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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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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Time Evolution for Interacting Particles [closed]
Let us consider a Hamiltonian consisting of some quantity times the tensor product of two Pauli Spin matrices. How does one deal with the time evolution operator in this context? We usually do the ...
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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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Why Magnetic Field doesn't annihilate itself?
Reasked, edited
Note: I don't get why it is not a good question according to some people, but as far as I can say it is well posed. If usefulness is of matter, the question is about why zero energy ...
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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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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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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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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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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 ...
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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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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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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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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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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}$$ Is it true that the unitary operator generated by $H(t)$ is a ...
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How to calculate pressure $p$ in the Navier-Stokes equation to simulate the time evolution of a fluid?
I wanted to simulate the motion of a fluid (continuously filling the entire space) in a given space, say $3$-dimensional Euclidean space. To calculate the dynamics of fluid motion, I used the Navier-...
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Question about the time-ordered exponential operator
I learned that a unitary matrix generated by time-dependent Hamiltonians is written down as
\begin{equation}
U(t) = \mathcal{T}\exp\Big(-i\int_0^t H(t') dt' \Big),\tag{1}
\end{equation} where $\...
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Can a non-unitary process be made unitary using a transformation or by expanding the phase space?
Suppose I have a matrix differential equation:
$$ \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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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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Time evolution operator in classical mechanics?
Hamilton's equation can be written in terms of Poisson brackets, as follows:
$$\dot{q} = \{q,H\}$$
$$\dot{p} = \{p,H\}$$
where $H$ is the Hamiltonian of the system. Now, wikipedia says that the ...
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Formula from wikipedia relating Heisenberg with Schrödinger pictures [duplicate]
I have a pretty naive question about a formula from wikipedia relating operator $A_H $ in Heisenberg picture with it's equivalent $A_S $ in Schroedinger picture:
$$ \frac{d}{dt}A_\text{H}(t)=\frac{...
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Discrete-time-dependent Hamiltonian from sequence of unitaries
Consider a system evolving in discrete time with the time evolution given by a some unitary operator $\hat{U}(t)$ that advances the system by one time step, i.e. $$\psi(t+1) = \hat{U}(t)\psi(t).$$ The ...
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Self-adjointness, time evolution operator and the role of domains
Introduction
Let us consider a Hamiltonian $\hat{H}$ for a certain system and suppose that I would like to know if it is possible to define it (i.e. its domain) in such a way that it results self-...
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Commutator in the derivation of the Runge-Gross Theorem
In the derivation of the Runge-Gross theorem, a theorem that states a one-to-one correspondence between potential and density in an evolving system, a commutator appears that I seem to have trouble ...
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Time Evolution Operator Explained in The Theoretical Minimum: Quantum Mechanics book
In section 4.4 and 4.5 of "The Theoretical Minimum: Quantum Mechanics" book, there are a explanation about time evolution operator U(t).
The author explains that:
$U^\dagger (t)U(t) = I$
In ...
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Time evolution of non-interacting field operators [closed]
I learned that for a non-interacting tight-binding system $H=\sum_{n}\varepsilon_na^\dagger_n a_n$, the time evolution of $a_n$ is simply $a_n(t)=e^{-i\varepsilon_nt}a_n$. I tried to prove this:
\...
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Behavior of the temporal part of the wavefunction in an infinite well
I have a question about the behavior of the temporal part of the wavefunction in an infinite well. For reference, the solution given in my textbook for the wavefunction of a particle in an infinite ...
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Evolution of conformal primary operator in spacetime
Let $P$ be the element of conformal algebra (including Poincare) corresponding to infinitesimal spacetime translation. Let $\phi(x)$ be a primary conformal operator, defined as $\phi(0)|0\rangle=|\phi\...
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Time evolution of perturbation
This is from Noelle Pottier's book. I am unclear on how equation 15.1.10 follows from the information that she's given. Is the equation a consequence of inserting equation 15.1.9 in the commutator?
...
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Clarification on interaction picture in QFT
Say we want to calculate $\langle f(t_2)|O|i(t_1)\rangle$. Where $O$ is an arbitrary operator. We can treat the states as stationary and then evolve the operator
$$\langle f(0)|O(t)|i(0)\rangle\\O(t) =...
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Free particle propagator, wavefunction not moving problem
I'm learning the QM propagator and the first example is of course the free particle: $\hat{H}=\frac{p^2}{2m}$, then the new wavefunction is found by:
$$\psi(x,t)=\int dx_0\;K(x,t;x_0,t_0)\;\psi(x_0,...
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Is Hamiltonian being a time evolution operator a consequence of minimizing action?
When I write down the Lagrangian for a quantum field, I can derive the equation of motion for the field. Therefore, Lagrangian specifies how field evolves with time completely. Can I derive ...
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Time evolution of Gaussian wave packet in free space with initial condition Dirac delta function [duplicate]
I learn something about the Time evolution of Gaussian wave packet in free space.
if the initial condition is a Dirac delta function at $t=0$, then the wave function is
$$
\psi (x,t)=\frac{1}{\sqrt{2\...
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Double change for frame of reference
Is it possible to change frame of references 2 times? Being more specific, if I have a Hamiltonian $$H = H_0 + H_a + H_b,$$
i can go in the interaction picture and have my new interaction picture ...
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Time evolution operator for 2-level system hamiltonian
I'm trying to compute the time evolution operator of a system with the following hamiltonian:
$$ H(t) = g(t)[\sigma^+ e^{i \omega t } + \sigma^-e^{-i \omega t }] $$
I tried to use the Magnus expansion,...
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What's the connection between the product of advanced and retarded Green functions and system dynamics in quantum mechanics?
As written in the book Atom-photon interaction by Claude Cohen-Tannoudji, the time-evolution operator can be expressed as the Fourier transform of the Green's function
$$
U(\tau)=e^{-iH\tau}=\frac{1}{...
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Second-order Trotter error involving an unbounded Hamiltonian
I have an Hamiltonian of this form:
\begin{equation}
H = \frac{p^2}{2m} + V(x),
\end{equation}
I would like to approximate the time evolution for a time $\tau$ of a known initial Gaussian state $|\...
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Time evolution operator in quantum mechanics
One of the postulates of quantum mechanics is that, given a quantum state $\psi_{0}$ at time $t=0$, the state of the system at a posterior time $t > 0$ is given by $\psi_{t} = e^{-iHt}\psi_{0}$, ...
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Showing that interaction and Schrödinger pictures are equivalent -- does the result hold without commutativity?
I am trying to show that the interaction picture represents a valid picture. We use this picture when our Hamiltonian takes the form $H = H_0 +H_1$. "Valid" means that it should reproduce ...
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Are von Neumann entropies of complementary but physically distinct subsystems in time-dependent settings identical?
We assume a quantum system AB with subsystems A and B and take the Schmidt decomposition of a state $\vert\psi_{AB}\rangle=\sum_i\lambda_i \vert a_i\rangle\vert b_i\rangle$ defined on the compsite ...
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The evolution of observables in the Heisenberg picture
Suppose that a system $S$ undergoes the following time-evolution:
$t_{1} \rightarrow t_{2}: |x_{a}\rangle \rightarrow \sum c_{i}|x_{i}\rangle$,
where $|x_{a}\rangle$ and each of $|x_{i}\rangle$ is a ...
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