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).

Filter by
Sorted by
Tagged with
0
votes
0answers
32 views

Finding the solution of time-dependent Schrodinger equation

I want to solve the following Equation: $$\frac{d G(\tau,\tau')}{d\tau}=H(\tau)G(\tau,\tau')+\delta(\tau-\tau').$$ When $\tau \neq \tau'$, then, if $\tau>\tau'$, $$G(\tau,\tau')=T e^{\int_{\tau'}^{\...
2
votes
2answers
74 views

Time dependence of operators

In Griffiths's Introduction to Quantum Mechanics, while studying the time evolution of the expectation value of position, the author wrote: $$\langle x\rangle=\int_{-\infty}^{+\infty}x|\Psi(x,t)|^2\,...
1
vote
1answer
63 views

Getting the time evolution relationship in QM

Considering the standard evolution for a generic quantum state $\psi(t)$, setting $\hbar=1$ we have: $$| \psi(t) \rangle = U |\psi(0) \rangle \hspace{5em} \text{where}\hspace{1em} U=\exp[-iH(t-t_0)] $...
1
vote
1answer
69 views

Subspace of Hilbert space as manifold for variational state

I have a question on the geometrical description of time-dependent quantum states and variatioanl states. I will outline my problem and ask questions along the way. Assume you have a time-dependent ...
2
votes
1answer
82 views

Dirac image and free fields in QFT

In QFT, when we take the scattering matrix $S$ and work it out to compute amplitudes, it is usually said that $S$ is written in the interaction (Dirac) image such that the fields $\phi_I$ that $S$ ...
1
vote
1answer
61 views

If we know the inital state of a quantum field can we predict its later state?

If we have the wavefunctional $\Phi[\psi]$ which tells us the probability density for finding $\psi$. Let's say we know the exact field state at $t=0;$ $\psi(x,0)$. Can we use the wavefunctional $\Phi[...
1
vote
1answer
22 views

How can we relate the particles motion in $k$-space to $x$-space?

Suppose a particle's time evolution in a 2D $k$-space of first Brillouin zone is as shown in the figure. How can we interpret the motion of the particle in $x$-space? Any hint for interpretation is ...
2
votes
0answers
36 views

Meaning of time evolution of Floquet matrix

Consider the time-periodical Hamiltonian $H(t)=H(t+T)$. In the Floquet theorem, the Schrödinger equation has a solution of the form \begin{align} |\psi_\alpha(t)\rangle=e^{-i\epsilon_\alpha t}|\phi_\...
2
votes
2answers
45 views

How do we perform the time derivative of the perturbation series for the time-evolution operator?

The following image is from Greiner's book, Field Quantisation, where he carried out the derivative in question. The only way I could make sense of it, was that the derivative acts only on the last ...
1
vote
1answer
54 views

Finding the probability of measuring a particular eigenvalue of an operator for a system after time evolution

Consider a quantum system with Hamiltonian H and consider the measurement of an observable $a_n$ associated with a different operator A. Initially the system is an eigenstate $|\phi_n \rangle$ with ...
0
votes
2answers
33 views

Assumption that time evolution operator first order in $dt$

I'm having some trouble with this part of Sakurai's derivation of the time evolution operator in QM (page 70): Because of continuity, the infinitesimal time-evolution operator must reduce to the ...
3
votes
0answers
53 views

Time dependence of operators in QM

I have a question. I have seen Heisenberg's equation of motion for observables: $$\frac{dA}{dt}=\frac{1}{i\hbar}[A,H]+\frac{\partial A}{\partial t}.$$ However if I want to calculate for example the ...
2
votes
1answer
71 views

Unitary evolution of quantum states

In QM the unitary operator is everywhere and alway given $\hat U = e^{- i \hat H t/\hbar}$ so that a system evolves unitarily so that $\hat U |\psi(t_0) \rangle = |\psi(t) \rangle$ at another time $t$....
6
votes
2answers
131 views

“in” and “out” states in Weinberg's QFT

In Weinberg's QFT Page 109, he defines the "in" and "out" states as the 'in' and 'out' states* $\Psi_{\alpha}^{+}$ and $\Psi_{x}^{-}$ will be found to contain the particles ...
1
vote
2answers
36 views

Time derivative of path ordered monodromy matrix

I am currently gettting familiar with integrably systems and came the following statement in my literature: $U=U(x,\lambda,t)$ some matrix (Lax component) we define $$T(\lambda,t) = \mathcal{P} \exp \...
2
votes
1answer
46 views

Time ordering in correlation function in QFT

In Peskin and Shroeder, "An Introduction to Quantum Field theory", chapter 4, the author derives the 2 point correlation function: $$\langle \Omega|P{\phi(x)\phi(y)}|\Omega \rangle = \lim_{T\...
5
votes
3answers
144 views

How to understand the quantum adiabatic theorem intuitively?

The quantum adiabatic theorem states that: A parametric system remains in its instantaneous eigenstate (with a phase difference) if one of the parameters of the Hamiltonian changes slow enough. This ...
1
vote
1answer
40 views

How do you transform a time evolution function to work with any Cauchy surface?

If you have a time evolution function $K_t(\phi,\phi')$ which gives you the amplitude to go from a field state $\phi$ to a field state $\phi'$ in time $t$ this gives you all the information you ...
2
votes
0answers
21 views

How to show time evolution operator obeys causality?

If we are given a time evolution function $K_t(\phi,\phi')$ which give the amplitude for a field starting in confiruation $\phi$ to go to configuration $\phi'$ after time t. What is the condition that ...
4
votes
3answers
1k views

What is the Schrödinger equation used for exactly?

The Schrödinger equation is just another way of writing the conservation of energy, right? So how can you use it to find the quantum wavefunction? I mean in every example I've seen the wavefunction is ...
0
votes
0answers
3 views

In a Time series of a magnetic field (in RTN coordinates) what is the correlation in time and space of fluctuations?

My question is in a time series of a magnetic field (in RTN coordinates) what is the correlation time and also correlation in space of fluctuations? For example here:
5
votes
2answers
59 views

Time ordering operator if commutator is $c$-number function

I have a question concerning the time ordering operator. Let's suppose we have a time evolution generated by some Hamiltonian $H(t)$ given by $$ U(t)=T_\leftarrow\exp\left(-\mathrm{i}\int_0^t\mathrm{d}...
0
votes
1answer
54 views

What does it mean to say that the operators evolve in time in the Heisenberg picture?

I get that in the Schrödinger picture the wave function evolve in time and the quantum operators are independent of time. However, in the Heisenberg picture the operators evolve in time and the wave ...
6
votes
3answers
313 views

Paths in phase space can never intersect, but why can't they merge?

Page 272 of No-Nonsense Classical Mechanics sketches why paths in phase space can never intersect: Problem: It seems to me this reasoning only implies that paths can never "strictly" ...
0
votes
0answers
70 views

Time-evolution operator written through a commutator [duplicate]

I found this expression for the time-evolution operator: $$\begin{split} U(t) & = T_{\leftarrow}\exp\left[-i\int_0^t ds H(s)\right] \\ &= \exp\left[-\frac{1}{2}\int_0^t ds\int_0^t ds' [H(s),H(...
2
votes
1answer
24 views

Finite temperature greens function in grand canonical ensemble

I see this question was asked several times before but I don't think any answer can explain the issue perfectly. I am studying many body theory and encounters finite temperature Green's function. At ...
0
votes
1answer
41 views

Adjoint of the time-evolution operator

The time-evolution operator $\hat U$ is defined so that $\Psi(x,t)=\hat U(t)\Psi(x,0)$. In terms of the Hamiltonian, it is expressed as $\hat{U}(t)=\exp \left(-\frac{i t}{\hbar} \hat{H}\right)$. I'm ...
-1
votes
1answer
65 views

Expression of the time evolution operator $\hat U(t)$ in terms of the hamiltonian [closed]

I'm trying to derive the expression of the time evolution operator, $\hat U$, in terms of the Hamiltonian of a system, $\hat H$. This operator $\hat U$ is defined so that $\Psi(x,t)=\hat U(t)\Psi(x,0)$...
2
votes
1answer
33 views

Bose-Hubbard dimension arbitrary state

Suppose we have the Bose-Hubbard model with $N=2$ particles and $M=4$ sites. We can construct the Hamiltonian in the Fock basis $|u\rangle =|n_1,n_1,...,n_M\rangle$, where $n_i$ is number of particles ...
1
vote
1answer
105 views

Time Evolution of Wigner Function

The Wigner Function is defined as: $$W(x,p,t)=\frac{1}{2\pi\hbar}\int dy \rho(x+y/2, x-y/2, t)e^{-ipy/\hbar}\tag{1}$$ Where $\rho(x, y, t)=\langle x|\hat{\rho}|y\rangle$. I am supposed to find the ...
1
vote
1answer
36 views

Propagator in normal modes

I started with the Hamiltonian of coupled oscillators in a circular lattice(with $m=\hbar=1$ and $x_{a+N}=x_{a}$) $$H=\frac{1}{2}\sum_{a=0}^{N-1}\left[p_a^2+\omega^2 x_a^2+\Omega^2\left(x_a-a_{a+1}\...
2
votes
0answers
52 views

Doubt regarding interacting and free field in Schwartz

Pg no 85. Schwartz has mentioned that "The interaction picture fields are just what we had been calling (and will continue to call) the free fields". Later when he calculates the vacuum ...
0
votes
1answer
79 views

In non-relativistic Quantum Physics are there equations other than the Schrödinger equations that can be used to model wave functions?

The general form for the time-dependent Schrödinger equations is $$i\hbar\frac{\partial\Psi}{\partial{t}}=\hat{H}\Psi$$ with $\hat{H}$ being the Hamiltonian operator, which as I understand it ...
-2
votes
2answers
51 views

Probability of measuring eigen-energies?

I am trying to make sense of the underlined notes above. I don't understand how did the term $$\Large e^{-2i\frac{E_k t}{\hbar}}$$ got cancelled out? I understand the wave k function times its complex ...
14
votes
12answers
3k views

Is there an equivalent of computation of physical processes in nature? [closed]

I was watching a waterfall in the Austrian Alps. There were thousands of water droplets falling down, splattering on the stones below. I thought - how does nature find out so quickly where each ...
3
votes
1answer
53 views

Question about quantum mechanics formalism

Are degenerate states stationary in quantum mechanics? I know that stationary state means single wavefunction corresponding to single energy. But, what will be in the case of several wavefunctions ...
0
votes
1answer
51 views

Temperature evolution of a Heated Sphere (high Biot number)

A homogeneous sphere is uniformly heated (or cooled) while the boundary is kept at constant temperature. How does its temperature evolve in time and how is it distributed spatially?
0
votes
2answers
74 views

What is the analog of the equation $\frac{d^2\vec{r}(t)}{dt^2}=\frac{\vec{r}(t)}{\left|\vec{r}(t)\right|}f(\vec{r}(t))$ in Quantum Mechanics?

In Newtonian Physics the general equation for the acceleration when there is a central force is $$\frac{d^2\vec{r}(t)}{dt^2}=\frac{1}{m}\frac{\vec{r}(t)}{\left|\vec{r}(t)\right|}f(\vec{r}(t))$$ with $...
2
votes
1answer
45 views

Construction of propagator for time-dependent hamiltonian

In deriving a general propagator to the time-dependent ($H = H(t)$) Hamiltonian problem, Shankar works to first order in $\Delta = T/N$ (a small time interval for large $N$) and argues that by ...
0
votes
2answers
101 views

Time dependence of ladder operators in QFT

I'm currently going through Matthew D. Schwartz book Quantum Field Theory and the Standard Model, p. 23. For free (non interacting) field theories we are able to quantise the field by expanding our ...
0
votes
1answer
42 views

Example of analytic time evolution with a Pauli Hamiltonian

I'm looking for any (non-trivial*) time-independent Hamiltonian expressed in the Pauli basis (with analytically known real coefficients), which unitary time evolves some analytic initial state to some ...
1
vote
1answer
44 views

Generator of Time Shift in Classical and Quantum Mechanics

The time evolution of a point in phase space in classical mechanics can be described as \begin{equation}\label{eq:TmeShift} ( q_i(t + \Delta t),p_i(t + \Delta t) ) = \left( 1 - i\Delta t \hat{L}\...
2
votes
2answers
108 views

Do Maxwell's equations contain any information on the time evolution of the current density $J$?

The answers to Can the Lorentz force expression be derived from Maxwell's equations? make clear that Maxwell's equations contain only information on the evolution of the fields, and not their effects ...
3
votes
3answers
441 views

What's considered a small time step?

I was looking at the following identity that's often used in time evolution: $$ (e^{xA/n}e^{xB/n})^n \approx (e^{x(A+B)/n})^n$$ This holds when $(\frac{1}{2}(x/n)^2[A,B])^n$ is small. I'm wondering ...
4
votes
2answers
94 views

Time evolution of the operators vs. the expectation values

The time evolution of a quantum mechanical operator $A$ (without explicit time dependence) is given by the Heisenberg equation $$ \frac{d}{dt}A = \frac{i}{\hbar} \left[H,A\right] \tag{1}$$ where $H$ ...
0
votes
1answer
44 views

Is the density of an object time dependent?

If an untouched (an object whose wavefunction has never collapsed) charged mass has a probability cloud of radius $3*10^8 m + \Delta x$ after one second of observing, shouldn't the mass associated ...
3
votes
3answers
67 views

Commuting the time evolution operator [closed]

Given the time evolution operator $U(t, t_0)$, I don't understand why it is true that for a time-independent operator Q, $$[Q, U(t, t_{0})] = 0 \Leftrightarrow [Q, H(t)] = 0 $$ where H is the ...
1
vote
2answers
59 views

Super Adiabatic Evolution numerically

I'm doing some adiabatic and super adiabatic evolution using python but without success, the super adiabatic to be more precise. The problem is the way I've manage to write the super adiabatic ...
1
vote
1answer
41 views

Repeatedly measuring the same observable on the same system

Suppose we have got some operator $A$ which does not commute with the Hamiltonian, for example $\hat{p}_{x}$ and $\hat{H}$, where $\hat{p}_{x}$ does not commute with the $\hat{H}$ and we make the ...
2
votes
3answers
298 views

Time evolution of the standard deviation of an operator

How would I find the time evolution of the standard deviation of an operator? For example, how might I find the time evolution $\sigma_x (t)$ of the standard deviation $\sigma_x = \sqrt{ \langle \hat{...

1
2 3 4 5
10