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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39 views

Dependence of wave function with time, especially probability density function. And Continuity equation

I was learning Basic Quantum mechanics. I cam across the fluid equation in QM, which suggests $\Psi^*\Psi$ is probability density function. Consider the two statements below Probability will change ...
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1answer
93 views

How does time-translation symmetry morph into evolution in time?

I am reading Ballentine's textbook "Quantum Mechanics: A Modern Development". In it he transitions from discussing time-symmetry to discussing evolution (of the state) in time. I'm finding it ...
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Are $\hat x$ and $\hat p$ assumed to be time-independent operators?

In the book Quantum Mechanics by Cohen-Tannoudji, at $G_{III}$, it is given that and then in the comment section, it is also given that so I'm pretty confused in here, because in one side, they say ...
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71 views

Time dependence of the momentum operator for a free particle

I was studying Modern Quantum Mechanics by Sakurai, and at the page 85, it is given the analysis of a free particle. There, the author assumes that Hamiltonian is $$\hat H = \frac{ \hat p ^2}{ 2m},$$...
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2answers
64 views

Two ways to define wave function in Heisenberg picture

I found two ways to define a wave function in Heisenberg picture, $| \psi(t) \rangle_\mathrm{H}=\mathrm e^{\mathrm i H t/\hbar} | \psi(t) \rangle_\mathrm{S}$ which further gives $|\psi(t) \rangle_\...
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2answers
92 views

Does $i\hbar \frac{d \hat A }{d t} = [\hat A (t_0), \hat H]$ hold when $H$ is time-dependent, but $[H(t_0), H(t'_0)] = 0$?

It is known that - given in Sakurai, ch2.2, p83 - in Heisenberg's picture, for a Hamiltonian, $H$, independent of time, the time evolution of any operator $\hat A$ is given by $$i\hbar \frac{d \hat ...
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1answer
48 views

Example of time-dependent factorization of a Hilbert space

In these notes on entanglement in QFT it is pointed out that in the Heisenberg picture the factorization of a Hilbert space is time-dependent (pages 18 and 19): In the Schrödinger picture, it is ...
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52 views

Operators evolution

I have a little question about equation for creation/annihilation operators. Usually we obtain time evolution equation for these operators from Heisenberg equation. for example: $$\frac{da_l}{dt} = -...
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48 views

Calculating the evolution at any moment $t$ of a density matrix

I was reading the paper https://arxiv.org/abs/1303.4686, where we are given $N$ systems, all with the same Hamiltonian $$H=\sum_i \varepsilon_i \mid i\rangle\langle i\mid ~,$$ such that the joint ...
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1answer
104 views

Utility of the time-ordered exponential

Is the time-ordered exponential $$ \mathcal{T}\exp\left\{-i\int_{t_0}^tdt'V(t')\right\}\tag{1} $$ just a mnemonic device for the series $$ \begin{aligned} 1 + (-i)\int_{t_0}^tdt_1 \, V(t_1) +{} &...
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Time-dependent perturbation theory in a degenerate system

In the derivation of probability transition of time-dependent perturbation theory (see for example these notes, from Ben Simons from Cambridge University), I have only encountered treatments of non-...
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4answers
237 views

Does the Schrodinger wave function associated with a non-moving free particle change in time?

I'm a bit confused by an answer given on this question. In the answer with the animation of a moving free (chargeless) particle and a non-moving free particle (or a free particle with a non-zero ...
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1answer
75 views

Phase in time evolution operator for time-dependent Hamiltonian [duplicate]

In Quantum Mechanics, a state vector $|\psi\rangle$ will evolve in time according to $$|\psi(t)\rangle=e^{-\frac{i}{\hbar}\hat H t}|\psi(0)\rangle$$ Imagine we have a system such that, for a short ...
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34 views

Timescales of plasma recombination and fluorescence?

I am currently working on a very simple model for the radiation from electric arcs. As both fluorescence (internal electronic transition) and plasma recombination occur, I would like to compare the ...
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1answer
288 views

What does Ehrenfest's theorem actually mean?

I am told that Ehrenfest's theorem, applied to a physical observable $\hat A$, is: $$\frac{d\langle\hat A\rangle}{dt}= \frac{i}{\bar h}\langle[\hat H,\hat A]\rangle$$ I don't understand how to use ...
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44 views

Time dependence of expectation value $\hat O$ if $\frac{\partial \hat O}{\partial t} = 0$

I am given the following derivation in my lectures: $$\frac{\partial}{\partial t} \langle \hat O \rangle = \frac{\partial}{\partial t}\int_{-\infty}^{\infty} \psi^* \hat O \ \psi \ dx$$ $$\implies ...
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1answer
123 views

Why in quantum mechanics must orthogonal states stay orthogonal? [duplicate]

Given two states $|A(t)\rangle$ and $|B(t)\rangle$. If $\langle A(0)|B(0)\rangle=0$ then for all $t$, $\langle A(t)|B(t)\rangle=0$. This is a fundamental rule of quantum mechanics. And we can imply ...
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1answer
80 views

Is a quantum channel essentially either a unitary evolution or a measurement?

I'd like to understand exactly what people mean when they speak of quantum channels. As I understand it, we can represent a channel by a set of Kraus operators, $M_i$, which satisfy $\sum_{i}M^{\...
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477 views

Logarithm of Operators in Quantum Mechanics

In an operators algebra $\mathcal{A}$ one can consider a self-adjoint (i.e. real) operator $H$ and note that $$U=e^{iH}$$ exists and is unitary. A mathematical question will be whether any unitary ...
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1answer
34 views

What does “Real-time” mean?

In the context of describing Real-time dynamics of Lattice gauge theories, have they specifically mentioned real-time in order to differentiate it from imaginary-time. Or does it have any other ...
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1answer
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Time evolution of a free particle with a given initial state [closed]

My homework problem reads: Consider a free particle in one dimension. Write an expression for the wavefunction $\psi(x, t)$ given an initial state $\psi_0(x) = Ae^{-ax^2}$ at $t = 0$, where $A$ is ...
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Matrix elements of the free particle Hamiltonian

The Hamiltonian of a free particle is $\hat H = \frac{\hat p^2}{2m}$, in position representation $$ \hat H = -\frac{\hbar^2}{2m} \Delta \;. $$ Now consider two wave functions $\psi_1(x)$ and $\psi_2(x)...
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1answer
63 views

“General” for time evolution of quantum state

I am reading a book in which at some point they find the time-evolved wavefunction $\phi_0(\mathbf{r},t)$ from the static $\phi_0(\mathbf{r})$. They say that "employing the Heisenberg time evolution ...
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1answer
178 views

How come there are Schrödinger Picture operators with explicit time dependence?

In the Schrödinger picture, observables are said to be time independent (see Cohen, for example) operators. However, when deriving the Heisenberg Equation of Motion $$i\hbar\frac{d}{dt}A_H(t)=[A_H(t),...
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Time evolution of stationary states [closed]

Let's say we have a state $ \phi=\sum_i c_i \phi_i $ where the $ \phi_i $ denote energy eigen vectors with non degenerate eigen values. Now if a measurement of the energy is done this state ...
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1answer
337 views

Does Haags Theorem forbid Time-Evolution?

I didn't quite grasp the essence of Haags Theorem in the the way it is presented (for example on wikipedia), but the issue seems to be that if one wants to represent infinitely degrees of freedom ...
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2answers
227 views

What's the time derivative of the Annihilation operator?

I've been dealing with annihilation operator recently where you can see related information Time derivative of the state vector as expressed in abstract Hilbert space vs. as a wavefunction How to ...
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1answer
58 views

Does it make sense to ask: what is the probability of a particle being found in a certain state at time $t>0$?

I am dealing with a problem which involves a quantum system of orthonormal two states, $\left|\nu_1\right>$ and $\left|\nu_2\right>$, which are eigenstates of a time-independent Hamiltonian, ...
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1answer
74 views

What is the state of particle at time $t$ if at $t=0$ it is in an eigenstate of $\hat{A}$, and $\hat{A}$ commutes with $\hat{H}$?

EDIT: added (assuming $\lambda$ to be non-degenerate). Based on the specifics of the question, we don't in fact know whether this is the case, so it may be that $\left|\lambda\right>$ is not an ...
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1answer
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Can I use time evolving block decimation (TEBD) to simulate the dynamics for many body localized systems?

In the many-body localized phase, the system is described by quasi-local integrals of motion ("l-bits"). The entanglement does grow logarithmically with time. So if I use TEBD to get the real-time ...
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3answers
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What is meant by unitary time evolution?

According to the time evolution the system changes its state the with the passage of time. Is there any difference between time evolution and unitary time evolution?
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1answer
275 views

Time evolution operator in QM

I am reading a introduction to quantum mechanics right now. There is a part about the time evolution operator: \begin{align*} i\hbar \partial_t \,\psi(\vec r, t) = \hat H(t)\, \psi(\vec r,t) \end{...
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1answer
144 views

Heisenberg and Schrödinger pictures - clarification

Question related to The equivalence between Heisenberg and Schroedinger pictures. I understand what's explained in the link provided above. My textbook (Breuer and Petruccione's Theory of Open ...
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1answer
155 views

Solving the Schrodinger equation with a time-dependent Hamiltonian

I am trying to find the general solution to the Schrodinger equation with a time-dependent Hamiltonian: $$ i \frac{\partial}{\partial t}| \psi(t) \rangle = H(t) | \psi(t) \rangle.$$ My Hamiltonian ...
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4answers
856 views

How do base kets satisfy Schrödinger's equation in Schrödinger picture and why don't they evolve with time?

According to Sakurai, eigenvalue equation for an operator $A$, $A|a'\rangle=a'|a'\rangle$. In the Schrödinger picture, $A$ does not change, so the base kets, obtained as the solutions to this ...
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2answers
129 views

Are the fundamental concepts in Heisenberg Picture and Schrodinger Picture different?

In Heisenberg Picture, for a free particle, $[x_i(t),x_i(0)]=\frac{-i\hbar t}{m}$. This relation implies that even if the particle is well localized at t=0, its position becomes more and more ...
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150 views

Infinite square well and Heisenberg picture

The infinite square well is often a mainstay of introductory quantum physics courses. Its boundary conditions at the well-walls are easily solved to the find the Hamiltonian's eigenfunctions in the ...
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How do I add decoherence to an oscillating system

If a have an initial (two-qubit) system in the state $\rho_i= \begin{pmatrix} 0&0&0&0\\ 0&1&0&0\\ 0&0&0&0\\ 0&0&0&0 \end{pmatrix}$ and this state ...
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3answers
128 views

Time evolution of eigenstates superposition

If a system is in a state $\psi$ which is superposition of, let's say two, energy eigenfunction, namely $\psi_1$ and $\psi_2$, so that $$\psi(t)=\psi_1e^{-i\omega_1t}+\psi_2e^{-i\omega_2t}$$ (I am ...
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2answers
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Is there a unitary transformation such that the Hamiltonian in the time-dependent Schrödinger equation becomes real symmetric?

The time-dependent Schroödinger equation is given as (with $\hbar=1$): $$i\dfrac{d}{dt}\psi(t)=H(t)\psi(t)\ ,$$ where $\psi$ is some normalized column vector and $H(t)$ is a Hermitian matrix with time-...
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QM: Time evolution with $H = H(t)$

In order to calculate time evolution in QM we use Schrödinger equation \begin{align*} i \partial_t |\psi\rangle_t = H(t) | \psi\rangle_t. \end{align*} If $H\neq H(t)$ then \begin{align*} i \partial_t ...
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4answers
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Why can't two different quantum states evolve into the same final state?

Is it true that two different states cannot evolve into the same final state? Can they achieve this state at different times? If yes, what is the proof?
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Definition of Hamiltonian in Quantum Mechanics [duplicate]

Is there any particular reason that the Hamiltonian operator was defined in quantum mechanics to be $$\hat H := \frac{\hat p^2}{2m} + V$$ as opposed to $$\hat H := i\hbar \frac{\partial}{\partial t}?$$...
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Is the universe ~14 billion years old or that's the farthest photons which reached Earth?

I know that the universe: It's around 13.772 billion years old It expands But it's not clear to me if this is not merely the age of the farthest known photons which reached Earth.
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Utility of the Magnus expansion (preserving symplectic form?)

There are (at least) two ways to perturbatively solve a matrix initial value problem: the Dyson expansion and the Magnus expansion To be explicit, suppose you're solving for a density matrix $\rho(t)$...
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9answers
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How does a particle know how to behave? [duplicate]

How does a particle know it should behave in such and such manner? As a person, I can set mass is so and so, charge is so and so - then set up equation to solve its equation of motion but who ...
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3answers
283 views

Can a second-order Schrödinger equation preserve the norm?

Suppose we lived in a universe in which the Schrödinger equation contains second order time derivatives, $$i\hbar \partial_t^2|\varphi(t)\rangle = \mathbb{H} | \varphi(t)\rangle.$$ Would it be true ...
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3answers
132 views

Collapse of Wavefunction, and Subsequent Time Evolution

To keep it simple, suppose the system is the well-known particle in a 1D infinite potential well. Suppose the wavefunction is $ a|1\rangle + b|2 \rangle + c|3\rangle$, where the $|i\rangle$ are ...
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55 views

Phase of quantum state during propagation

Evolution of quantum state in time can be obtained from the time-dependent Schrodinger equation $$\hat{H} \psi(x,t) = i \frac{\partial}{\partial t} \psi(x,t).$$ For time-independent Hamiltonian, the ...
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Translating between classical treatment of non-autonomous systems and time evolution in quantum mechanics

When I read an introduction to (classical) dynamical systems, the system was considered in a phase space, and the state of the system evolving in phase space. For a non-autonomous system, an ...