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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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conductivity and Linear response theory in Quantum mechanics [closed]

Hello friends I have seen this formula in Linear response theory but I don't understand what is the role of delta function here I asked ChatGPT and said : This term ensures energy conservation, only ...
Lawerence's user avatar
2 votes
2 answers
88 views

Does time average induce phase space propability distribution?

Lets say we have a trajectory (positions and momenta) $(x(t), p(t))$ that is the solution of the equation of motion for a system with Hamiltonian $H(x,p)$. For some function $A(x,p)$, the time average ...
user403461's user avatar
4 votes
1 answer
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Inconsistency in transition rate derivation in "Introduction to the Quantum Theory of Scattering" by Rodberg and Thaler

I've been working through the derivation of the transition rate in the book "Introduction to the Quantum Theory of Scattering" by Leonard S. Rodberg and R. M. Thaler (Chapter 8, Section 4 &...
Frank's user avatar
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8 votes
0 answers
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Time evolution keeps a certain product state always a product state. Is there a time-independent factorizable evolution for this state?

I am typically thinking of quantum spin chains in the following of some length $L$. I am OK without any locality in the assumptions on $H$. I have a product state $|\psi\rangle$ and a potentially very ...
user196574's user avatar
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1 vote
1 answer
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Is it possible for a unitary transformation to align 2 spatially distinct wave packages into 1?

This is mainly meant as a more concise and more general formulation of the problems and realizations occurring to me while thinking about the apparatus I described in this question. The main problem ...
Zaph's user avatar
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1 vote
2 answers
82 views

Why is the time derivative of the wavefunction proportional to a linear operator on it? [closed]

I am currently trying to self-study quantum mechanics. From what I have read, it is said that knowing the wave function at some instant determines its behavior at all feature instants, I came across ...
Gauss_fan's user avatar
4 votes
2 answers
113 views

Experimental constraints on time evolution of quantum states

We have so many experiments on quantum systems; many of those regard superposition principle; tests of probabilities; entanglement; quantum communication protocols; and others are related to ...
Jip's user avatar
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0 answers
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Do solutions to the time-independent Schrödinger equation always (for any $V$) form a basis for solutions to the time-dependent equation?

Griffith's "Intro to Quantum Mechanics" shows that for $V(x)=x^2$ and $V(x)=0$, solutions to the SE can be constructed as a linear combination of stationary solutions. But is there a theorem ...
user56834's user avatar
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3 votes
1 answer
106 views

What do you get when you Taylor expand a Magnus expansion?

The Magnus expansion and Dyson series are very similar to each other, in that they both give a way to approximate a time-evolution operator as a series expansion $$U(t) = \mathcal{T}\left(\exp\left[-i\...
user34722's user avatar
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0 votes
2 answers
61 views

Constant of Motion in Quantum Mechanics for explicit time-dependent Operators

I was studying constants of motion in quantum mechanics, and at first, I don't understand the condition to be a constant of motion. Generally, the temporal variation of an operator $A$ is given by the ...
QuantumBrachistochrone's user avatar
1 vote
1 answer
76 views

How to deal with explicit time dependence in the Heisenberg picture?

I am studying for my test in Quantum Mechanics, and there is something I don't quite understand about the Heisenberg picture and Heisenberg's equation of motion. In the lecture, we derived Heisenberg'...
Shai Avr's user avatar
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0 answers
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Prove that a displaced thermal state evolves into a displaced thermal state

Considering the physical model of a harmonic oscillator at frequency $\omega_0$, which interacts with a heat bath at temperature $T$. The relevant Hamiltonians for this model are given as \begin{...
Yashovardhan Jha's user avatar
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3 answers
205 views

Basic doubt in quantum mechanics

Do entities like electrons, which are considered point particles in Classical Mechanics, actually have a definite position at a particular time (irrespective of it can be measured or not)?
Users's user avatar
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Functional analysis question about operator on quantum wave functions

If I have two time-independent wave functions $\psi_{t_{1}}$ and $\psi_{t_{2}}$ and define an operator $\hat{U}$ such that $$\psi_{t_{2}} = \hat{U}_{t_{1},t_{2}}(\psi_{t_{1}})$$ and $$\psi_{t_{2}}(x) =...
Adam Kabbeke's user avatar
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25 views

Time evolution using non-Hermitian (not a PT symmetric) Hamiltonian

I am currently dealing with non-Hermitian hamiltonian and dynamics using it. In general the diagonalizable non-Hermitian matrix might have complex eigenvalues and the eigenvectors may not be ...
user101134's user avatar
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1 answer
53 views

Does the Hamiltonian always commute with the Time Evolution Operator?

The time evolution operator $U(t, t_0)$ is given as the solution of the equation $$ i\hbar \dfrac{\text{d}}{\text{d}t} U(t, t_0) = HU(t, t_0)$$ whether or not the system is conservative. When the ...
zaphodxvii's user avatar
-1 votes
1 answer
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Derivation of ODE for $c_n(t)$ in Time-dependent Perturbation Theory [closed]

I'm going through the notes on Time-dependent perturbation theory from MIT OCW and want to make sure I understand how we're going from the first equality to the second in the picture attached below: ...
Keshav Balwant Deoskar's user avatar
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0 answers
33 views

Homogeneity of Schroedinger equation implies norm conservation

I am trying to understand how homogeneity of Schroedinger equation implies norm conservation. I know that we are considering the non-relativistic case, where particle number is conserved, so we do not ...
imbAF's user avatar
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1 vote
1 answer
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Estimate (time dependent) Hamiltonian from given time evolution of a density matrix

Basically the question is, whether you can give some estimation for the Hamiltonian of a system, given the time evolution of a density matrix $\rho$ under the assumption that it obeys the von-Neumann ...
Jurek's user avatar
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1 answer
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Generator of time shift when the Hamiltonian is time dependent

Let's consider the unitary group $\hat{S_{\tau}^†}$ such that :$$\hat{S^†_{\tau}}|\psi(t)\rangle=|\psi(t-\tau)\rangle$$ Since we know that: $$\hat{U}(t,t_0)|\psi(t_0)\rangle=|\psi(t)\rangle$$ Where ${...
davise's user avatar
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3 votes
2 answers
602 views

How would a system fall to the "ground state"?

This is a basic question but I struggle to grasp it really. In the Wikipedia page for Ground state, it's written According to the third law of thermodynamics, a system at absolute zero temperature ...
Kim Dong's user avatar
  • 688
2 votes
0 answers
162 views

Non-adiabatic evolution and time-dependent adiabatic parameter

I am dealing with the dynamics of a two-bands lattice system. The idea is that you have a lattice model of free fermions, with some hopping amplitudes and on-site energies.The lattice have two fermion ...
TopoLynch's user avatar
  • 487
0 votes
1 answer
35 views

An exercise to find the expectation of quantum states [closed]

Given $\vert\psi(0)\rangle=\vert 0 \rangle$, $ H =\hbar\omega(\vert 0\rangle \langle 1 \vert+\vert 1\rangle \langle 0 \vert) $, $p=\vert 0\rangle \langle 0\vert$, ask for $\langle \psi(t)\vert p \vert ...
ajowa's user avatar
  • 9
4 votes
2 answers
531 views

Time Evolution of Eigenkets in the Heisenberg picture

I'm reading Modern Quantum Mechanics by Jun John Sakurai and in section 2.2 he talks about Base Kets and Transition Amplitudes. He goes to show, that $|a',t\rangle=\mathcal{U}^\dagger|a'\rangle$, (...
Florpsiturtle's user avatar
2 votes
1 answer
56 views

On properties of open quantum dynamics and Lindbladians

It is well known that the open quantum dynamics is governed by the Lindblad master equation $$\partial_t{\rho}=\mathbb{L}(\rho)=-\frac{i}{\hbar}[H, \rho]+\sum_i \gamma_i\left(L_i \rho L_i^{\dagger}-\...
ironmanaudi's user avatar
2 votes
0 answers
52 views

The limit of time evolution operator

Through reading Nenciu's rigorous proof on the Adiabatic Theorem and Gell-Mann-Low Theorem, I found: Since the limit $t_0\to-\infty$ does not exist for $U(t,t_0,\epsilon)$, in order to make use of ...
Sakana's user avatar
  • 21
4 votes
1 answer
678 views

Why does the Dyson series have a 1/n! factor?

This is the explanation from Wikipedia: Is there a more rigorous proof or explanation of how reducing the integration region to these sub-regions introduces a $\frac{1}{n!}$ factor? I am confused ...
pll04's user avatar
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0 answers
30 views

How long does quantum Zeno effect last?

Everywhere I read about the quantum Zeno effect, the phrase used is: <"immediately" after the measurement, the system remains in the observed state>. What does "immediately" ...
Abc2000ro's user avatar
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2 votes
1 answer
80 views

Contractivity of the Lindblad Generator Adjoint

In the context of the Lindblad equation in the Heisenberg picture, the adjoint of the Lindblad generator, denoted as $\mathcal{L}^\dagger$, is known to be non-contractive in different cases. I would ...
Testina's user avatar
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1 vote
0 answers
30 views

How are propagators for splitting methods applied to time-dependent Hamiltonians derived?

Splitting methods are defined to approximate the solution of the differential equation $$ y'(t) = (X+Y)y(t), \ \ \ \ \ \ \ t \in (t_0,T) \tag{1}\label{eq:1} $$ where $X$ and $Y$ are non-commuting ...
Idieh's user avatar
  • 71
0 votes
2 answers
85 views

Connection of a concrete Hamiltonian to the generator of time-translations

In a Quantum-mechanics lecture I am hearing we defined the Hamiltonian of a quantum system (a system with an observer) as the generator of the time translation-operation of the system under ...
Hermann Gessler's user avatar
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0 answers
56 views

Do energy eigenstates evolve in energy eigenstates for a time-dependent Hamiltonian?

Consider a time-dependent Hamiltonian $H=H(t)$, in Schrödinger's picture. Let's say that at time $t_0$ the system is in a state ${\mid\alpha(t_0)\rangle}$ such that $$ H(t_0){\mid\alpha(t_0)\rangle} = ...
HomoVafer's user avatar
  • 298
0 votes
1 answer
57 views

Time evolution of mixed state?

Suppose I have a quantum statistical mechanics system in the grand-canonical ensemble. It is given by some Hamiltonian $H = H_{0} + V$, where $H_{0}$ is the free part and $V$ an interaction. The state ...
MathMath's user avatar
  • 1,123
1 vote
1 answer
148 views

The problem of time in classical general relativity

It is well known that the Hamiltonian of General Relativity is a linear combination of constraints. This poses a challenge in quantum gravity. If a state $\psi$ solves the constraints ($\hat C_\alpha \...
Thorstein's user avatar
  • 157
4 votes
2 answers
161 views

A version of Maxwell's Demon using Norton's dome to erase information for free?

Norton's dome is a thought expriment that shows Newtonian mechanics is non-deterministic. It has the shape of a dome (see exact details of its construction on the linked page) with a rather peculiar ...
Hadi Khan's user avatar
  • 521
1 vote
0 answers
48 views

Time evolution of monitored quantum dynamics

Let us assume that the quantum system consists of Hilbert space $H$ with basis vectors $|i\rangle$. The initial state $|\Psi(0)\rangle$ evolves in discrete time steps such that $|\Psi(t+\delta t) \...
Young Kindaichi's user avatar
1 vote
0 answers
63 views

Non-Hermitian Hamiltonian in the Heisenberg Picture

I am trying to study a system whose Hamiltonian, after some transformations can be written as \begin{equation} \hat{H} = \hat{N}_1(\omega-i\mu)+\hat{N}_2(\omega +i\mu)+\omega\hat{\mathbb{I}}, \end{...
Jim Charamis's user avatar
1 vote
1 answer
113 views

Time evolution vs. Time translation in QFT

There is a certain sign mismatch between the time translation operator and time evolution operator in quantum field theory which I hope someone can illuminate. From my understanding, a Poincaré ...
Mishary Al Rashed's user avatar
1 vote
0 answers
153 views

Fermi Golden Rule: Why interaction picture is fundamental for proving it? [closed]

Sakurai and Wikipedia proove the golden rule in the interaction picture. As the reason of that choice is not clear for me, I ask you what are the difficulties that rise up when you try to get the rule ...
Grande Rocco's user avatar
2 votes
1 answer
55 views

Time evolution of scale factor for FRW solutions

I'm reading TASI Lectures on Inflation(https://arxiv.org/pdf/0907.5424.pdf). On page 20, it says ... also called the Friedmann Equations $$\tag{21} \boxed{H^2=\left(\frac{\dot{a}}{a}\right)^2=\frac13\...
Photon's user avatar
  • 139
0 votes
1 answer
66 views

How we construct the Gaussian wave packet at $t=0$ with given avarage coordinate and momentum? Does it satisfy any Schrödinger equation? [duplicate]

I've begun delving into quantum mechanics and encountered a point of confusion. In classical mechanics, we define an initial position and initial momentum, which can take on any values. However, in ...
dmitrii's user avatar
0 votes
1 answer
187 views

Gaussian wave packet time evolution

I am currently studying quantum mechanics and I am struggling to obtain the time evolution of a Gaussian wave packet. We know that the wave function of a free particle is: $$\Psi(x,t)=\frac{\sqrt{a}}{(...
user353399's user avatar
2 votes
2 answers
282 views

How to take derivative of density operator?

I was just trying to confirm to myself that the following density operator $$\rho(t) = e^{-iHt/\hbar} \rho(0) e^{iHt/\hbar}$$ fulfills the Liouville-von Neumann equation: $$\frac{d}{dt}\rho(t) = - \...
Physchem16's user avatar
0 votes
0 answers
148 views

Time evolution operator with chemical potential

In the Bruus and Flensberg textbook, section 1.5, it is mentioned that Basically, the result obtained from the canonical ensemble is carried over to the grand canonical ensemble by the substitution $...
Bio's user avatar
  • 843
2 votes
1 answer
226 views

A problem with the analytic solution to Rabi oscillation of a two-level system

Rabi oscillation of a two-level system is colloquially discussed in textbooks about quantum mechanics (especially quantum optics), so I think I'm good with just stating the results. The Hamiltonian is ...
Len's user avatar
  • 179
-3 votes
1 answer
80 views

Time-ordering and Minkowski metric's negative sign [closed]

I'm coming at the following question from a mostly lay perspective (i.e. barely-undergrad physics), so please bear with the weirdness of it if possible. I've generally been uncomfortable with the ...
allidoiswin's user avatar
5 votes
2 answers
792 views

Quantum time evolution after position measurement

Consider a free particle with hamiltonian $\hat{H}=\frac{\hat{p}^2}{2m}$ and propagator $\hat{U}(t) = e^{-\frac{i}{\hbar}\hat{H}t}$: we can compute the time evolution of a position wavefunction as: $$ ...
cmatteo's user avatar
  • 244
0 votes
2 answers
137 views

Minimal Time for Quantum System to Reach Orthogonal State [closed]

I am trying to determine the minimal time $t$ where a single qubit system (as detailed below) reaches the orthogonal state $|1\rangle$. I have arrived at an answer, but I am not entirely sure whether ...
2307's user avatar
  • 103
2 votes
1 answer
154 views

Confusion regarding the $S$-matrix in Quantum Field Theory

In his Harvard lectures on QFT, Sidney Coleman defines the $S$-matrix as, $$ S \equiv U_{I}(\infty, -\infty) $$ Where $U_{I}(-\infty, \infty)$ is the time evolution operator in the interaction picture....
ShKol's user avatar
  • 312
0 votes
2 answers
95 views

Is the time evolution of the universe cyclic? [closed]

If we can assume that quantum mechanics does not have a bound on its applicability, i.e. there are no inherently classical properties of the universe, we can represent the physical state of the entire ...
Joel Järnefelt's user avatar

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