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).

Filter by
Sorted by
Tagged with
0
votes
1answer
26 views

Interaction picture Sakurai

I’m going through Sakurai and got stuck with the following in the interaction picture subsection $$i \hbar \frac{\partial}{\partial t}\left|\alpha, t_{0} ; t\right\rangle_{I}=i \hbar \frac{\partial}{\...
0
votes
2answers
50 views

Solution of Time-dependent Schrodinger Equation for Unitary Operator

While reading Quantum Mechanics Book by Sakurai, I found the time-dependent Schrodinger equation for Unitary Operator. $$i\hbar \frac{\partial}{\partial t}\mathcal{U}(t,t_0)=H\mathcal{U}(t,t_0).$$ ...
1
vote
1answer
28 views

Difference between time series and trajectory terminology

What is the difference between trajectory and time series? To me both seem the same thing. In the 3D diagram (cube picture on left of Fig.2 from the paper titled “Review and comparative evaluation of ...
0
votes
0answers
36 views

Do all unitary operations manifest from time-evolution?

Let $|\psi\rangle$ be an element of a Hilbert space $\mathcal{H}$ and $U$ a unitary operator on $\mathcal{H}$. I am concerned with the actual physical manifestation of such a unitary operator in the ...
8
votes
1answer
584 views

Is the evolution operator well-defined mathematically?

We know that in order to solve the time-dependent Schrodinger equation $i\partial_t \psi = H(t) \psi$, we need the evolution operator $$U(t) = T \exp{\left(-i\int_0^t H(t')dt'\right)}$$ where $T$ is ...
1
vote
1answer
94 views

Does time evolution preserve parity?

Let $\psi(t_0, \cdot)$ be the state of a quantum system corresponding to a Hamiltonian $H$ in the position representation at time $t_0$. Assume $\psi(t_0, -x) = \psi(t_0,x)$, that is $\psi(t_0, \cdot)$...
1
vote
1answer
62 views

Why don't expectation values for a stationary state evolve over time?

I have an observable $O$ with operator $\hat{O}$. $\Psi_1$ is a wave function in an energy eigenstate, and $\psi_1$ is the corresponding spatial wave function. $E$ is the corresponding energy. It is ...
1
vote
0answers
30 views

Total hamiltonian is time independent in interaction picture

There is this general statement in Ashok which, if it's true, could someone explain why it is true? Regarding the interaction picture: Since $H_0$ is time independent in the interaction picture (...
1
vote
1answer
40 views

First order wave function in adiabatic approximation

If the Hamiltonian is slowly varying in time and suppose the initial state is the n-th eigenstate of the initial Hamiltonian H(0), the adiabatic theorem says that the state will still on the n-th ...
1
vote
0answers
25 views

Time-order evolution operator: Wei-Norman form and unitarity

I'm reading a paper[1] in which the propagator is calculated for this kind of Hamiltonian \begin{align} \hat{H}(t) = \omega(t)\hat{J}_3 + \Omega^*(t)\hat{J}_{+} + \Omega(t)\hat{J}_-. \end{align} ...
2
votes
1answer
110 views

Dyson Series Iteration - Gives Exact Solution?

When we derive the Dyson series for usage as the time evolution operator in the case of a time dependent Hamiltonian, we start with the equation: \begin{align}\hat{U}_I(t,t_i) = 1 - \frac{i}{\hbar}\...
1
vote
1answer
75 views

I really wonder about the time derivative of creation and annihilation operators in the derivation of LSZ

In Schwartz book, they assume that $$\lim_{t \to \pm\infty}\partial_0 a_p(t)=0.\tag{1}$$ But I thought that is just assumption. so we have to construct the mathematical description. I found the Gell-...
0
votes
1answer
30 views

Probability density of time-dependent wave functions

Why is it so that probability density of eigenfunctions of time-dependent schrodinger equation are time independent while that of general wave functions (which are a combination of the eigenfunctions) ...
3
votes
1answer
76 views

Significance of energy in a time dependent quantum box

The Hamiltonian for a particle in a finite box is $$H = \frac{p^2}{2m} + V(x)$$ which will give time evolution as $$ i\hbar d/dt|{\psi(t)}\rangle = H|{\psi(t)}\rangle \, .$$ However, if I do a ...
0
votes
1answer
57 views

Are superposition and time-evolution of a quantum system unrelated?

Consider a single particle (a single qubit if you will) in some arbitrary state $|\psi\rangle$ and an eigenvector $|\lambda\rangle$ corresponding to the eigenvalue $\lambda.$ Consider the time ...
4
votes
1answer
64 views

Examples of non-Hermitian Hamiltonians in open systems?

I have often heard the statement that non-Hermitian Hamiltonians can be used to describe open systems, since the dynamics are non-unitary. However, I have not been able to find any examples of a non-...
1
vote
1answer
47 views

What are the ways to carry out time propagation numerically?

There are many ways to solve the time-dependent Schrodinger equation (TDSE) and find the wavefunction $|\Psi(t)\rangle$ for a given Hamiltonian. For example, consider a tight binding type Hamiltonian: ...
0
votes
0answers
40 views

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: ...
2
votes
1answer
90 views

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 ...
0
votes
0answers
25 views

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 ...
0
votes
1answer
76 views

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 ...
1
vote
0answers
39 views

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 ...
0
votes
1answer
28 views

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 ...
0
votes
0answers
33 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?
1
vote
0answers
30 views

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 ...
2
votes
2answers
106 views

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 ...
9
votes
2answers
701 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 ...
3
votes
1answer
76 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 ...
3
votes
4answers
130 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 $\phi(t)$ ...
1
vote
1answer
56 views

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$ ...
0
votes
2answers
42 views

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 ...
1
vote
1answer
44 views

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 ...
2
votes
0answers
42 views

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 ...
0
votes
0answers
42 views

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 ...
2
votes
0answers
65 views

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{...
1
vote
1answer
82 views

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 $\mathcal{H}$ ...
5
votes
1answer
184 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 ...
1
vote
2answers
190 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 ...
1
vote
1answer
108 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 ...
7
votes
2answers
653 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)\...
0
votes
0answers
54 views

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 ...
2
votes
2answers
114 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 ...
1
vote
1answer
72 views

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\...
2
votes
4answers
120 views

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 ...
-2
votes
1answer
54 views

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 ...
0
votes
0answers
63 views

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 ...
1
vote
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 ...
1
vote
2answers
231 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 ...
0
votes
1answer
65 views

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 $\Psi$ ...
0
votes
2answers
62 views

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 ...