ is a Hermitian matrix with time-...
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### QM: Time evolution with $H = H(t)$

In order to calculate time evolution in QM we use Schrödinger equation \begin{align*} i \partial_t |\psi\rangle_t = H(t) | \psi\rangle_t. \end{align*} If $H\neq H(t)$ then \begin{align*} i \partial_t ...
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### Why can't two different quantum states evolve into the same final state?

Is it true that two different states cannot evolve into the same final state? Can they achieve this state at different times? If yes, what is the proof?
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### Definition of Hamiltonian in Quantum Mechanics [duplicate]

Is there any particular reason that the Hamiltonian operator was defined in quantum mechanics to be $$\hat H := \frac{\hat p^2}{2m} + V$$ as opposed to $$\hat H := i\hbar \frac{\partial}{\partial t}?$$...
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### Is the universe ~14 billion years old or that's the farthest photons which reached Earth?

I know that the universe: It's around 13.772 billion years old It expands But it's not clear to me if this is not merely the age of the farthest known photons which reached Earth.
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### Utility of the Magnus expansion (preserving symplectic form?)

There are (at least) two ways to perturbatively solve a matrix initial value problem: the Dyson expansion and the Magnus expansion To be explicit, suppose you're solving for a density matrix $\rho(t)$...
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### How does a particle know how to behave? [duplicate]

How does a particle know it should behave in such and such manner? As a person, I can set mass is so and so, charge is so and so - then set up equation to solve its equation of motion but who ...
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### Can a second-order Schrödinger equation preserve the norm?

Suppose we lived in a universe in which the Schrödinger equation contains second order time derivatives, $$i\hbar \partial_t^2|\varphi(t)\rangle = \mathbb{H} | \varphi(t)\rangle.$$ Would it be true ...
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### Collapse of Wavefunction, and Subsequent Time Evolution

To keep it simple, suppose the system is the well-known particle in a 1D infinite potential well. Suppose the wavefunction is $a|1\rangle + b|2 \rangle + c|3\rangle$, where the $|i\rangle$ are ...
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### Phase of quantum state during propagation

Evolution of quantum state in time can be obtained from the time-dependent Schrodinger equation $$\hat{H} \psi(x,t) = i \frac{\partial}{\partial t} \psi(x,t).$$ For time-independent Hamiltonian, the ...
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### Translating between classical treatment of non-autonomous systems and time evolution in quantum mechanics

When I read an introduction to (classical) dynamical systems, the system was considered in a phase space, and the state of the system evolving in phase space. For a non-autonomous system, an ...
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### Two Time Correlation function calculated from Born rule

Update Below I'm having a hard time reconciling two different calculations of the quantum two time correlation function. Consider quantum operator $A$ with eigenvectors $\{|\phi_i\rangle\}$ and ...
UPDATE: See update at bottom of post Short Version: $\sigma^-_H$ in the Heisenberg picture. $U = e^{+i\omega (|e\rangle \langle e)_H}$. Prove that $$\frac{d \sigma^-_H}{dt} e^{+i\omega t} = U^{\... 1answer 80 views ### Time evolution of a projected mixed state Suppose a quantum system (non-interacting) at finite temperature (\beta^{-1}). I want to know how to compute the transition probability between two degrees of freedom (u and v) at two different ... 2answers 413 views ### Two Level Atom Rotating Frame Introduction I am trying to work out the Rabi problem. In particular I am trying to work it out from a perspective focusing on the operators rather than the kets. Think Heisenberg picture rather than ... 1answer 73 views ### Time evolution of probability in the Heisenberg Picture How does the probability of a system being in a state change with time in the Heisenberg picture if the state vector is time independent? 1answer 75 views ### Is it an assumption or truth for spatial and temporal variables separation if Hamiltonian is time-indepdent in Schrodinger equation? For the derivation from time-depdent Schrodinger equation to time-indepdent Schrodinger equation, if the Hamiltonian is time-indepdent, we assume the spatial and temporal variables in the wave-... 0answers 83 views ### Expectation value of time dependent Hamiltonian Let us assume that we have a time dependent Hamiltonian, precisely a Hamiltonian of a harmonic oscillator with time dependent frequency. \hat{H}=\frac{\hat{p}^2}{2m}+\frac{1}{2}m\... 2answers 207 views ### Is there an equivalent to the Schrodinger equation for quantum mechanics over the reals? Many people have considered alternatives to standard quantum mechanics in which the Hilbert space is over the real instead of the complex numbers - see e.g. here, here, here, here, here, and here. In ... 0answers 32 views ### Why can we use time-dependent perturbations when evaluating the S-matrix? Suppose we have Hamiltonian H_0 + V. When working in the interaction picture we may derive the evolution operator of |\psi_I(0)\rangle which is given by$$S(t,t_0) = T\left[\exp \left( -i \int_{...
In case of a time dependent Hamiltonian of the sort $$H=\frac{p^2}{2m}+\frac{1}{2}m \omega(t) x^2$$ I have solved for the time evolution operator using the Schrodinger equation and got $U(t,0)$. If, I ...