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).
918 questions
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Why do physical laws depend only on velocity but not acceleration? [duplicate]
Newton's Laws can be presented as a statement regarding the acceleration and time evolution of objects:
$$\ddot {\bf x} = F(\dot {\bf x},{\bf x}) $$
In other words, the time evolution of a physical ...
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How do we define time-ordering operations in QFT?
Given two bosonic operators $A$, $B$ (in the Heisenberg picture) in a QFT, the time-ordered product of $A$ and $B$ is defined as
$$
T\{A(t_1)B(t_2)\}=\theta(t_1-t_2)A(t_1)B(t_2)+\theta(t_2-t_1)B(t_2)A(...
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Dyson series expression for the two-point Green function
On Chapter 7 of Fetter & Walecka, the authors prove Dyson formula for the (imaginary time) propagator $U(t,t_{0}) = e^{H_{0}t_{0}}e^{-H(t_{0}-t)}e^{-H_{0}t}$, where I am ommitting the $\hbar$'s. ...
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Is it truly possible to adiabatically achieve the flat band limit?
The conditions for adiabatic evolution is that for every possible $n$, we must have
$$ \sum_{n\neq m}\frac{\langle n(t) | \dot{H} | m(t) \rangle }{E_n(t)-E_m(t)} \to 0$$
which is achievable either by ...
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Time evolution of mechanical momentum operator in electromagnetic field [closed]
Follow-up question to How to deal with explicit time dependence in the Heisenberg picture?
Time evolution of an operator A (in Heisenberg picture) is given as:
$$\frac{\mathrm{d}\hat{A}_H}{\mathrm{d}t}...
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How to reconcile the reversible nature of microscopic physics with the irreversible nature of macroscopic reality? [duplicate]
This might be a conceptual question more than a technical one, however it would be very useful to see different conceptual approaches to address this problem physically/mathematically.
The question ...
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How to switch from the interaction picture to the Heisenberg picture?
In the Schrödinger, Heisenberg, and interaction pictures, the time evolution of an operator $A$ is defined differently.
In the $\textbf{Heisenberg picture}:$
\begin{equation}
A_H(t) = e^{i(H_0 + V)t} ...
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Formula for a moving Schrodinger gaussian wave packet in three space dimensions [closed]
I am trying to time evolve quantum systems in the presence of a gravitational potential. To do this I need a free localised 3+1 dimensional wave packet with velocity so that I can perturb with a weak ...
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If there are time-dependent coefficients on the left side of the equation, and a wave function on the right side of the equation, is that okay?
The state of a quantum system at time $t$ is given by the orthonormal base states $k$ and $j$, with probability amplitudes:
\begin{equation}
C_k(t_0 + \Delta t) \: = \: \langle{k} \vert \psi(t_0) \...
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Time evolution of the uncertainty principle
I've written the following proof:
Let $A,B$ be observables (Hermitian operators). define $ ΔA:=A-<A> $.
the uncertainty principle is: $$ ⟨(ΔA)^2⟩⟨(ΔB)^2⟩ ≥|⟨[A,B]⟩|^2/4 $$
The Heisenberg ...
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How to calculate the Floquet state creation operator in Heisenberg picture? [closed]
I am following the The Floquet Engineer’s Handbook (https://arxiv.org/pdf/2003.08252).
The formula (36) in page 10 is trying to calculate the time evolution of the Floquet creation/anni. operator in ...
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Meaning of Derivative in time evolution equations [duplicate]
In wave mechanics, the time evolution equation is given as
$$
i \hbar \frac{\partial}{\partial t} \psi(\vec{r},t)= \hat{H} \psi(\vec{r},t)
$$
Here $\psi(\vec{r},t)$ is a function from space and time ...
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Energy eigenstates in time dependent potential
Suppose the potential of the quantum particle is time dependent , $V(x,t)$ . Therefore the hamiltonian can be written as :-
$$\hat H=-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}+V(x,t)$$
And we ...
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In Euclidean QFT, what is the generator of $\tau$ translational symmetry?
In Minkowski space, the KG Lagrangian is time translation invariant. The corresponding Noether charge is the Hamiltonian. Because this is a symmetry it is generated by a unitary operator
$$|\psi\...
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Paradox regarding propagation of a pulse wave
In optics, we often use the complex amplitude to specify the electric and magnetic fields at any instant.
A rightward-propagating plane wave can be expressed as
$$
\exp(ikz)
$$
and
$$
\exp(-ikz)
$$
is ...
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