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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 $\psi(t) = e^{-iHt} \psi(t_0)$ ...
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[-\...