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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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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: ...
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Energy Interpretation of Quantum Effective Action From Weinberg's “The Quantum Theory of Fields”

In section 16.3 of Weinberg, he attempts to prove that the effective potential energy $V(\phi)$ is equal to the minimum energy density of a state with field expectation value $\phi$. I am confused ...
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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 ...
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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 ...
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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 ...
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Any method that can show the time evolution of a open many body system?

the master equation seems is a choice but this method seems only give a mean field result which can not show obviously the effect of specific interaction between particles. So, I am wondering is there ...
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31 views

Generically, why do we want to evolve states with unitary operators? [duplicate]

Why is it so important that operators that evolve states are unitary?
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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 ...
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What is the mathematical reasoning behind Schrodinger's equation preserving its normalization, with the evolution of time?

I am currently in high-school, currently working on a physics research on the normalization of the Schrodinger's equation. I was quite interested on how we can mathematically explain preservation of ...
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698 views

Do atomic orbitals “pulse” in time?

I understand that atomic orbitals are solutions to the time-independent Schrödinger equation, and that they are are analogous to standing waves ("stationary states"). However, even a standing ...
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1answer
71 views

Non-Hermitian Hamiltonian for electron conductance in electric field?

Electron conductance in a solid state is usually driven by electric field - making some direction of jumps more likely. It makes (e.g. Hubbard's) Hamiltonian no longer self-adjoint, how to simulate ...
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4answers
119 views

Understanding intuition behind time translation in classical mechanics

In V.Arnold book "Mathematical Methods of Classical Mechanics" he says that invariance with respect to the time for isolated systems means that "the laws of nature remain constant", i.e., if $\phi(t)$ ...
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Problem with understanding Time Evolution of a Quantum State [closed]

I was given the following task and I'm having some troubles with understanding a few things about it: There is given a system with Orthonormal basis $ |u_1 \rangle , |u_2 \rangle, |u_3 \rangle$ ...
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Wave function of a particle under $V(x)$ (QM)

Suppose I have a particle with mass $m$ and it's under potential of a certain $V(x)$. (NOT an infinite or finite potential well) Also given is the wave function at time $t=0$, $\psi(x,0)$. What is ...
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How to understand the kernel as a transition amplitude?

Consider the time evolution operator $U(t_f, t_i)$ that controls the evolution of a wave function according to $|\psi(t_f \rangle = U(t_f, t_i) | \psi(t_i) \rangle$. As I understand it, the Born ...
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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 ...
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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 ...
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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{...
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Is the Process of Projection of a Generic State Onto a Subspace Impossible? [closed]

I can define (in a standard way) the process of projection of a generic state onto the subspace $\mathcal{G}$ as a process that takes a generic state $|\psi\rangle$ of the Hilbert space $\mathcal{H}$ ...
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1answer
161 views

Fine Tuning of the Universe

I'm an A level student looking into the fine tuning of various constants. Physicists explain the extensive effects that would happen if these constants were to be changed/different and hence, how this ...
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2answers
150 views

Baker-Campbell-Hausdorff (BCH) Formula for the Time Evolution Operator

In following Prof. Toyer's Computational Quantum Physics lecture notes, I came across the following: In computing the Schrödinger equation in real space, one can make a "split operator" Ansatz, for ...
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1answer
92 views

Wave function evolution of an electron [closed]

In many basic quantum mechanics books the wave packet of an electron is described. It will say that the wave packet will broaden as time evolves because of dispersion. But suppose the electron just ...
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644 views

Probabilities in non-stationary states

I'm confusing myself. Let's represent some state in the eigenbasis for Hydrogen: $$|\psi\rangle = \sum_{n,l,m}|n,l,m\rangle\langle n,l,m|\psi\rangle.$$ Now denote the initial state by $\psi(t=0)\...
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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 ...
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2answers
112 views

Why don't the oscillator coherent states disperse in time?

A Gaussian wavepacket is made of a continuum of frequencies (or energies) and stretches in time due to the phenomenon of dispersion: the different plane wave components with different frequencies ...
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1answer
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Derivation of von Neumann Equation for Density Matrices

Consider an ensemble of systems where each system is in one of a set of states $|\alpha_i\rangle$, with proportions $w_i$, such that the density operator is $$ \hat{\rho} = \sum_i w_i |\alpha_i\...
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4answers
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Why do wavefunctions for stationary states include $e^{-iEt/\hbar}$? [duplicate]

Stationary states are separable solutions with $\Psi(x, t)=\psi(x)e^{-iEt/\hbar}$. But why is that there? Griffiths (Section 2.1 Stationary states, equation 2.8) says that observables for these states ...
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1answer
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Time difference between all particles and waves [closed]

Since all elementary particles and waves were created simultaneously in the big-bang (t0) would there be any time difference between any interacting elementary particles and/or waves after t0? I'm ...
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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 ...
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1answer
39 views

What's the minimum condition for time evolution operator to be written as $U(t,t_0)=e^{-\frac{i}{\hbar} \int_{t_0}^t H(t') dt'}$?

Is $\frac{d }{dt} e^{H(t)}=H(t)' e^{H(t)}$ the minimum condition for time evolution operator to be written as $U(t,t_0)=e^{-\frac{i}{\hbar} \int_{t_0}^t H(t') dt'}$? Further, what's the minimum ...
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167 views

How does a Hamiltonian 'generate' a unitary?

I know that the unitary (propagator) is given by $$U=e^{iHt}\tag{1}.$$ But I actually never saw how a Hamiltonian translates into a unitary. For example when I consider a two-level rotation in a ...
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What people mean by “state evolving with the interacting/free theory”?

This is a quite basic question but I confess it is something I didn't get up to this point. When defining the Moller operators and hence the $\cal{S}$-matrix one usually considers "states $\Psi$ ...
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Are quantum mechanical orbits specified uniquely by a Hamiltonian and initial state?

EDIT: This is completely wrong, don't bother reading. Consider a finite dimensional quantum mechanical system, say an $N$-qubit system so that $\text{dim}(\mathcal{H})=2^{N}$. Let's prepare our ...
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1answer
34 views

Dependence of wave function with time, especially probability density function. And Continuity equation

I was learning Basic Quantum mechanics. I cam across the fluid equation in QM, which suggests $\Psi^*\Psi$ is probability density function. Consider the two statements below Probability will change ...
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69 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 ...
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Are $\hat x$ and $\hat p$ assumed to be time-independent operators?

In the book Quantum Mechanics by Cohen-Tannoudji, at $G_{III}$, it is given that and then in the comment section, it is also given that so I'm pretty confused in here, because in one side, they say ...
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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 $$\hat H = \frac{ \hat p ^2}{ 2m},$$...
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Two ways to define wave function in Heisenberg picture

I found two ways to define a wave function in Heisenberg picture, $| \psi(t) \rangle_\mathrm{H}=\mathrm e^{\mathrm i H t/\hbar} | \psi(t) \rangle_\mathrm{S}$ which further gives $|\psi(t) \rangle_\...
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Does $i\hbar \frac{d \hat A }{d t} = [\hat A (t_0), \hat H]$ hold when $H$ is time-dependent, but $[H(t_0), H(t'_0)] = 0$?

It is known that - given in Sakurai, ch2.2, p83 - in Heisenberg's picture, for a Hamiltonian, $H$, independent of time, the time evolution of any operator $\hat A$ is given by $$i\hbar \frac{d \hat ...
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Example of time-dependent factorization of a Hilbert space

In these notes on entanglement in QFT it is pointed out that in the Heisenberg picture the factorization of a Hilbert space is time-dependent (pages 18 and 19): In the Schrödinger picture, it is ...
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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} = -...
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Calculating the evolution at any moment $t$ of a density matrix

I was reading the paper https://arxiv.org/abs/1303.4686, where we are given $N$ systems, all with the same Hamiltonian $$H=\sum_i \varepsilon_i \mid i\rangle\langle i\mid ~,$$ such that the joint ...
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1answer
90 views

Utility of the time-ordered exponential

Is the time-ordered exponential $$ \mathcal{T}\exp\left\{-i\int_{t_0}^tdt'V(t')\right\}\tag{1} $$ just a mnemonic device for the series $$ \begin{aligned} 1 + (-i)\int_{t_0}^tdt_1 \, V(t_1) +{} &...
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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-...
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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 ...
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1answer
182 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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1answer
42 views

Time dependence of expectation value $\hat O$ if $\frac{\partial \hat O}{\partial t} = 0$

I am given the following derivation in my lectures: $$\frac{\partial}{\partial t} \langle \hat O \rangle = \frac{\partial}{\partial t}\int_{-\infty}^{\infty} \psi^* \hat O \ \psi \ dx$$ $$\implies ...
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114 views

Why in quantum mechanics must orthogonal states stay orthogonal? [duplicate]

Given two states $|A(t)\rangle$ and $|B(t)\rangle$. If $\langle A(0)|B(0)\rangle=0$ then for all $t$, $\langle A(t)|B(t)\rangle=0$. This is a fundamental rule of quantum mechanics. And we can imply ...