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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294 views

Proof of quantum correlation functions

I'm reading through David Tong's lecture notes on QFT. On pages 76-77, he gives a proof about correlation functions. See the below link: QFT notes by Tong I'm following the proof steps to obtain ...
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603 views

What's the relation between path integral and Dyson series?

If one solves the Schrodinger equation $$i\hbar\partial_tU(t,0) = H U(t,0)$$ for time evolution operator $U(t,0)$, one can get the following Dyson series $$U(t,0) = \sum_n(\dfrac{-i}{\hbar})^n\...
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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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62 views

Time evolution $x(t)$, $p(t)$

For the one-dimensional harmonic oscillator, if $U(t,0)$ is the time-evolution operator, why $$ x(t_{0})=U(t_{0},0)\cdot x(t)\cdot U(-t_{0},0) $$ $$ p(t_{0})=U(-t_{0},0)\cdot p(t)\cdot U(t_{0},0) $$ ...
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76 views

Schrödinger-like equation for destruction operator: Fermionic Case

In my previous question QM Continuity Equation: Many-Body Version for Density Operator? a board member showed me the proof, that the Continuity Equation for single-particle QM can be directly ...
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QM Continuity Equation: Many-Body Version for Density Operator?

I am trying to brush up my rusty intuition on second quantization and many-particle systems and i came across the following problem: In 1-particle QM we have the continuity equation $$ \frac{\...
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66 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 ...
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522 views

Phase factors of eigenstates for a time-dependent Hamiltonian

For a time-dependent Hamiltonian, the Schrödinger equation is given by $$i\hbar\frac{\partial}{\partial t}|\alpha;t\rangle=H(t)|\alpha;t\rangle,$$ where the physical time-dependent state $|\alpha;t\...
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Heisenberg equation with time-dependent Hamiltonian

It is the root of quantum mechanics that Heisenberg picture and Schrödinger picture are equivalent? In most textbooks and wikipedia, the equivalence is proved with a time-independent Hamiltonian. ...
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1answer
151 views

Time dependence of wave packets without eigenfunctions

In general, to obtain the time dependence of an arbitrary wave packet $\left| \phi(x)\right>$ in the Schödinger picture, we expand the wave packet in the energy eigenfunction basis $\left| \psi_n(x)...
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What's the reasons to use time-ordering operator?

I have met the time ordering operator $T$ in many places, such as in the Dyson series $$U(t) = T\exp{\left(-\dfrac{i}{\hbar}\int_0^tdt'H(t')\right)},$$ or in the definition of single particle ...
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56 views

Exactly solvable quantum dynamics problems [duplicate]

By exactly solvable model, people generally mean models whose eigenstates or eigenenergies can be solved analytically. Simple examples are the harmonic oscillator, the infinitely deep square well ...
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367 views

Why time ordering (and not space ordering)?

I am trying to self-learn QFT and thermal field theory, starting from Srednicki's book. Eqns-8.14-8.15 of Srednicki's book on QFT shows the following. $$\langle 0 | T \phi(\mathbf{x}_1) \phi(\...
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74 views

Moving point particles in quantum mechanics

Given a quantum system with a hamiltonian $H$, if initial state of a physical object is $|\psi\rangle $, then state after time $dt$ is $e^{iH\cdot dt}|\psi\rangle$. It is easy to see that $|\langle\...
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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 ...
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479 views

What is the equation analogous to the Schrödinger equation in classical mechanics?

I have a question that asks me for an equation in classical mechanics which would be analogous to the quantum mechanical Schrödinger equation.
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133 views

Evolution operator [closed]

Please, what are the requirements, the Hamiltonian should satisfy before we can use the evolution operator: $$U(t) =\exp\left[-~\mathrm i \int_0^t \mathcal{H}(t^\prime)~\mathrm dt^\prime\right]$$ ...
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Example of non-linear time evolution in quantum mechanics

Preamble: I am a mathematician and not a physicist. From what little I know about quantum mechanics, Schrödinger's equation is a linear PDE that describes the time-evolution of a system. In general ...
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Time evolution of an eigenstate and probabilities

Suppose we have some Hamiltonian $H$ with at least one normalized eigenstate $v$ with real eigenvalue $\lambda$. The time evolution operator is given by $$ U(t,0) = e^{- i \frac{H}{\hbar}t} \ .$$ Now ...
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Possible Error in Assumption - Griffiths Quantum Mechanics

In "Introduction to Quantum Mechanics" by Griffiths, right at the beginning of section 9.1.1 (Time-Dependent Perturbation Theory, The Perturbed System), Griffiths states: Now suppose we turn on a ...
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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 \begin{equation} e^{-iHt}, \end{equation} where $H$ ...
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Time-ordering operator acting on exponential?

Looking into quantum field theory I have come across the time-ordering operator, $T$, defined such that (ignoring the sign associated with fermion opeartors): $$T(a_1(t_1)...a_n(t_n))=a_{\pi(1)}(t_{\...
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146 views

Can the solution to a time-independent Schrödinger equation be considered as a limiting state?

As discussed here, the quantum state $S$ that corresponds to the solution $\psi(t,x)$ of a time-independent Schrödinger equation remains constant wrt. time. I assume that the the solution of a time-...
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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 ...
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208 views

Time evolution of 2-particle states in QFT in the Heisenberg picture

I have a rather basic question about quantum field states. Usually the Heisenberg picture is used in QFT whose effect is that quantum states are time-independent and the operators on the states carry ...
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734 views

Does angular momentum of hydrogen atom imply motion of electron around the nucleus?

Does the non-zero orbital angular momentum (or z-component of angular momentum) of a stationary state of hydrogen atom imply motion of electron (or at least the probability density $|\Psi|^2$) around ...
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90 views

Time resulting of a thermodynamics effect?

I heard in a documentary that : Irreversibility of time is linked to Heat Dissipation in thermodynamics. The theorist says "passage of time" is only a macro sensation, the underlying effect ...
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1answer
355 views

Using Dyson formula in Schrodinger picture

From Time-ordering and Dyson series and what I learnt, Dyson formula is used in the situation of interaction picture: $$i\frac{dU_I}{dt} = H_{I}(t)U_I$$ where $H_I(t)$ is interaction Hamiltonian ...
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271 views

Time dependence of canonical variables

As far as I understand it, at least in scalar QFT, the canonical variables are the field operator $\hat{\phi}(x)$ and its conjugate momentum $\hat{\pi}_{\phi}(x)=\frac{\partial\mathcal{L}}{\partial\...
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1answer
468 views

Does the Hamiltonian time-evolution operator actually change the state of the system?

According to my understanding of things, the time evolution operator in QM looks something like this, $$U = \exp(-iHt/\hbar)$$ Which acts on the state vector / wave-function of the system to ...
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169 views

Derivation involving finite unitary transformation [closed]

Hi I just want to confirm a short derivation involving a particular finite unitary transformation which is important in QM. My working is as follows: Given the finite unitary transformation defined ...
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4answers
2k views

Schrodinger equation for a Hamiltonian with explicit time-dependence?

Can I write a Schrodinger equation for time-dependent Hamiltonian like this: $$i\hbar\frac{d}{dt}\psi(t) = H(t)\psi(t)$$ and then perform Euler integration like this: $$\psi(t+\Delta t) = (1-\frac{...
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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{...
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1answer
83 views

Why is the energy operator special?

Only the energy operator controls the time dependence of a quantum system, but not the others, why is that?
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1answer
88 views

The $T\rightarrow \infty $ limit in quantum field theory

I am new to quantum field theory. Prior to this, I have been using quantum mechanics for a few years. I am reading the book by A. Zee, ''quantum field theory in a nutshell'', 2nd Ed.. On page 18, ...
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1answer
164 views

Writing operator evolution as a quantum dynamical map

In the Heisenberg picture we have the evolution of the operator in time given by: $$A(t)=U^+A(0)U$$ I was looking into the theory of open quantum systems where we introduce the concept of a quantum ...
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How is time evolution done in numerical GR?

Suppose we're simulating what happens when a fairly massive object falls into a black hole. Say the object starts far away, so that the initial condition is that the metric is the Schwarzschild metric ...
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1answer
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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 ...
3
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1answer
564 views

Magnus Expansion in Floquet theory [closed]

I wonder how to obtain the second equality as follows in Eq. (44) of Universal high-frequency behavior of periodically driven systems: from dynamical stabilization to Floquet engineering. M Bukov, ...
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1answer
130 views

time evolution of the state $|a'\rangle$ with Hamiltonian $H=|a'\rangle\delta\langle{a}''|+|a''\rangle\delta\langle{a}'|$ [closed]

Reference to Chapter2, Problem8.b, Modern Quantum Mechanics, Sakurai: $|a'\rangle$,$|a''\rangle$: eigenket of the hermitian operator $A$ and the Hamiltonian, $$ H=|a'\rangle\delta\langle{a}''|+|a''\...
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1answer
378 views

Expectation Value of Unitary Time Evolution Operator in Quantum Mechanics

Does the expression $\langle \Psi_i|U(t)|\Psi_i\rangle$ have a specific meaning, where $U(T)$ is the unitary time evolution operator of $\Psi$, and $\Psi_i$ is the initial state of $\Psi$? If so, ...
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1answer
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Time-Evolution of a 3-State System [closed]

The Hamiltonian for a three-state system is, in some basis $|1\rangle ,|2\rangle,|3\rangle$ $$\hat{H}= \left( \begin{array}{ccc} E_0 & 0 & A \\ 0 & E_1 & 0 \\ A & 0 & E_0 \end{...
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821 views

Time derivative of a function in Phase Space

Consider a function $\mathcal{H}(q_i,p_i;t)$ such that it obeys the equation: $$ \frac{d\mathcal{H}}{dt}=\frac{\partial\mathcal{H}}{\partial t}$$ What does this equation imply (read: mean), physically?...
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1answer
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Hamilton's Equations

The last step of this derivation of Hamilton's Equations is what's making me doubt it. It is as follows: Assuming the existence of a smooth function $\mathcal{H}(q_i,p_i)$ in $(q_i(t), \,p_i(t))$ ...
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2answers
648 views

Schroedinger Picture more general than Heisenberg Picture?

When thinking about the two pictures, what I found to be strange was: I can write the postulate of time-evolution in the Schroedinger picture by: \begin{align} i \hbar \frac{d}{dt} \lvert \Psi(t) \...
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1answer
945 views

Relationship between the Lindblad Equation and Redfield Equation

Both the Lindblad and Redfield Equation both model the open quantum system dynamics given a Hamiltonian and some operators. What is the relationship between the two equations? How can they transformed ...
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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 ...
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2answers
229 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 ...
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61 views

Quantum transitions between energy states [closed]

When a quantum system is acted on by time dependent perturbation, the initial state evolves according to the new time-dependent Hamiltonian and grows to some superposition of states. During the time ...
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2answers
826 views

Are the eigenstates of an operator time independent?

In the Schrodinger picture, are the eigenstates of an operator time independent? Is it their expectation values that evolve in time rather than the actual eigenstates? For example, say I have an ...