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).

71 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
8
votes
0answers
820 views

How to do time evolution of operators in the Heisenberg Picture while staying in the Heisenberg Picture

Consider the time evolution of an operator in the Heisenberg picture: $$\tag{1}i\hbar \frac{d}{d t} \hat{A}_{H}(t) = \left([ \hat{A}_S(t), \hat H_S (t)] + i\hbar \frac{d}{d t} \hat{A}_S(t) \right)_H.$...
5
votes
1answer
208 views

Time-dependence of free and interacting Hamiltonians

Consider an interacting field theory with Hamiltonian $$H=H_0+V$$ where $H_0$ is the Hamiltonian of the free theory and $V$ is the added interaction. Now, I know the full Hamiltonian $H$ should be ...
4
votes
0answers
94 views

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 ...
4
votes
1answer
2k views

Time Evolution Operator in Interaction Picture (Harmonic Oscillator with Time Dependent Perturbation)

1. The problem statement, all variables and given/known data Consider a time-dependent harmonic oscillator with Hamiltonian $$\hat{H}(t)=\hat{H}_0+\hat{V}(t)$$ $$\hat{H}_0=\hbar \omega \left( \...
3
votes
1answer
75 views

Significance of energy in a time dependent quantum box

The Hamiltonian for a particle in a finite box is $$H = \frac{p^2}{2m} + V(x)$$ which will give time evolution as $$ i\hbar d/dt|{\psi(t)}\rangle = H|{\psi(t)}\rangle \, .$$ However, if I do a ...
3
votes
1answer
78 views

How does time-translation symmetry morph into evolution in time?

I am reading Ballentine's textbook "Quantum Mechanics: A Modern Development". In it he transitions from discussing time-symmetry to discussing evolution (of the state) in time. I'm finding it ...
3
votes
0answers
835 views

What approximation does Tamm-Dancoff approximation (CI singles) correspond to in real time Time-Dependent Density Functional Theory?

Starting from equations of motion for time-dependent density functional theory (in real time) $$ \frac{ {\rm d} \rho_{nn} }{ {\rm d} t} = i \left[ \rho_{nn}^{(1)}, h^{\rm KS} \right] \quad\text{...
3
votes
0answers
98 views

Picture-independence of quantum mechanics

I've been thinking about the equivalence of the Heisenberg and Schrödinger pictures of quantum mechanics in the following terms lately: a quantum system is a Hilbert space $\mathcal{H}$ equipped with ...
3
votes
1answer
99 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 ...
2
votes
0answers
40 views

Virtual terms in the Dyson series (time dependent perturbation theory)

Let the interaction evolution operator in the interaction picture be $$U_I(t,t_0)=T \exp \Big( -i \int_{t_0}^t dt_1 H_I(t_1) \Big) ,$$ where $T$ is the time order operator and $H_I=H-H_0$ is the ...
2
votes
0answers
60 views

Time-independent and time-dependent perturbation theory yield different results

First, here's the problem statement. Suppose you have an infinite square well of length $a$, where the box extend from $x=0$ to $x=a$. At $t=0$, you add a perturbation $H'$ of the form: \begin{...
2
votes
0answers
146 views

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 ...
2
votes
0answers
16 views

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 ...
2
votes
1answer
304 views

What is the unitary matrix that diagonalizes the Hamiltonian?

If $H = H_0 + g H_1$ is our (free + interaction) Hamiltonian, and we assume that we have a basis of states $\{ | i \rangle \}$ under which $H_0$ is diagonal, then we may diagonalize $H$ by some ...
2
votes
0answers
115 views

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 ...
2
votes
0answers
159 views

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 ...
2
votes
0answers
229 views

Time evolution of squeezed states

I cannot find anywhere on the web or on some books the esplicit expression for the time evolution of squeezed states (defined as $|\xi\rangle = S(\xi)|0\rangle = e^{\frac{1}{2}(\xi^*a^2-\xi (a^\dagger)...
2
votes
0answers
160 views

Wave packets in Dirac equation

Gaussian wave packets remain Gaussians after evolution in case of the Schrodinger equation. It is a very useful property of these wave packets. I don't think the same is true for a Gaussian wave ...
2
votes
1answer
69 views

Dyson Series Iteration - Gives Exact Solution?

When we derive the Dyson series for usage as the time evolution operator in the case of a time dependent Hamiltonian, we start with the equation: \begin{align}\hat{U}_I(t,t_i) = 1 - \frac{i}{\hbar}\...
1
vote
0answers
56 views

I really wonder about the time derivative of creation and annihilation operators in the derivation of LSZ

In Schwartz book, they assume that $\lim_{t \to \pm\infty}\partial^0 a_p(t)=0$. But I thought that is just assumption. so we have to construct the mathematical description. I found the Gell-Mann and ...
1
vote
0answers
36 views

When would an open system reach the steady state calculated from master equation?

From the master equation for density matrix, it seems that one can have steady state solution requiring the derivative of density matrix equals to zero, but I want to know whether a real open system ...
1
vote
0answers
28 views

What happens to the time evolution equations in canonical quantum gravity?

Many expositions on canonical quantum gravity start from a 3+1 type formalism, where spacetime is foliated along the time dimension. The Einstein equations then decompose into constraint equations on ...
1
vote
0answers
50 views

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-...
1
vote
0answers
33 views

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 ...
1
vote
0answers
43 views

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 $\rho(t)$...
1
vote
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-...
1
vote
0answers
38 views

A shaped pulse as a sum of rectangular pulses

I have a pulse with lineshape $L(ω)=\frac{1}{π}\frac{\frac{1}{2}Γ}{((ω−ω_0)^2+(\frac{1}{2}Γ)^2)}$ in the frequency domain where $\Gamma$ is the pulse width and $\omega_0$ is the resonant frequency ...
1
vote
0answers
89 views

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 ...
1
vote
0answers
105 views

Time evolution (spin-1 particle)

The state ket of a spin-1 particle with orbital angular momentum is given by $$\left|\alpha,0\right\rangle=\frac{f\left(r\right)}{4\sqrt2}\left[\sqrt2\left|1-1\right\rangle_L \left|11\right\...
1
vote
0answers
117 views

Time evolution in tensor product Hilbert space

Having the Hamiltonian $$H=\beta\left(\sigma^{(1)}_1 -\sigma^{(2)}_1 \right)^2\in\mathcal{H}=\mathcal{H}_{(1)}\otimes\mathcal{H}_{(2)},$$ where $$\sigma_1^{(1)}\equiv\sigma_1\otimes 1\!\!1_2, \;\;\; \...
1
vote
0answers
77 views

Time evolution operator of Klein-Gordon field

If $U(t)=e^{itH}$ is the time evolution operator. And $|\phi \rangle$ is a state of a field at particular time $t_1$ and $|\phi' \rangle$ is the state of a (free) Klein Gordon field at a time $t_2$. ...
1
vote
0answers
139 views

Origin of the Schrodinger equation by L. D. Landau and E. M. Lifshitz

In the book "Quantum Mechanics" by L. D. Landau and E. M. Lifshitz, it is mentioned that, "The wave function Ψ completely determines the state of a physical system in quantum mechanics. This means ...
1
vote
0answers
62 views

Characteristic time for changes in the Hamiltonian

Just a short query, given an electron at rest at the origin in the presence of a magnetic field whose magnitude is constant but whose direction is rotating around a cone at constant angular velocity $...
1
vote
0answers
265 views

Beta Decay in Time Dependent Perturbation Theory

I'm trying to find the probability of an electron jumping from the 1s to the 2s state due to Beta decay, where $Z\rightarrow Z\pm1$. My idea is that $H' = -\frac{1}{4\pi\epsilon_0}\frac{(Z\pm1)e^2}{r}...
1
vote
0answers
63 views

Time-dependent quantum mechanical picture

As we know,there are three kinds of pictures in quantum mechanics, namely the Schrodinger picture, Heisenberg Picture and also the Dirac picture. If you look at the Wikipedia about the explanation of ...
1
vote
0answers
161 views

How do I derive and use the effective hamiltonian for plasmon-exciton coupled system to obtain operator equations of motion?

Consider the interaction of a single surface plasmon mode in a metal nanoparticle with a dipole emitter(atom) under the classical driving field $ E_i = E_0e^{-i\omega t} + c.c $ , with the following ...
1
vote
1answer
84 views

A derivation problem about the Lewis-Riesenfeld theory

I am read the original paper. There is one step I cannot understand. Namely, the highlight sentence below. How to understand $i \frac{\partial }{\partial t}$ as a Hermitian operator? How to prove ...
1
vote
2answers
228 views

Time-dependence of density operator in quantum statistical mechanics

I'm struggling to understand a couple of textbook explanations relating to the density operator in quantum statistical mechanics. Firstly, in Huang's book "Statistical Mechanics" it says that "The ...
1
vote
1answer
343 views

Lindblad equation solution

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 ...
1
vote
0answers
149 views

Find Equation of Motion given Hamiltonian

So I am given a harmonic oscillator in an electric field. At $t=0$, we are given that the oscillator is in the ground state. The Hamiltonian is: $$H=\hbar \omega[a^{\dagger}a+\frac12+\kappa E_0\cos(...
1
vote
0answers
156 views

Time evolution of quantum states

Time evolution of a quantum state is fully described by a one parameter family of unitary operators. What I can't seem to understand is, given some unitary operator acting on some Hilbert space, can ...
0
votes
1answer
27 views

Probability density of time-dependent wave functions

Why is it so that probability density of eigenfunctions of time-dependent schrodinger equation are time independent while that of general wave functions (which are a combination of the eigenfunctions) ...
0
votes
0answers
38 views

Hamiltonian flows and Heisenberg picture of Quantum Mechanics

I am a math bachelor student studying Quantum Mechanics and I was very briefly introduced to the Heisenberg picture. (Hence many of the following may be trivial) In particular what I know is that: ...
0
votes
0answers
21 views

Evolution of the propagator in the Interaction picture?

The evolution operator in the interaction picture is defined as $U_I=e^{iH_0t}e^{-iH_St}e^{-iH_0t}$ Where $H_S=H_0+V$ I am trying to find the evolution of the operator $U_I$. In literature it is ...
0
votes
0answers
61 views

A question in imaginary time Green's function

I am learning many-body quantum field theory with Bruus and Flensberg's Introduction to Many-body Quantum Theory in Condensed Matter Physics, there is a derivation that confuses me a lot. To add ...
0
votes
0answers
41 views

Discrete time evolution in a non-Euclidean space?

The time independent schrödinger equation can be written as $$i\frac{\partial \psi}{ \partial t}=H\psi$$ if we consider the case of a 1D particle we can evolve it in time by discretising the ...
0
votes
0answers
48 views

Solving time evolution equations in Hamiltonian formalism

I have 4 time evolution equations and the Hamiltonian $H(X_{1},X_{2},P_{1},P_{2})$ that generates the time evolution depends on 4 canonical coordinates but I would like to solve the differential ...
0
votes
0answers
60 views

Interaction picture: why the Hamiltonian describing the dynamic doesn't change with the same law as other observables?

First: what happens in a general change of picture? If I have the following equation: $$ A | x \rangle = | y \rangle .$$ To do a change of picture is to apply a unitary $U$ on all vectors of the ...
0
votes
0answers
70 views

Time dependence of the momentum operator for a free particle

I was studying Modern Quantum Mechanics by Sakurai, and at the page 85, it is given the analysis of a free particle. There, the author assumes that Hamiltonian is $$\hat H = \frac{ \hat p ^2}{ 2m},$$...
0
votes
0answers
52 views

Operators evolution

I have a little question about equation for creation/annihilation operators. Usually we obtain time evolution equation for these operators from Heisenberg equation. for example: $$\frac{da_l}{dt} = -...