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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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91 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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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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0answers
36 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
478 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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1answer
54 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
135 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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2answers
496 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
36 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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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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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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1answer
66 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
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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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2answers
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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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4answers
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Why does the time-evolution operator $U(t)$ depend explicitly on time in the Schrodinger picture?

Schrodinger's picture is that operators are time-independent. But time evolution operator $U(t)$ is time-dependent. Why is that?
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2answers
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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
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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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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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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
306 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
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Why is time-evolution unitary (the sequel)?

One foundational postulate of QM is that a closed physical system at one instant of time, say $t$, is completely described by a wavefunction $\psi \in S^1\subset H$ (where $H$ is a Hilbert space and $...
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1answer
180 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
870 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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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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2answers
246 views

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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3answers
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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
76 views

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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0answers
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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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2answers
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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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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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0answers
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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
149 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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2answers
56 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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1answer
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Two Time Correlation function calculated from Born rule

Update Below I'm having a hard time reconciling two different calculations of the quantum two time correlation function. Consider quantum operator $A$ with eigenvectors $\{|\phi_i\rangle\}$ and ...
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1answer
155 views

Operator Transformation and Time Evolution

UPDATE: See update at bottom of post Short Version: $\sigma^-_H$ in the Heisenberg picture. $U = e^{+i\omega (|e\rangle \langle e)_H}$. Prove that $$ \frac{d \sigma^-_H}{dt} e^{+i\omega t} = U^{\...
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2answers
521 views

Two Level Atom Rotating Frame

Introduction I am trying to work out the Rabi problem. In particular I am trying to work it out from a perspective focusing on the operators rather than the kets. Think Heisenberg picture rather than ...
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4answers
2k views

Schrodinger equation for a Hamiltonian with explicit time-dependence?

Can I write a Schrodinger equation for time-dependent Hamiltonian like this: $$i\hbar\frac{d}{dt}\psi(t) = H(t)\psi(t)$$ and then perform Euler integration like this: $$\psi(t+\Delta t) = (1-\frac{...
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1answer
95 views

Time evolution of a projected mixed state

Suppose a quantum system (non-interacting) at finite temperature ($\beta^{-1}$). I want to know how to compute the transition probability between two degrees of freedom ($u$ and $v$) at two different ...
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0answers
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Plausibility of Heisenberg equation for the canonical momentum:

In this question, I want to to know wether my reasoning on the plausibility of the Heisenberg equation is flawed: Let's say I want to describe my system in the quantum-mechanics framework: ...
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1answer
83 views

Time evolution of probability in the Heisenberg Picture

How does the probability of a system being in a state change with time in the Heisenberg picture if the state vector is time independent?
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Is it an assumption or truth for spatial and temporal variables separation if Hamiltonian is time-indepdent in Schrodinger equation?

For the derivation from time-depdent Schrodinger equation to time-indepdent Schrodinger equation, if the Hamiltonian is time-indepdent, we assume the spatial and temporal variables in the wave-...
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2answers
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Is there an equivalent to the Schrodinger equation for quantum mechanics over the reals?

Many people have considered alternatives to standard quantum mechanics in which the Hilbert space is over the real instead of the complex numbers - see e.g. here, here, here, here, here, and here. In ...
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0answers
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Why can we use time-dependent perturbations when evaluating the S-matrix?

Suppose we have Hamiltonian $H_0 + V$. When working in the interaction picture we may derive the evolution operator of $|\psi_I(0)\rangle$ which is given by $$S(t,t_0) = T\left[\exp \left( -i \int_{...
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1answer
582 views

Magnus Expansion in Floquet theory [closed]

I wonder how to obtain the second equality as follows in Eq. (44) of Universal high-frequency behavior of periodically driven systems: from dynamical stabilization to Floquet engineering. M Bukov, ...
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1answer
186 views

Time evolution of operators with explicit time dependence in case of time dependent Hamiltonian

In case of a time dependent Hamiltonian of the sort $$H=\frac{p^2}{2m}+\frac{1}{2}m \omega(t) x^2$$ I have solved for the time evolution operator using the Schrodinger equation and got $U(t,0)$. If, I ...
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1answer
84 views

A fundamental question about Time-dependent Hamiltonians

I have a fundamental question about Quantum Mechanics or even mechanics in general. I am aware that there are stationary solutions and non-stationary solutions. The stationary solutions solve ...
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2answers
546 views

Does the state of a quantum system change between measurements?

I've recently started studying QM and I was told that the quantum state in which a particle is will change as its wave function evolves. So let us say I have a source that emits one photon at a time. ...
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1answer
172 views

Problem involving Dirac notation and Ehrenfest's theorem

Ehrenfest's theorem in one guise says (omitting hats for vectors) for suitable operators A $$\frac{\mathrm d}{\mathrm dt}\langle A\rangle~=~\bigg\langle\frac{\partial A}{\partial t}\bigg\rangle+\bigg\...