The Stack Overflow podcast is back! Listen to an interview with our new CEO.

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
32 views

How to find the probability of a state from non-degenerate spectral decomposition

I have just begun learning the topics of time evolution in quantum mechanics, and I'm having trouble understanding how to calculate the probabilities of certain eigenvalues of an operator at a later ...
1
vote
1answer
31 views

What's the necessary and sufficient condition for the expectation value of an measurement to be constant?

For operator $A$, the famous equation from Heisenberg picture (see: Time Variation of Expectation Value ) stated that, since $$\frac{d A}{dt}=\frac{i}{\hbar}\left[H,A\right]+\left(\frac{\partial A}{\...
4
votes
1answer
80 views

Why is time-evolution unitary - The Heisenberg-picture Version

There are various versions of this question already on this site, which attempt to justify / make plausible that the time evolution of quantum mechanical observables is unitary. Most of these ...
1
vote
2answers
58 views

Representation of time evolution operator and spectral decomposition

I was preparing for my quantum mechanics exam, and I came to think about this question regarding the spectral representation of time evolution operator. Let's say we are given the Hamiltonian: $\hat{...
2
votes
2answers
82 views

Does a quantum channel being time-translation invariant imply that its Kraus operators commute with the Hamiltonian?

Let $\mathcal E\in\mathrm{T}(\mathcal X,\mathcal Y)$ be a quantum channel (i.e. a completely positive, trace-preserving linear map sending states in $\mathrm{Lin}(\mathcal X,\mathcal X)$ into states ...
4
votes
0answers
50 views

Why can time-translation invariant quantum operations never increase coherence between energy eigenspaces?

Set $\hbar =1$. Let $U(t) = e^{-itH}$ be evolution under a Hamiltonian $H$ (for convenience let's assume $H$ is not degenerate). A time-translation invariant quantum operation $\mathcal{E}$ is one ...
0
votes
0answers
20 views

$S$-matrix and in and out states

So, I have a short one. When observing scattering, we say that the amplitude for transition from one interacting state to some other interacting state same as this amplitude for free hamiltonian ...
2
votes
2answers
73 views

Can any linear but non-unitary “time-evolution operator” be normalized to a unitary one?

A comment to this answer to another question states I would imagine that for any linear non-unitary time-evolution operator, I can find a unitary one that will yield the same expectation values for ...
2
votes
0answers
14 views

Formalism for an open system with non-adiabatic (non periodic) time dependence

Most non-equilibrium statistical processes of open time dependent systems are approached by Markovian dynamics of a system where time dependence of the system is assumed to be adiabatic (if Floquet ...
1
vote
0answers
37 views

“Imaginary-time” argument in high energy physics

In many-body physics, there are many "imaginary-time" techniques, such as Matsubara Green's function, imaginary-time path integral and others. It seems that these concepts are frequently used in ...
-2
votes
1answer
32 views

Do the determination of conserved quantity in QM can use other operator instead of Hamiltonian?

The conserved quantity in quantum mechanic determine by $$\frac{d}{dt} \langle Q\rangle = \frac{-1}{i\bar{h}}\langle\psi|[H,Q]|\psi\rangle+\langle\psi|\frac{dQ}{dt}|\psi\rangle$$ First: From the ...
-3
votes
1answer
67 views

A simple explanation of the Born rule (v.2)? [duplicate]

Please post further comments or answers to A simple explanation of the Born rule? The probability that an initial quantum state $|\psi_i\rangle$ evolves to become the final quantum state $|\psi_f\...
0
votes
2answers
75 views

What is evolving with time?

In Griffiths, we are told that the expansion coefficients of the stationary states are simply complex numbers: $$\Psi(x, \ t) \ = \ \displaystyle\sum_{n} c_n e^{-iEt/\hbar} \psi_n(x)$$ How do we ...
1
vote
1answer
142 views

When should we consider “reverse Heisenberg” evolution of operators?

In Quantum Mechanics, the Heisenberg evolution of an observable $\hat{o}$ is defined as $$ \hat{o}(t) = U(t,0)^{\dagger} \hat{o} U(t,0) $$ where $U(t,0)$ is the unitary time-evolution operator from ...
0
votes
2answers
109 views

A simple explanation of the Born rule?

The probability that an initial quantum state $|\psi_i\rangle$ becomes the final quantum state $|\psi_f\rangle$ is given by \begin{eqnarray} P(i \rightarrow f) &=& |\langle\psi_f|\psi_i\...
3
votes
0answers
52 views

Given two abitrary operators satisfying the canonical commutation relations at all times, can I show that they obey the same time evolution?

To be more clear: I take two operators, $A$ and $B$, which are functions of time, $A(t)$ and $B(t)$. I assume their time development to be unitary (but not neccessarily the same): $A$'s time evolution ...
1
vote
1answer
33 views

Time-dependent pertubation theory assigning the order of expansion to squares of the solution

What is the square of a solution from time dependent pertubation theory? Assume we have found the corrections up to second order such that $$ |\psi(t)\rangle \approx |\psi^0(t)\rangle + |\psi^1(t)\...
1
vote
0answers
52 views

Could Time-Evolution be antiunitary?

There are serveral Arguments for Time-Evolution to be unitary, for example, time-evolution should preserve the norm of each given state (because elseways the probabillity Interpretation would not work)...
0
votes
1answer
32 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
64 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
35 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
605 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
103 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
64 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
48 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
132 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
83 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
76 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
80 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
63 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
103 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
51 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
43 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
91 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
27 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
97 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
41 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
30 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
34 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
31 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
115 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
711 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
83 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
142 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
57 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
55 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 ...