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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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:
$$
...
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Equation of motion for time evolution operator in the interaction picture
Consider a system with Hamiltonian $\hat{H}=\hat{H_0}+\hat{V}$. We define the interaction picture kets $|\psi(t)\rangle _I$ by
$$\tag{1} |\psi(t)\rangle _I=\exp\left(\frac{i}{\hbar}\hat{H}_0(t-t_0)\...
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Is unitary time evolution the same as obeying the Schrödinger equation?
In this question, the answer says that unitary time evolution means that probability is conserved. Is this the same as saying that a system obeys the Schrödinger equation?
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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 ...
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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....
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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 ...
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Time Variation of Momentum Expectation Value [duplicate]
When deriving the expression of momentum expectation value one gets to what's shown in the picture. However, every text I've seen so far simply neglect the first term inside the integral. I would ...
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It seems that expectation value of $H$ on coherent states is independent of time? But why?
Let's say the particle is in the state $| \psi(0) \rangle = \exp(-i\alpha p/\hbar) |0 \rangle$, where $p$ is the momentum operator.
I have to show that $| \psi(0) \rangle$ is a coherent state and to ...
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State collapse in the Heisenberg picture
I've been studying quantum mechanics and quantum field theory for a few years now and one question continues to bother me.
The Schrödinger picture allows for an evolving state, which evolves through ...
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Integral form of solution of Dyson series and differentiation of the exponential form
The solution to time dependent hamiltonian equation is:
$$\frac{\partial}{\partial t}U(t) = -\frac{i}{\hbar}H(t)U(t)$$
The immediate integral form solution is
$U(t) = I - \frac{i}{\hbar}\int_{0}^{t}...
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Prove that the Dyson series solves the differential equation
I have a short question about Dyson series.
Prove that $$U\left(t, 0\right) = \sum_{n}^{} \left(-i\right)^{n} \int_{0}^{t} dt_{1} \int_{0}^{t_{1}}dt_{2} ... \int_{0}^{t_{n - 1}} dt_{n} H_{I}\left(t_{...
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Why is time evolution replaced by strongly continuous unitary representation of the Poincaré group in relativistic QM?
One of the basic postulates in quantum mechanics is the one of time evolution: a state $\psi$ on a Hilbert space $\mathscr{H}$ at $t_{0}\ge 0$ will evolve to a state $\psi_{t} := e^{-i(t-t_{0})H}\psi$ ...
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Changing the time parameter and finding the corresponding hamiltonian
I'm dealing with a problem where I have a (classical) Hamiltonian $H(q,p)$ such that, for any scalar function $f(p,q)$,
$$
\dot{f} = \frac{\mathrm{d} f}{\mathrm{d}t} =\{ f,H \}
$$
If I change the time ...
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$S$-matrix in Dirac picture
Let's define the interaction Hamiltonian as
$$\hat{H}(t) = \hat{H}_{\text{S}}+\hat{V}_{\text{S}}(t)\tag{1}$$
Where $\hat{V}_{\text{S}}\in \mathcal{L}(\mathcal{H})$ represents time-dependent ...
CommunityBot
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Where $(x,y,z)$ will a spaceship be in a next second? [closed]
Developing a physics-based video game in Unity.
Given a spaceship position, velocity vector, mass, and force vector currently applied to the ship body, I would like to calculate where it will be in ...
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Time Evolution of Heisenberg Operators $x$ and $x^2$
In the Heisenberg picture, operators evolve according to
$$
\partial_t A = \frac{1}{i\hbar} [A,H].
$$
My question is, does the following relation hold?
$$
(X_H)^2 = (X^2)_H
$$
The system (Hamiltonian) ...
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Why $U(t)A_{H}(t)|\psi_{H}\rangle = A_{I}(t)|\psi_{I}(t)\rangle$?
Can anyone explain why $$U(t)A_{H}(t)|\psi_{H}\rangle = A_{I}(t)|\psi_{I}(t)\rangle~?$$
This equation is on Piers Coleman's book, Introduction to Many-body physics. It is used to prove
$$
A_{H} = U^{\...
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Time evolution of operators in the Heisenberg picture
In the Schrodinger picture, a state $|{\psi_{S}(t)}\rangle$ at a time $t$ is given by applying the time-evolution operator $\hat{U}(t)=e^{-\frac{i\hat{H}t}{\hbar}}$ to the state $|{\psi_{S}(0)}\rangle$...
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Evolution of volumes in Phase-Space
Liouville's theorem states that the volume occupied by an ensemble does not change as the ensemble evolves. My question regards the volume of the smallest sphere that contains the ensemble. Is there a ...
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How come there are Schrödinger Picture operators with explicit time dependence?
In the Schrödinger picture, observables are said to be time independent (see Cohen, for example) operators. However, when deriving the Heisenberg Equation of Motion $$i\hbar\frac{d}{dt}A_H(t)=[A_H(t),...
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Solving time-dependent two-levels hamiltonian [closed]
I would like to solve the time-dependent Schrodinger equation for a two-levels system with a time-dependent Hamiltonian ($ \hbar = 1 $)
$$ H(t) = \frac{\Omega_R}{2} i (\sigma_+ e^{i t \Delta}-\sigma_- ...
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Exact solution of the evolution operator generated by a linearly driven quantum harmonic oscillator
Consider a linearly driven quantum harmonic oscillator with the time-dependent Hamiltonian $H(t) = \hbar \omega a^{\dagger} a + i \hbar \left[ \varepsilon(t) a^{\dagger} - \varepsilon^*(t) a \right]$, ...
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Why is this Heisenberg EOM true even for time-dependent magnetic fields (spin dynamics)?
In what follows, I will use primes to denote Heisenberg picture operators (non-primed operators will be Schrödinger picture).
In his Chapter 12.1 on spin dynamics, Ballentine has us first consider a ...
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Cohen Quantum Mechanics Derivation? [closed]
I dont understand the argument on page 38 eq. (C-6) of Cohen's quantum mechanics. Could someone break down for me what is $g(k)$? Is it the initial condition?
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What physical processes other than scattering are accounted for by QFT? How do they fit into the general formalism?
For background, I'm primarily a mathematics student, studying geometric Langlands and related areas. I've recently been trying to catch up on the vast amount of physics knowledge I'm lacking, but I've ...
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Eigenvalues of an time-ordered exponential operator
Let's consider a simple 1-qubit time-dependent Hamiltonian: $$H(t) = \delta(t) \sigma_x + \sigma_z \ ,$$
where $\delta(t)$ is some time-continuous (real-valued) function.
Evolving $H(t)$ continuously ...
Buzz♦
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Time-evolution of momentum eigenstate under harmonic oscillator Hamiltonian
I'm wanting to understand the dynamics of a momentum eigenstate $| p \rangle$ governed by a harmonic oscillator Hamiltonian. Consider $\hat{H} = \hat{p}^2 + \hat{q}^2$. Then inserting a completeness ...
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Causal expansion of EM fields
When an electron and positron are created at the same time and location, is it correct to say that the electromagnetic field due to the two particles is zero when $r>ct$?
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Is there a physical reason why the time evolution in quantum mechanics is given by $e^{-itH}$? [duplicate]
Let $H$ be the Hamiltonian operator. Since $H$ is self-adjoint, by Stone's theorem there is a strongly continuous one-parameter unitary group $U(t)$ such that $U(t) = e^{-itH}$. Mathematically this ...
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Understanding adiabatic elimination in three-level system coupled to EM field
I am having some difficulties understanding the "adiabatic elimination" in the context of atomic physics.
In particular, consider a three-level system with states labeled by $|g_1\rangle$, $|...
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Factorization of density matrices
I'm currently reading through the following document about quantum noise and open quantum systems: https://courses.cs.washington.edu/courses/cse599d/06wi/lecturenotes13.pdf. On page 6 of the document, ...
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How to describe time evolution in relativistic QFT?
I must confess that I'm still confused about the question of time evolution in relativistic quantum field theory (RQFT). From symmetry arguments, from the representation of the Poincare group through ...
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Non-formal expression for the classical propagator
I'm studying classical molecular dynamics and have come across an object called the classical propagator in the following context.
Let $\mathcal{A}(t) = \mathcal{A}(\vec{x}(t))$ be a function on phase-...
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Time-ordered exponential operator generated by two commuting Hamiltonians
Define a time-dependent Hamiltonian $$H(t) = H_1(t) + H_2(t),\tag{1}$$ where $$[H_1(t), H_2(t)] = 0 ~ \forall t \in [0,T].\tag{2}$$ Is it true that the unitary operator generated by $H(t)$ is a ...
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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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Question about the time-ordered exponential operator
I learned that a unitary matrix generated by time-dependent Hamiltonians is written down as
\begin{equation}
U(t) = \mathcal{T}\exp\Big(-i\int_0^t H(t') dt' \Big),\tag{1}
\end{equation} where $\...
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How to calculate pressure $p$ in the Navier-Stokes equation to simulate the time evolution of a fluid?
I wanted to simulate the motion of a fluid (continuously filling the entire space) in a given space, say $3$-dimensional Euclidean space. To calculate the dynamics of fluid motion, I used the Navier-...
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Liouville's theorem and the preservation of topology
What might be a simple proof showing that the time evolution of the phase space volume can't lead to splitting off of the phase space volume?
By Liouville's theorem, the total phase space volume is ...
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Schrödinger equation as a limit of von Neumann equation
How would I derive the Schrödinger Equation as a limit of the von Neumann equation?
The quantum Liouville equation (von Neumann equation) is given by
$$i \hbar \: \partial_t \rho = [ H, \rho ] \quad .$...
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Can a non-unitary process be made unitary using a transformation or by expanding the phase space?
Suppose I have a matrix differential equation:
$$ \frac{d\mathbf{x}}{dt} = A\mathbf{x}$$
The solution to this is
$$\mathbf{x}(t) = e^{At}\mathbf{x}(0)$$
If $A^{\dagger}=-A$, then the time evolution ...
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Equations of motion and infinitesimal canonical transformations
Currently, I'm diving into infinitesimal canonical transformations, with a particular focus on using the infinitesimal change $\epsilon=\delta t$ and $H$ as our generating function. So, in this ...
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Interpretation of second-order term in time-dependent perturbation series (Dyson series)
$\newcommand{\ket}[1]{\left \lvert #1 \right \rangle}$
Context
Consider a system described by
$$H(t) = H_0 + V_0 v(t) \mathcal{O}$$
where $V_0$ defines the strength of a time dependent perturbation, $...
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Is the propagator the same as the matrix elements of the time evolution operator?
So Sakurai in their QM book defines the propagator in wave mechanics as:
$$K(x'',t;x',t_0)=\sum_{a'}\langle x''\vert a'\rangle \langle a'\vert x'\rangle \exp\left[\dfrac{-iE_{a'}(t-t_0)}{\hbar}\right]....
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The spreading of the wave function and Causality violation
In Hegerfeldt, 1998 paper "Instantaneous Spreading and Einstein Causality in
Quantum Theory" he states that,
"In nonrelativistic quantum mechanics the immediate spreading of wave ...
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Relationship between the Lindblad Equation and Redfield Equation
Both the Lindblad and the Redfield Equation 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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Conditions for the existence of a steady state in time-driven system
Consider an open quantum system given by the Hamiltonian
$$H = H_B + H_S(t) + H_{SB}$$
with $B$ denoting the noninteracting bath, $H_S(t)$ the time-dependent noninteracting system and $H_{SB}$ is a ...
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Intuition behind the different collapse terms of the Lindbladian?
A common way to treat dissipative quantum systems is through the use of the Lindblad master equation. Compared to the Schrodinger equation, it has extra non-unitary collapse/jump operators that ...
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How can time-evolution lead a Nèel-ordered state to an entangled state?
I am working with a system governed by the Following Hamiltonian:
H = $\sum_{i<j} J_{ij}(\sigma_i^+ \sigma_j^- + \sigma_i^- \sigma_j^+) + B\sum_j\sigma_j^z$
The Time evolution is:
$|\psi(t)\rangle =...
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Is quantum gravity compatible with unitary evolution?
I am thinking that they aren't strictly compatible. I have the following logical argument for this:
The unitary evolution postulate says that the state of a system is given by a time-depending state ...
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In what sense is this a feedback control?
I am reading a couple of papers on control enhanced parameter estimation. One is titled "Optimal Feedback Scheme and Universal Time Scaling for Hamiltonian Parameter Estimation" (arXiv ...
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