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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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223 views

Integral of Schrodinger equation for a time-dependent Hamiltonian

I am given the following Hamiltonian, $H=H_1=\frac{p^2}{2m}+\frac{1}{2}m\omega_1^2x^2$ for $t<0$ and $H=H_2=\frac{p^2}{2m}+\frac{1}{2}m\omega_2^2x^2$ for $t\geq 0$. Now I want to integrate my ...
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2answers
2k views

The formal solution of the time-dependent Schrödinger equation

Consider the time-dependent Schrödinger equation (or some equation in Schrödinger form) written down as $$ \tag 1 i \partial_{0} \Psi ~=~ \hat{ H}~ \Psi . $$ Usually, one likes to write that it has a ...
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1answer
42 views

State of a system at previous time

If I am given the state of a quantum system at $t=0$ as $| \psi \rangle$ and I know the Hamiltonian $H$ of the system for time $t<0$, how can I write the state of the system at some time $t<0$?
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126 views

Dyson series for Hamiltonian with $c$-number commutator

I am trying to derive the evolution operator for a time dependent Hamiltonian which satisfies the commutator $$[H(t_1), H(t_2)]=I f(t_1,t_2)$$ Where $I$ is the identity operator, and $f(t_1,t_2)$ is ...
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148 views

Time Evolution of Asymptotic Free States in QFT

In equation (4.70) of Peskin, he states that $$_{out}\langle \mathbf{p_1, p_2, \cdots} | \mathbf{k_A,k_B}\rangle_{in} = \lim_{T\rightarrow \infty}\langle \mathbf{p_1, p_2, \cdots} | e^{-iH(2T)} |\...
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1answer
211 views

Quantum Mechanics - time evolution after a measurement?

A non-degenerate two-level system is described by a Hamiltonian $\hat H$ with $\hat H|n\rangle = \epsilon_n|n\rangle$, where $n = 1, 2$. An observable $\hat B$ has eigenvalues $\pm 1$, with the ...
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2answers
74 views

Could the perceived increasing rate of expansion of the universe be explained by time accelerating, rather than space

If speed = distance / time then as time accelerates, speed must also increase. So, when we look at distant objects in the sky, and they appear to be accelerating away from us, could we actually be ...
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1answer
25 views

How to describe time dependence piece of a stationary state with average phase

I am reading this paper on the synchronization of atomic clocks via entanglement (https://arxiv.org/pdf/quant-ph/0004105.pdf) and can't figure out how they are using $\Omega = (E_{1} - E_{o})/\hbar$ ...
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46 views

A shaped pulse as a sum of rectangular pulses

I have a pulse with lineshape $L(ω)=\frac{1}{π}\frac{\frac{1}{2}Γ}{((ω−ω_0)^2+(\frac{1}{2}Γ)^2)}$ in the frequency domain where $\Gamma$ is the pulse width and $\omega_0$ is the resonant frequency ...
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26 views

Evolution of a quantum state due to a pulse

I have some Lorentzian pulse with equation $L(\omega)=\frac{1}{\pi}\frac{\frac{1}{2}\Gamma}{((\omega-\omega_0)^2+(\frac{1}{2}\Gamma)^2)}$ with Fourier transform $F(t)=e^{(-2\pi it\omega_0-\Gamma\...
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1answer
104 views

Momentum space in real computation

I took quantum where I learnt the usage of momentum space. I then understood why momentum space could be so useful in computation. However, what I still did not understood was the mean to define a ...
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56 views

Step explain in Quantum Field Theory

In my QFT homework, I am supposed to justify every step of a derivation. However, I'm confused about this step: $$\langle u_b|\int_{-\infty}^0 dx_2 e^{i(H_0-E)x_2}e^{-\epsilon |x_1+x_2|}H_1 \dots \...
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1answer
125 views

Are there 'closed' solution of Schroedinger equation?

Are there closed solution for the wave function of schroedinger equation? i mean solutions in the form $ \Psi (x,t)= f(x-t,y,z,t) $ that are not given by infinite series. For example for the 1+1D ...
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3answers
317 views

Why does time evolution preserve the norm of a wavefunction?

I saw an awesome derivation of Schrodinger's equation on Wikipedia. Part of it relies on: Since time-evolution must preserve the norm of the wave-function, it follows that $U(t', t)$ must be a ...
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92 views

How to numerically implement a Wick rotation?

I'm solving a Schroedinger-type differential equation using numerical methods (RK4 for precision, explicit Euler to get a rough idea). I have an initial condition to start. I understand that replacing ...
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1answer
53 views

Visualising traveling waves as solutions of $\partial_t ^{2} u - \partial_x ^{2} u=0$

I have seen that there are particular solutions for the wave equation called traveling waves. I have also seen stationary waves, but I would like to understand the physical meaning of the former. ...
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3answers
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Why is the Hamiltonian in QFT the generator of time evolution?

In non-relativistic Quantum Mechanics one can derive that the time translation operator that acts on quantum states is given (in natural units) by \begin{equation} e^{-iHt}, \end{equation} where $H$ ...
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1answer
81 views

Time-dependent Hamiltonian in interaction picture

A spin-$1/2$ particle is subject to an external magnetic field $$\mathbf{B}\left(t\right)=B\left(\mathbf{i}\cos{\omega t}-\mathbf{j}\sin{\omega t}\right) + B_0\mathbf{k} \; \left(B,B_0\in\mathbb{R^...
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110 views

Time evolution (spin-1 particle)

The state ket of a spin-1 particle with orbital angular momentum is given by $$\left|\alpha,0\right\rangle=\frac{f\left(r\right)}{4\sqrt2}\left[\sqrt2\left|1-1\right\rangle_L \left|11\right\...
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Why do excited states decay if they are eigenstates of Hamiltonian and should not change in time?

Quantum mechanics says that if a system is in an eigenstate of the Hamiltonian, then the state ket representing the system will not evolve with time. So if the electron is in, say, the first excited ...
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133 views

Time evolution in tensor product Hilbert space

Having the Hamiltonian $$H=\beta\left(\sigma^{(1)}_1 -\sigma^{(2)}_1 \right)^2\in\mathcal{H}=\mathcal{H}_{(1)}\otimes\mathcal{H}_{(2)},$$ where $$\sigma_1^{(1)}\equiv\sigma_1\otimes 1\!\!1_2, \;\;\; \...
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1answer
141 views

Time evolution of a state

At $t=0$, a spin-$1/2$ particle has the wave function $$\psi=\frac{R(r)}{\sqrt{3}}\big[\sqrt{2} Y_1^{-1}(\theta,\phi) + Y_1^0(\theta,\phi) \big] \chi_{1/2}^{1/2},$$ that, in Dirac notation and using ...
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1answer
342 views

Time evolution with rotation Hamiltonian

At $t=0$, the wave function of a particle with Hamiltonian $$\mathcal{H}=\mu B L_y \equiv \omega L_y$$ is given by $$\left \langle \mathbf{r}|\alpha \right \rangle \equiv \psi\left ( \mathbf{r} \...
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1answer
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What's the relation between path integral and Dyson series?

If one solves the Schrodinger equation $$i\hbar\partial_tU(t,0) = H U(t,0)$$ for time evolution operator $U(t,0)$, one can get the following Dyson series $$U(t,0) = \sum_n(\dfrac{-i}{\hbar})^n\...
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2answers
106 views

If $|\beta \rangle = A(t) |\alpha \rangle$ in Heisenberg picture, then doesn't $|\beta \rangle$ depend on time?

We know that states are time independent in Heisenberg picture. However, if I apply an operator to a state in Heisenberg picture, $$|\beta\rangle= A(t)|\alpha\rangle \equiv \exp(itH) A(0)\exp(−itH)|\...
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Extending the ergodic theorem to non-equilibrium systems

I try to make this as short and concise as possible. For equilibrium systems in statistical mechanics, we have the Liouville's theorem which says that the volume in phase space is conserved when the ...
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0answers
165 views

Another way to calculate the time constant of a system approaching thermal equilibrium

I derived a formula for the time constant $\tau$ by which a toy-system of identical particles having two energy levels $E_1$ and $E_2$ approaches equilibrium. I'd like to ask if this derivation may be ...
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1answer
353 views

Dyson series derivation

$$ i\hbar \frac{dU_I(t, t_i)}{dt} = \hat{V}_I(t)\hat{U}_I(t,t_i) \tag{10.32} $$ The solutions of this equation, with the initial condition $\hat{U}_I(t_i,t_i)$, are given by the integral equation $...
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1answer
71 views

Time evolution operator for system -environment interaction

I am reading a paper https://journals.aps.org/prb/pdf/10.1103/PhysRevB.96.224302. In this paper the initial state of the system and environment is given as \begin{equation} |\Psi(0)\rangle=|\phi_{s}(...
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1answer
366 views

Time evolution coefficients of harmonic oscillator

I want to calculate the function $$\varphi(x,0) = \frac15(3 \Psi_0 + 4 \Psi_1)$$ for a later point in time. I know the formula for the time evolution is $$\Psi(x,t)=\sum_{n=0}^N c_n \Psi_n e^{-i \...
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879 views

How is Schrödinger evolution experimentally verified?

This is surely a pretty ignorant question, but I'm just starting out learning physics. In quantum theory the evolution of a particle's wavefunction is supposed to have two stages: deterministic ...
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2answers
524 views

Commutation relation under time ordering

Consider a quantum system with the following Hamiltonian: $$H(t)=H_0+H_1(t),\tag{1}$$ where $H_0$ is a noninteracting Hamiltonian and $H_1(t)$ a time-dependent perturbation. To formulate the linear ...
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1answer
155 views

Why is time evolution unitary in $PT$-symmetry?

I have a question on the time evolution for a $PT$-symmetric Hamiltonian. So far I have only read that time evolution was unitary because $H$ commutes with $PT$ and the newly constructed operator $C$ ...
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2answers
243 views

Determining Heisenberg equations of motion from a given Hamiltonian

Given the Hamiltonian $$\hat{H} = \frac{1}{2} \hat{p}^2 + \hat{p}\hat{q}^4.\tag{1}$$ I would like to know how do I find the Heisenberg equation for the operators $\hat{p}$ and $\hat{q}$. I know that ...
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1answer
236 views

Connecting the Schrodinger equation, unitarity and norm-preserving of states in time evolution

This may be really basic but I'm having trouble connecting the following issues: 1) The 2-norm for state vectors is preserved during time evolution 2) The Hamiltonian is a Hermitian operator. With ...
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4answers
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Liouville's theorem and the preservation of topology

What might be a simple proof showing that the time evolution of the phase space volume can't lead to splitting off of the phase space volume? By Liouville's theorem, the total phase space volume is ...
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1answer
92 views

Unitarity of Quantum Mechanical Systems

I was reading this lecture notes "Black holes from A to Z" by Andrew Strominger. In the first chapter Introduction the following statement is made: "If we know the present, there are laws that ...
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How is the hamiltonian a hermitian operator?

My book about quantum mechanics states that the hamiltonian, defined as $$H=i\hbar\frac{\partial}{\partial t}$$ is a hermitian operator. But i don't really see how I have to interpret this. First of ...
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2answers
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How is Liouville's theorem important to statistical mechanics?

I have come across Liouville's theorem in the first chapter of many statistical mechanics textbooks, still I don't quite get how it is important to statistical mechanics. How is it related to ...
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2answers
225 views

Ordinary vs. partial derivatives of kets and observables in Dirac formalism

I'm a bit confused as to when ordinary and partial derivatives are used in the Dirac formalism. In the Schrödinger equation, for instance, Griffiths [3.85] uses ordinary derivatives: $$ i \hbar \...
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How is time evolution done in numerical GR?

Suppose we're simulating what happens when a fairly massive object falls into a black hole. Say the object starts far away, so that the initial condition is that the metric is the Schwarzschild metric ...
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1answer
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How to solve Schrodinger equation back in time to find past wavefunction from which present wavefunction has been evolved?

How to solve Schrödinger equation back in time to find past wavefunction from which present wavefunction has been evolved? i.e. Suppose, at present or at this moment I know $\psi_{present}(r)$. ...
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1answer
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What's a reasonable 'translation' of the Schrödinger equation?

For this form of the equation: $$\hat H|\psi(t)\rangle = i \hbar \frac{\partial{}}{\partial{t}}|\psi(t)\rangle.$$ For instance: "The total energy of a quantum state at time t is equal to $i\hbar$...
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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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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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3answers
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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
853 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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3answers
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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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153 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 $\rho (t)$...