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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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Time evolution operator of Klein-Gordon field

If $U(t)=e^{itH}$ is the time evolution operator. And $|\phi \rangle$ is a state of a field at particular time $t_1$ and $|\phi' \rangle$ is the state of a (free) Klein Gordon field at a time $t_2$. ...
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Origin of the Schrodinger equation by L. D. Landau and E. M. Lifshitz

In the book "Quantum Mechanics" by L. D. Landau and E. M. Lifshitz, it is mentioned that, "The wave function Ψ completely determines the state of a physical system in quantum mechanics. This means ...
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1answer
188 views

Time Evolution of Velocity Operator

In the Heisenberg picture, I can define the velocity Operator $\hat{V}$ as the operator which satisfies $\hat{V}(t) = \frac{\partial \hat{x}}{\partial t}(t)$ for all $t$. The Heisenberg equation then ...
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437 views

Is it possible to study a time-dependent Hamiltonian in Schrödinger picture?

Operators in Heisenberg picture are time-dependent while those in Schrödinger picture are time-independent, and they are related by $$A_H(t)=U^\dagger(t,t_0)A_S(t_0)U(t,t_0)$$ where $U(t,t_0)$ is the ...
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Time evolution of an eigenstate and probabilities

Suppose we have some Hamiltonian $H$ with at least one normalized eigenstate $v$ with real eigenvalue $\lambda$. The time evolution operator is given by $$ U(t,0) = e^{- i \frac{H}{\hbar}t} \ .$$ Now ...
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1answer
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A question in Dirac article about Dirac equation about a sentence

Why it is said $W$ should be linear partial time derivative so that wave function could be determined by initial wave function?
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103 views

Quantum Mechanics: Pictures description

This is a question related to the Schrodinger and Heisenberg picture. Consider a physical system. There are two states- initial and final. Now this is the explanation from Schrodinger and Heisenberg ...
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Time-evolution with a time-dependent Hamiltonian [closed]

Consider a quantum mechanical system whose Hilbert space of states is $\mathbb{C}^2$, and has Hamiltonian $$\hat{H}= \begin{pmatrix} E_0e^{t/w_0} & E_1 \\ E_1 & E_0e^{t/w_0} \end{pmatrix}$$ ...
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1answer
68 views

In quantum mechanics, what is the probability of $B$ given that $A$ has happened?

I get that the probability of event $A$ is $\langle A|A\rangle$. Given that $A$ has happened what is the probability of B happening at $t$ seconds in the future? My first guess is: $$ P(A|B) \...
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What's the evolution of a state under an adiabatic evolution? [duplicate]

For an initial state $|\Psi\rangle_0$ as the ground state of a Hamiltonian $H(0)$, if it undergoes an adiabatic evolution $H(t)$ to reach the ground state $|\Psi\rangle_1$ of $H(1)$. Then what's the ...
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Easy question on math of diffusion equation

I have the following well-known diffusion equation: $$\frac{\partial{\sigma}}{\partial t}=D\nabla^2\sigma$$ where $\sigma$ is the hydrostatic stress. I also know the relationship between stress and ...
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Does $U_S(t_2,t_1)$ have any meaning in the Heisenberg or interaction pictures?

In the Schrödinger picture of QM, the time-evolution operator $\hat U_S(t_2,t_1)=e^{-i \hat H_S(t_2-t_1)}$ has the following action: \[\newcommand{\p}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\...
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Why are we using the interaction picture?

I know the interaction picture states and operators: \begin{align} \lvert\psi_I(t)\rangle &=e^{i\hat{H}_0t}\lvert\psi_S(t)\rangle,\\ \hat{O}_I(t) &=e^{i\hat{H}_0t}\hat{O}_Se^{-i\hat{H}_0t},\\ \...
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Time Evolution Operator in Interaction Picture (Harmonic Oscillator with Time Dependent Perturbation)

1. The problem statement, all variables and given/known data Consider a time-dependent harmonic oscillator with Hamiltonian $$\hat{H}(t)=\hat{H}_0+\hat{V}(t)$$ $$\hat{H}_0=\hbar \omega \left( \...
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1answer
136 views

Non-autonomous Hamiltonian flow in phase space is volume preserving

How does one prove that for a system whose Hamiltonian is dependent explicitly on time ($H (q,p,t)$), the volume of an element in phase space is conserved i.e. $\frac{d V}{dt} = 0$ ? In what follows ...
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1answer
179 views

Time evolution operator for Hamiltonien with scalar commutator at different times

Let $H(t)$ be a time-dependent Hamilton-operator and assume that $[H(t),H(t')] = f(t,t')\, \mathrm{id}_\mathcal{H}$. Is there a closed formula for its time-evolution operator? I tried deducing an ...
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Who is doing the normalization of wave function in the time evolution of wave function?

In the Schrödinger equation, at any given time $t$ we should jointly add another sub equation, like $$||\psi_t(x)|| = 1$$ where $\psi_t(x) = \Psi(x,t)$, and then try to solve the two equations ...
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Time-ordering and Dyson series

In Dyson series we use a time-ordered exponential by arguing that Hamiltonians at two different instants of time do not commute. Why is that so? Can anyone explain with an example why should the same ...
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2answers
166 views

Is it possible to use Noether's theorem to prove that Hamiltonian is the time invariance in quantum mechanics?

On page 46 of Sarkurai's Modern QM, he defined momentum as the generator of infinitesimal translation of a QM system. Later with similar methods, he defined the generator of time evolution of QM ...
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1answer
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Why do galaxies over time become more refined? [duplicate]

Why do galaxies over time become more refined, ordered and defined instead of more random and disordered?
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1answer
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An identity of time-ordered operators that intertwines between the Schrödinger picture and the interaction picture

Let $V(t)$ and $H_0$ be two operators where $V(t)$ has explicit time dependence while $H_0$ is time independent. I am trying to prove the interesting identity, $$T(e^{-i\int_{t_{0}}^{t} dt' (H_0+V(t'...
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1answer
385 views

Is it obvious that the Hamiltonian observable in Quantum Mechanics should also be the Energy observable?

In Quantum Mechanics, the Hamiltonian observable is defined as the generator of time translations. It's easy to show that if we take this to be the definition of the Hamiltonian, then it is of the ...
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3answers
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Schrödinger equation for time dependent Hamiltonian and conjugation

The Schrödinger equation for the evolution operator reads: $$ \frac{\partial U}{\partial t} = -\frac{i}{\hbar}HU $$ where for a time dependent Hamiltonian which need not commute with itself at ...
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What exactly does the Hamiltonian operator tell us?

I'm confused about how energy and time are linked. On the one hand, the Hamiltonian seems to describe the time evolution of the system because in the time dependent Schrodinger equation, $$ \hat H \...
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Time-evolution of localised particle [duplicate]

I am interested in the question of, if a particle is initially localised at some position $x_0$ what it will evolve to at a later time assuming a free Hamiltonian $H = p^2 /2m$. Long story short, I ...
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Heisenberg equation with time-dependent Hamiltonian

It is the root of quantum mechanics that Heisenberg picture and Schrödinger picture are equivalent? In most textbooks and wikipedia, the equivalence is proved with a time-independent Hamiltonian. ...
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3k views

Heisenberg equation of motion

In the Heisenberg picture (using natural dimensions): $$ O_H = e^{iHt}O_se^{-iHt}. \tag{1} $$ If the Hamiltonian is independent of time then we can take a partial derivative of both sides with respect ...
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Why is time-evolution operator unitary?

When we shift the system's time from $t=0$ to $t = t$, we can define the following operator $\hat{U}$. $$\hat{U} = e^{- i \hat{H} t / \hbar} \, .\tag{1}$$ So many (as far as I read, almost all of) ...
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Which derivative with respect to time is which in the Heisenberg picture of quantum mechanics?

For an observable $A$ and a Hamiltonian $H$, Wikipedia gives the time evolution equation for $A(t) = e^{iHt/\hbar} A e^{-iHt/\hbar}$ in the Heisenberg picture as $$\frac{d}{dt} A(t) = \frac{i}{\hbar} ...
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4answers
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Why do we search for stationary solutions to the Schrodinger equation for potential wells?

When considering potential wells textbooks simply say that we search for the stationary solutions of the schrodinger equation. Why do we do this? What tells us that the wavefunction will be ...
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Hamiltonian covariant time translation

I am working on vector fields in curved manifolds and arrive at the following question: Why is it that we demand the Hamiltonian to generate time translations: $$[i\mathcal{H}, A_\mu] = \partial_t ...
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Characteristic time for changes in the Hamiltonian

Just a short query, given an electron at rest at the origin in the presence of a magnetic field whose magnitude is constant but whose direction is rotating around a cone at constant angular velocity $...
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Exactly solvable time-dependent Schroedinger equation [closed]

We know there are some (many actually) exactly solvable models, like the Hydrogen atom, the harmonic oscillator, etc. But these models are solvable often only in the sense that the eigenstates or ...
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1answer
390 views

Convert time operator from momentum space to position space

I'm trying to transform the time evolution operator from momentum space to position space. I know that $$ U(t) = e^{-iHt/h} = \int_{-\infty}^\infty e^{-ip^2t/2uh} | p \rangle \langle p | dp $$ and ...
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281 views

Beta Decay in Time Dependent Perturbation Theory

I'm trying to find the probability of an electron jumping from the 1s to the 2s state due to Beta decay, where $Z\rightarrow Z\pm1$. My idea is that $H' = -\frac{1}{4\pi\epsilon_0}\frac{(Z\pm1)e^2}{r}...
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1answer
255 views

Coherent state evolution for a given non usual Hamiltonian

I am trying to compute the temporal evolution of a coherent state $|\alpha\rangle$ using a given hamiltonian of the form:$$\hat{H}=\hbar \omega(\hat{a}^{\dagger}\hat{a}-\alpha(\hat{a}+\hat{a}^{\dagger}...
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2answers
597 views

How do I find the time evolution of a ket?

I have a question which reads: Let \begin{bmatrix} {E_0} & 0 & A \\ 0 & E_1 & 0 \\ A & 0 & E_0 \end{bmatrix} be the matrix representation of ...
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1answer
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Evolution operator for time-dependent Hamiltonian

When I studied QM I'm only working with time independent Hamiltonians. In this case the unitary evolution operator has the form $$\hat{U}=e^{-\frac{i}{\hbar}Ht}$$ that follows from this equation $$ i\...
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1answer
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Equivalent representations of stationary states in Quantum Mechanics

The time-dependent Schrodinger equation is given as $$i \hbar \frac{\mathrm d}{\mathrm dt}| \psi(t) \rangle = \hat{H} | \psi(t) \rangle. $$ To find how the states evolve in time we want to find the ...
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661 views

Time evolution of operators, derivation

I don't understand something about the Heisenberg and interaction picture, in my notes the time evolution of operators for the Heisenberg and interaction picture is derived, by inserting them into the ...
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0answers
126 views

Operators in Heisenberg picture [duplicate]

An operator $\hat{Q}(t)$ can be written as $\hat{Q}(t)= e^{iHt} \hat{Q(0)} e^{-iHt}$ in Heisenberg picture. Let us choose $\hat{Q(0)} = |{m(0)}\rangle \langle{m(0)}|$. Then $e^{iHt} \hat{Q(0)} e^{-...
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1answer
541 views

Commutator of spin and linear momentum

More specifically, what is $[S_z, p^2]$? This came up in a time-evolution problem for $\hat{S}_z(t)$, knowing that that it commutes with the non-kinetic part of some Hamiltonian $\leftrightarrow [S_z, ...
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670 views

What is a “picture” in quantum mechanics?

One of the basic ingredients of quantum mechanics is the possibility of working in different "pictures". Thus, while we normally work in the Schrödinger picture, in which states evolve according to ...
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1answer
178 views

Why are we using Heisenberg equation of motion for non-observable $a$ and $a^{\dagger}$?

The author in one of my textbooks derived the time dependency of $a(t)$ and $a^{\dagger}(t)$ through the equation of motion. Is that allowed?
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How does time evolution go between degenerate states?

What's the time evolution process between two different degenerate states? Is it also described by Schrodinger equation?
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233 views

Liouville's Theorem. True or False?

In my quantum theory course, there is a question ask for checking whether the expectations in quantum and classical Liouville theory are identical. Here is the original version: "Assume the system ...
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2answers
50 views

What is the correct *first* interpretation of the time derivative of some measurable quantity?

For example, take the position function $x(t)$. When I take $(d/dt)(x(t))$, I know that I must ultimately conclude that the result is the velocity function $v(t)$. But this feels like a ''jump ahead'' ...
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2answers
66 views

Do we suspect that any 2 seemingly identical experiments actually change under the passage of time?

For example, let's say that I set up 2 consecutive identical experiments where I know that the conditions are exactly the same (go through whatever difficulties you need to). The only thing I can't ...
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1answer
141 views

Dirac equation, why not unitary, why not single-particle formalism?

I am reading the first chapter of Akhiezer, Berestetskii QED (1981). They state that Dirac was wrong to assume that the evolution of the wave function is described by $\psi(t) = e^{-iHt} \psi(t_0)$ ...
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1answer
172 views

Product of two Dyson series

Is ${\cal T}\exp\left[\frac{i}{\hbar}\int_0^tH(t')dt'\right]{\cal T}\exp\left[-\frac{i}{\hbar}\int_0^tH(t')dt'\right]$ equal to 1? I do not think so. I know that \begin{align} {\cal T}\exp\left[-\...