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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The reasoning of the definition of $S$-matrix

The definition of the $S$-matrix is given by $$S=\lim_{t_{f}\rightarrow\infty}\lim_{t_{i}\rightarrow-\infty}U(t_{f},t_{i}).$$ Where $U(t_{f},t_{i})$ is the evolution operator, given by the $$U(t_{f},...
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In Quantum Mechanics is it possible to apply time evolution operator to wavefunction?

If I consider a wavefunction that is the superposition of Hamiltonian eigenfunctions, for example like: $$\psi(x)=\frac{1}{\sqrt{2}}\psi_1(x)+\frac{1}{\sqrt{2}}\psi_2(x)$$ with $\hat{H}\psi_1(x)=E_1\...
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Schrodinger equation of linear combination of quantum states

We know that the solution for $i\hbar \frac{\partial}{\partial t}|\psi (t) \rangle = H|\psi (t)\rangle $ where $H$ is time-independent Hamiltonian, is $|\psi(t)\rangle = e^{-iHt/\hbar}|\psi(t=0)\...
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Which interaction-picture evolution operator is correct for a shifted Harmonic oscillator?

Which way of writing the evolution operator for a shifted harmonic oscillator is correct? The Hamiltonian for a shifted (or driven) harmonic oscillator is $$ H = \underbrace{\omega a^\dagger a}_{H_0} +...
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Relation $\Delta x \Delta p \approx \hbar$. Show that the condition $\Delta p \ll p$ guarantees that the packet does not spread

Consider a wave packet that satisfies the relation $\Delta x \Delta p \approx \hbar$. Show that the condition $\Delta p \ll p$ guarantees that the packet does not spread appreciably in the time it ...
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Product rule for bras and kets

For the time evolution of expectation value of an operator $\Omega$, we can write $$\frac{d}{dt}\langle\psi|\Omega |\psi\rangle=\langle\dot\psi|\Omega|\psi\rangle+\langle\psi|\dot\Omega|\psi\rangle+\...
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Solving Periodic Time Dependent Hamiltonians

For a general time dependent Hamiltonian, if the Hamiltonian at two different times $t_1,\,t_2$ satisfies $$\left[ \hat{H}(t_1),\hat{H}(t_2) \right]=0,$$ then the time evolution operator is $\hat{U}(t)...
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A wave function normalized for a given time $t=0$ is normalized for every time $t \gt 0$ [closed]

Given $\Psi(x,t)$ a wave function such that $$1=\int_{-\infty}^{\infty}\Psi^{*}(x,0)\Psi(x,0)dx$$ Prove that $\Psi(x,t)$ is normalized for every $t \gt 0$. My approach on this has been the following: ...
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What is the correct general form of Hamilton's equation?

Usually, Hamilton's equations of motion are given by: $$ (1)\;\; \frac{dp}{dt} = -\frac{\partial H}{\partial q} \;\;\; \text{ and } \;\;\;(2)\;\; \frac{dq}{dt} =\frac{\partial H}{\partial p}.$$ ...
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Is the set of reachable arrangements of an indeterministic universe (of a given material substrate) sensitive to its initial arrangement?

Suppose two equally massive universes have an identical material substrate of the same fundamental particles. Suppose then that these two universes initially have different arrangements of these same ...
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Is the set of reachable states of an indeterministic universe sensitive to its initial conditions?

Suppose two universes with the same amount of mass-energy and evolving according to the same natural laws, but having different initial conditions. Is the set of states that are reachable by the ...
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Schrödinger equation obtain $ψ(x,t)$ from $ψ(x,0)$

In this answer of the post "Wave packet expression and Fourier transforms" it is said that for the S.E. we have this property: If we start with an initial profile $ψ(x,0)=e^{ikx}$, then the ...
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Can the time-evolution operator be factorised if the Hamiltonian is a sum of two commuting operators?

Let the time-dependent Hamiltonian $H(t) = A(t) + B(t)$ for some quantum system be given as the sum of two time-dependent operators $A(t)$ and $B(t)$. Further, assume that $A(t)$ and $B(t)$ commute, ...
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Dynamical pictures in quantum mechanics

I know that pictures in quantum mechanics are different ways to deal with a quantum problem. What keeps me confused is how to decide what picture I have to use?, i.e., for which quantum system do I ...
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Time evolution of Gaussian packet

From Shankar's QM book pg. 154: Consider the Gaussian wave packet at time $t=0:$ $$\psi(x',0)=e^{ip_0x'/\hbar} \frac{e^{-x'^2/2\Delta^2}}{(\pi\Delta^2)^{1/4}}.$$ Using the propagator $U(t)$ in the ...
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State time-evolution in the Interaction picture

What is the Schrödinger-like equation $$i\frac{d}{dt}|\psi(t)\rangle_I=V_I|\psi(t)\rangle_I$$ telling us for the behavior of the interaction picture state vectors, $|\psi(t)\rangle_I$, at infinity/...
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Electron in varying magnetic field

Lets consider an electron that is placed in an existing, constant in space near the electron, magnetic field. Electron is stationary. Magnetic field over time gradually reduces to zero. I assume ...
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Conservation of flux in scattering problem

Consider a localised potential which becomes 0 after some distance $a$. So, we are considering a wave coming from infinity along z direction, so for $r>>a$, $\psi_{incoming}=e^{ikz}$ Now for ...
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Why is the time evolution of occupied and unoccupied levels in a band under the influence of applied fields the same?

I am studying the semiclassical model for solid state physics as described in Chapter 12 of Ashcroft & Mermin. While I am familiar with the concept of holes, a formal argument is made in this ...
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2 votes
1 answer
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What is the spatial shape of a wavefunction? [closed]

Let's say a particle is localized in space and its momentum is observed. Does its wavefunction look like a ball spreading in distance with time? How can this be shown mathematically?
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What is the meaning of the time evolution of a product of coherent states of the QHO?

I am trying to analyze the dynamics of a coupled quantum harmonic oscillator (cQHO) system. The Hamiltonian of the system is given by: \begin{equation} \hat{H}_{Coupled}=\frac{1}{2m}\sum_{j}\hat{p}_{j}...
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7 votes
2 answers
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Does Heisenberg picture only work for time-dependent Schrödinger equation not Klein-Gordon equation?

For a Klein-Gordon field, our QFT lecture notes say we use the following relationship to define the Heisenberg picture. $$i \frac{dQ}{dt} = [Q,H]$$ which leads to $$Q(t) = e^{iHT}Q(0)e^{-iHt}$$ ...
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Schwartz book: Heisenberg equation of motion for quantum fields (equation 7.28)

In his book "Quantum field theory and the standard model", section 7.2 "Hamiltonian derivation" (of the Feynman rules), Schwartz states that the equations of motion $i\partial_t\...
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Time evolution operator of the free particle

I never came across an example of what the explicit form of a time evolution operator looks like. So when $H=\frac{p^2}{2m}$ does the time evolution operator look like $$ U(t,t_0)=\exp\Big(-\frac{i}{\...
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5 votes
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What is the idea behind quantum speed limits?

Could someone please explain to me how the very basic idea behind existence of a quantum speed limit arises? I think I understand (if it's correct) how it arises naturally between two pure states ...
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Simple time evolution for quantum relativistic particle

I am considering the time evolution of a relativistic particle in 1D, with the time-evolution governed by the equation: \begin{eqnarray} \left(\begin{matrix} \mathrm{i}\partial_{x} && m_{0}\...
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1 vote
4 answers
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Are stationary states only eigenstates of $H$?

are stationary states only eigenstates of $H$? If I have an hermitian operator $O$ that commutes with $H$ so that is a constant of motion, are the eigenstates of $O$ also stationary states since a ...
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Commuting with the Hamiltonian operators are conserved quantities [closed]

In this question What does "commuting with the Hamiltonian" mean?. I've read that if an operator commutes with the Hamiltonian it is a conserved quantity, this means that the average value ...
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Can we physically reverse a dispersing Gaussian wavepacket? [duplicate]

A Gaussian wavepacket can be considered a superposition of infinite sinusoidal waves, each with different frequency and amplitude. The propagation velocity of each constituent wave can be frequency ...
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Evolution operator as a Laurent series of coupling constant

Let the Hamiltonian be $H_{0}+gV$, where $g$ is the coupling constant. In the interaction picture, the equation for the evolution operator is $i\frac{dU}{dt}=gV_{I}U$. What I am going to do is assume $...
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Why did Heisenberg (in his 1925 paper) assume his observables had the following time dependence? [closed]

Many of the assumptions made by Heisenberg in his revolutionary 1925 paper could be justified in some form or another (although they are not by any means obvious), like for example his matrix ...
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Two questions regarding the quantum mechanical wavefunction

I'm just starting with quantum mechanics and I've got some questions. Long after measurement of a position of a particle, does the wavefunction return to the same form, or does an entirely different ...
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Question about Dyson series and integrating factor

We have the following evolution equation in the interaction picture $$i\frac{dU}{dt}=H_{int}U\tag{1}$$ Where $H_{int}$ is the interacting Hamiltonian. Usually the answer is given by Dyson series: $$U(...
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Is a coin toss pseudo-random or truly random? [duplicate]

I wonder if a coin toss is pseudo-random or truly random. Sure, you could say that a coin toss is pseudo-random because you don't know the speed of the coin or its rotation, but if you were to include ...
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Exact dynamics of spin in varying magnetic field

Consider an uncharged particle with spin one-half moving with speed $v$ in a region with magnetic field $\textbf{B}=B\textbf{e}_z$. In a certain length $L$ of the particle's path, there is an ...
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2 votes
1 answer
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Show that evolution of uncertainty is parabolic [closed]

Consider the case of a free particle with wavefunction $$\Psi(x, t=0) = \left(\frac{a}{\pi}\right)^{1/4} e^{-ax^2/2} \quad a>0$$ Prove that the uncertainty product $(\Delta x)(\Delta p)$ is ...
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Free particle wave function at $t=0$ with constant term

What does it mean for a free particle wave function at $t=0$, to have a form $$Ψ(x,0)=C+Ce^{ikx}?$$ If the aim is to construct time evolution of the particle through its representation in $k$-space (i....
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What does "coherent evolution" of an $N$-body quantum system mean?

In classical physics we know of coherence of waves and in quantum physics we identify coherent states. While those are clearly defined concepts/terms, in literature we regularly encouter also that a $...
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How to calculate the exact evolution of a one-body fermionic operator?

To simulate fermionic systems on a quantum computer, one has to do transformation from fermionic to bosonic operators. This transformation can be done for example with the Jordan-Wigner transformation....
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2 votes
1 answer
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Time evolution Operator: write expression for $t>0$

I have a question related to time evolution operator. I was analyzing my teacher solution after solving the problem myself, but there is a detail I dont get. I have a hamiltonian that is represented ...
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How to derive Gödel universe metric?

I've always really liked the Gödel universe metric, and I like the parallels with time travel, but I couldn't take advantage of the mathematics of it completely because I don't know where the shape of ...
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2 votes
1 answer
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If antiparticles are particles moving back in time, would messages from the future be possible?

So, here's the idea: if Feynman's interpretation of antiparticles is true and antiparticles are particles moving back in time, then they are carrying with them information from the future, no? If that'...
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Question about the units of an operator related to the time evolution operator

Let $U(t, t_0)$ be the unitary time evolution operator. Define an operator $\Lambda (t):= \partial_t [U(t, t_0)] U^\dagger (t, t_0)$. This operator turns out to be independent of $t_0$. Now, the ...
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1 vote
0 answers
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Interacting field operator at diffrent time in Peskin and Schroeder'sbook [duplicate]

Related questions Expanding field operators at a fixed time $t_0$ (from Peskin/Schroeder) About an expression of Peskin and Schroeder I saw this question has been asked a couple of times in different ...
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Unitary Time Evolution and Reversibility

I have a system as follows: $$\frac{d\mathbf{x}}{dt} = -iA \mathbf{x},$$ which describes the time evolution of variables $\mathbf{x}$ according to the matrix $A$. Now, $A$ is not Hermitian, but has ...
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2 votes
1 answer
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Hamiltonian in the Heisenberg picture with explicit time dependence

My question is about something I read in Zwiebach's "A First Course in String Theory", 2nd edition. On page 220, just below eqn (11.24) he says $$\alpha(t) = e^{iHt}\alpha e^{-iHt}.$$ Here, $...
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Unitary time evolution operator and the corresponding Hamiltonian

Could someone tell me how to find the eigenvalues $(E)$ of a time independent Hamiltonian $(H)$ if the eigenvalues of the corresponding unitary time evolution operator $U$ $\left(=e^{itH/\hbar}\right)$...
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Truncated Completely Positive Trace Preserving (CPTP) maps

Let us consider the the Liouville equation of a level $N$-system with density matrix $\rho$ together with its standard properties (positive semi-definite, unit trace, etc). The evolution of the system ...
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Varying the Hamiltonians between two fixed states

Let us have a Hamiltonian $H_0$ and 2 states which can time evolve into each other via this Hamiltonian. In this particular situation, say one of the states evolves into the other in time $t_0$ . Now ...
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3 votes
1 answer
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Form of the interaction representation time evolution operator

I am a bit confused about the interaction representation picture. Consider the time independent Hamiltonian $H = H_0 + V$. My question concerns the interaction representation time evolution operator: $...
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