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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99 views

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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2answers
819 views

How can I prove that the wave function remains normalized as time goes?

Exploiting Schrödinger equation and its conjugate we can show that $$ \dot{\Psi} = \frac{i \hbar}{2m} \nabla^2 \Psi - \frac{i}{\hbar} U \Psi $$ $$ \dot{\Psi}^* = -\frac{i \hbar}{2m} \nabla^2 \Psi^* + \...
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3answers
98 views

Understanding the time evolution of a quantum state

I am trying to understand this equality below, but I can't seem to wrap my head around it. The Hamiltonian is defined as $\vec H=-\frac {\gamma B \hbar}{2}\sigma_x$, which gives the eigenvalues $\pm \...
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Time ordering operator and derivative with respect to time

In the book Quantum field theory and the standard model from Schwartz, it is written on page 87 some results using time ordering operator. We have the following operator: $$ U(t,t_0)=T \exp\biggl(-\...
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1answer
816 views

Matrix representation of hamiltonian [closed]

I am studying spin-copuling and looking at an uncoupled harmonic oscillator Hamiltonian of the form: $$H = \hbar \omega (a^{\dagger}a +1/2) + B \sigma_z.$$ I would like to write this up in a $2 \...
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Why is the momentum-space wavefunction for a free particle not a function of time?

Suppose the initial wave function of a free particle is given by $\psi(x,0)$. Now to find how the wave function evolves with time we generally do the Fourier transform of the wave function at $t=0$. ...
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152 views

Time evolved density matrix

Working with an uncoupled harmonic oscillator Hamiltonian: $H = H_A + H_B$, where $H_A = \hbar \omega (a_+ a_{-} + 1/2 )$ and $H_B = B \sigma_z$. I'm trying to calculate the density matrix $\rho (t)$...
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80 views

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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143 views

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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2answers
158 views

Time dependence of a function of an operator

Suppose I know the time evolution of an operator is given by $\dot{\hat{O}} = \frac{i}{\hbar}[\hat{H}(t), \hat{O}(t)]$. Now I want to look at a function $\hat{f}(\hat{O}$, and I want to know the time ...
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1answer
184 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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423 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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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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100 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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2k views

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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67 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
196 views

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
513 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
134 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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175 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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145 views

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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2answers
160 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
28 views

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
44 views

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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237 views

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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927 views

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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39 views

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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3k views

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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2k views

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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4answers
918 views

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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1answer
359 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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65 views

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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2answers
505 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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66 views

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
377 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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278 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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136 views

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
244 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
541 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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74 views

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
645 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
141 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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123 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
527 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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3answers
647 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
169 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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174 views

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?