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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51 views

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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66 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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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
453 views

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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1answer
131 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
171 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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Eigenkets in Interaction Picture

Let us consider a system. In Schrodinger picture, its Hamiltonian $H$ is given by $H = H_0 + V(t)$, where $H_0$ is the unperturbed Hamiltonian and $V(t)$ is the time-dependent perturbation. In ...
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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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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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Time evolution of squeezed states

I cannot find anywhere on the web or on some books the esplicit expression for the time evolution of squeezed states (defined as $|\xi\rangle = S(\xi)|0\rangle = e^{\frac{1}{2}(\xi^*a^2-\xi (a^\dagger)...
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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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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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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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Heisenberg Picture with a time-dependent Schrödinger Hamiltonian

So when the Hamiltonian is time-independent, we can define the Heisenberg state vectors by evolving the Schrödinger state vectors back in time: $$ | \psi \rangle_H = \hat{U}^\dagger (t)|\psi(t) \...
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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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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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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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473 views

General solution of states of time dependent Hamiltonian

Given a time dependent Hamiltonian which commutes at different times, we have the time evolution operator given by $$\mathcal{U}(t,0) = \text{exp}\bigg[-\bigg(\frac{i}{h}\bigg)\int_{0}^{t}dt' H(t')\...
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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
359 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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265 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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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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1answer
232 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
504 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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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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1answer
622 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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2answers
138 views

Stationary state as initial condition for a free particle

I am trying to figure out what will happen to my particle, if it is initially in the ground state of an infinite square well, and suddenly becomes free. $$V(x) = 0, -\frac{a}{2} \leq x \leq \frac{a}{...
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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
508 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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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
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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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2answers
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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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2answers
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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 ...
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1answer
135 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
366 views

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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Derivative and logarithm of Dyson series

With \begin{align} {\cal T}\exp\left[-\frac{i}{\hbar}\int_0^tH(t')dt'\right] &= I - \frac{i}{\hbar} \int_{0}^{t} dt' H(t') + \left(-\frac{i}{\hbar}\right)^2 \frac{1}{2} \mathcal{T}\left(\int_{...
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1answer
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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[-\...
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1answer
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Matrix mechanics [closed]

So, I am given a Hamiltonian of a system, represented in the $|e_{i}\rangle$ basis as: $$H=\begin{bmatrix}1&0&0\\0&0&1\\0&1&0\end{bmatrix}$$ where, $|e_{1}\rangle = \begin{...
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2answers
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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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3answers
159 views

Time-dependence in QM

When you have a system governed by some Hamiltonian H that starts in some state $\lvert a \rangle$, the way to calculate its time-dependence is to decompose into a linear combination of eigenstates of ...
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1answer
90 views

The $N$th-order identity of the time-ordered exponential in Quantum Mechanics

In Quantum Mechanics, one often defines the time ordered exponential like e.g. here. Now my question is how the factor of $N!$ arises. I know the simplex volume as the following integral: \begin{...
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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. ...