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).
934 questions
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Do we have any example of non-autonomous system where the trajectory in phase space cuts itself?
In classical mechanics, for an autonomous dynamical system, trajectories in phase space cannot intersect themselves because the state uniquely determines the future evolution (given by a time-...
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How do we motivate $S$-matrix elements from quantum mechanics? [closed]
In quantum mechanics, if we want to find the probability amplitude of a state $\psi_i$ propagating to the state $\psi_f$, we evaluate the quantity $\langle\psi_f|\hat U(t_2-t_1)|\psi_i\rangle$, where $...
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Time dependent eikonal equation
I am studying wave theory and optics and I want to derive the eikonal equation from the general wave equation:
$${\displaystyle {\frac {\partial ^{2}u}{\partial t^{2}}}=c^{2}{\frac {\partial ^{2}u}{\...
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Is the evolution of an observable's eigenstate, via Schrodinger's equation, consistent with the postulates of Quantum Mechanics?
Given an observable $L$, suppose the state of the quantum system is an eigenvector
$|\lambda\rangle$ of $L$, corresponding to $L$'s eigenvalue $\lambda$. (Suppose for simplicity of notation that each ...
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Basis eigenvectors for the expansion of the state ket in the interaction picture
I'm approaching for the first time the study of the matter-radiation interaction, in particular I'm learning about time dependent perturbation theory in non relativistic quantum mechanics. I'm ...
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Path integral representation of the linear response/correlation function
I'm currently reading Xiao-Gang Wen's QFT of Many-Body Systems and I'm struggling to understand what happens in 2.1.9. Let $H=H_0+f(t)V$ and $|0\rangle$ be the ground state of $H_0$. We assume $f(t)$ ...
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Phase evolution rates of quantum sub-systems
Do the phases of the wavefunctions for non-interacting quantum systems evolve faster if those systems are conceived of as components within some larger system rather than each being treated separately?...
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Do existing quantum mechanical models provide a physical mechanism for finite-time quantum state transitions?
In standard quantum mechanics, the formalism successfully predicts transition probabilities between quantum states, but the transition itself is often treated as instantaneous or left without a ...
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Does time-ordered exponential still have the BCH lemma?
I am reading Shaul Mukamel's Principles of Nonlinear Spectroscopy. When talking about time evolution in Liouville space, he says in page 61 that
$$\mathscr{U}_{jk,j^\prime k^\prime}(t,t_0)=U_{jj^\...
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Fourier transform of fully nonlinear Heisenberg equations for coupled Kerr modes
I am investigating the dynamics of two interacting bosonic modes with self-Kerr and cross-Kerr nonlinearities. The Heisenberg equations of motion are:
$$
\frac{d\hat a}{dt} = - i\omega_a \hat a + \...
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Instantaneous eigenstates and real solutions of Schrödinger Equation [closed]
As we know, instantaneous eigenstates of energy are the states in which the wavefunction collapses instantly after we measure energy. And we also know that instantaneous eigenstates don't satisfy the ...
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Quantum harmonic oscillator. Application of evolution operator on position eigenstate yields a momentum eigenstate. How to find the phase factors?
Sorry about the stupid title. I have no idea how to summary the following content into a concise title.
This is related to exercise 5.8 of Cohen-Tannoudji's QM textbook.
Let us consider a 1D harmonic ...
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Do maps between pure states of a quantum system correspond to elements of the projective unitary group?
A very standard and widely discussed fact in quantum mechanics is that pure quantum states are only defined up to an overall phase factor.$^1$ That is, the state vectors $|\psi\rangle$ and $e^{i \...
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Confusion about the definition of the interaction-picture potential $V_I(t)$
I am reading a derivation of the interaction picture in subsection 7.2.4 from Schwartz's QFT book, and I am confused about how the interaction-picture potential $V_I(t)$ is being defined and used.
At ...
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Measuring energy (when the potential is time varying)
When the potential is time varying,there are no energy (total energy) eigenstates.So what energies values we obtain when we measure the energy of a particle in such a potential?Does "not having ...
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