# 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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48 views

### Solution of Time-dependent Schrodinger Equation for Unitary Operator

While reading Quantum Mechanics Book by Sakurai, I found the time-dependent Schrodinger equation for Unitary Operator. $$i\hbar \frac{\partial}{\partial t}\mathcal{U}(t,t_0)=H\mathcal{U}(t,t_0).$$ ...
28 views

### Difference between time series and trajectory terminology

What is the difference between trajectory and time series? To me both seem the same thing. In the 3D diagram (cube picture on left of Fig.2 from the paper titled “Review and comparative evaluation of ...
36 views

### Do all unitary operations manifest from time-evolution?

Let $|\psi\rangle$ be an element of a Hilbert space $\mathcal{H}$ and $U$ a unitary operator on $\mathcal{H}$. I am concerned with the actual physical manifestation of such a unitary operator in the ...
579 views

### Is the evolution operator well-defined mathematically?

We know that in order to solve the time-dependent Schrodinger equation $i\partial_t \psi = H(t) \psi$, we need the evolution operator $$U(t) = T \exp{\left(-i\int_0^t H(t')dt'\right)}$$ where $T$ is ...
92 views

### Does time evolution preserve parity?

Let $\psi(t_0, \cdot)$ be the state of a quantum system corresponding to a Hamiltonian $H$ in the position representation at time $t_0$. Assume $\psi(t_0, -x) = \psi(t_0,x)$, that is $\psi(t_0, \cdot)$...
62 views

### Why don't expectation values for a stationary state evolve over time?

I have an observable $O$ with operator $\hat{O}$. $\Psi_1$ is a wave function in an energy eigenstate, and $\psi_1$ is the corresponding spatial wave function. $E$ is the corresponding energy. It is ...
30 views

### Total hamiltonian is time independent in interaction picture

There is this general statement in Ashok which, if it's true, could someone explain why it is true? Regarding the interaction picture: Since $H_0$ is time independent in the interaction picture (...
40 views

### First order wave function in adiabatic approximation

If the Hamiltonian is slowly varying in time and suppose the initial state is the n-th eigenstate of the initial Hamiltonian H(0), the adiabatic theorem says that the state will still on the n-th ...
24 views

### Time-order evolution operator: Wei-Norman form and unitarity

I'm reading a paper[1] in which the propagator is calculated for this kind of Hamiltonian \begin{align} \hat{H}(t) = \omega(t)\hat{J}_3 + \Omega^*(t)\hat{J}_{+} + \Omega(t)\hat{J}_-. \end{align} ...
110 views

### Dyson Series Iteration - Gives Exact Solution?

When we derive the Dyson series for usage as the time evolution operator in the case of a time dependent Hamiltonian, we start with the equation: \begin{align}\hat{U}_I(t,t_i) = 1 - \frac{i}{\hbar}\...
75 views

### I really wonder about the time derivative of creation and annihilation operators in the derivation of LSZ

In Schwartz book, they assume that $$\lim_{t \to \pm\infty}\partial_0 a_p(t)=0.\tag{1}$$ But I thought that is just assumption. so we have to construct the mathematical description. I found the Gell-...
30 views

### Probability density of time-dependent wave functions

Why is it so that probability density of eigenfunctions of time-dependent schrodinger equation are time independent while that of general wave functions (which are a combination of the eigenfunctions) ...
76 views

### Significance of energy in a time dependent quantum box

The Hamiltonian for a particle in a finite box is $$H = \frac{p^2}{2m} + V(x)$$ which will give time evolution as $$i\hbar d/dt|{\psi(t)}\rangle = H|{\psi(t)}\rangle \, .$$ However, if I do a ...
57 views

### Are superposition and time-evolution of a quantum system unrelated?

Consider a single particle (a single qubit if you will) in some arbitrary state $|\psi\rangle$ and an eigenvector $|\lambda\rangle$ corresponding to the eigenvalue $\lambda.$ Consider the time ...
64 views

### Examples of non-Hermitian Hamiltonians in open systems?

I have often heard the statement that non-Hermitian Hamiltonians can be used to describe open systems, since the dynamics are non-unitary. However, I have not been able to find any examples of a non-...
47 views

### What are the ways to carry out time propagation numerically?

There are many ways to solve the time-dependent Schrodinger equation (TDSE) and find the wavefunction $|\Psi(t)\rangle$ for a given Hamiltonian. For example, consider a tight binding type Hamiltonian: ...
40 views

### Hamiltonian flows and Heisenberg picture of Quantum Mechanics

I am a math bachelor student studying Quantum Mechanics and I was very briefly introduced to the Heisenberg picture. (Hence many of the following may be trivial) In particular what I know is that: ...
90 views

### Energy Interpretation of Quantum Effective Action From Weinberg's “The Quantum Theory of Fields”

In section 16.3 of Weinberg, he attempts to prove that the effective potential energy $V(\phi)$ is equal to the minimum energy density of a state with field expectation value $\phi$. I am confused ...
24 views

### Evolution of the propagator in the Interaction picture?

The evolution operator in the interaction picture is defined as $U_I=e^{iH_0t}e^{-iH_St}e^{-iH_0t}$ Where $H_S=H_0+V$ I am trying to find the evolution of the operator $U_I$. In literature it is ...
62 views

### A question in imaginary time Green's function

I am learning many-body quantum field theory with Bruus and Flensberg's Introduction to Many-body Quantum Theory in Condensed Matter Physics, there is a derivation that confuses me a lot. To add ...
39 views

### When would an open system reach the steady state calculated from master equation?

From the master equation for density matrix, it seems that one can have steady state solution requiring the derivative of density matrix equals to zero, but I want to know whether a real open system ...
27 views

### Any method that can show the time evolution of a open many body system?

the master equation seems is a choice but this method seems only give a mean field result which can not show obviously the effect of specific interaction between particles. So, I am wondering is there ...
33 views

### Generically, why do we want to evolve states with unitary operators? [duplicate]

Why is it so important that operators that evolve states are unitary?
30 views

### What happens to the time evolution equations in canonical quantum gravity?

Many expositions on canonical quantum gravity start from a 3+1 type formalism, where spacetime is foliated along the time dimension. The Einstein equations then decompose into constraint equations on ...
105 views

### What is the mathematical reasoning behind Schrodinger's equation preserving its normalization, with the evolution of time?

I am currently in high-school, currently working on a physics research on the normalization of the Schrodinger's equation. I was quite interested on how we can mathematically explain preservation of ...
701 views

### Do atomic orbitals “pulse” in time?

I understand that atomic orbitals are solutions to the time-independent Schrödinger equation, and that they are are analogous to standing waves ("stationary states"). However, even a standing ...
76 views

### Non-Hermitian Hamiltonian for electron conductance in electric field?

Electron conductance in a solid state is usually driven by electric field - making some direction of jumps more likely. It makes (e.g. Hubbard's) Hamiltonian no longer self-adjoint, how to simulate ...
129 views

### Understanding intuition behind time translation in classical mechanics

In V.Arnold book "Mathematical Methods of Classical Mechanics" he says that invariance with respect to the time for isolated systems means that "the laws of nature remain constant", i.e., if $\phi(t)$ ...
56 views

### Problem with understanding Time Evolution of a Quantum State [closed]

I was given the following task and I'm having some troubles with understanding a few things about it: There is given a system with Orthonormal basis $|u_1 \rangle , |u_2 \rangle, |u_3 \rangle$ ...
42 views

### Wave function of a particle under $V(x)$ (QM)

Suppose I have a particle with mass $m$ and it's under potential of a certain $V(x)$. (NOT an infinite or finite potential well) Also given is the wave function at time $t=0$, $\psi(x,0)$. What is ...
44 views

### How to understand the kernel as a transition amplitude?

Consider the time evolution operator $U(t_f, t_i)$ that controls the evolution of a wave function according to $|\psi(t_f \rangle = U(t_f, t_i) | \psi(t_i) \rangle$. As I understand it, the Born ...
42 views

### Virtual terms in the Dyson series (time dependent perturbation theory)

Let the interaction evolution operator in the interaction picture be $$U_I(t,t_0)=T \exp \Big( -i \int_{t_0}^t dt_1 H_I(t_1) \Big) ,$$ where $T$ is the time order operator and $H_I=H-H_0$ is the ...
42 views

### Discrete time evolution in a non-Euclidean space?

The time independent schrödinger equation can be written as $$i\frac{\partial \psi}{ \partial t}=H\psi$$ if we consider the case of a 1D particle we can evolve it in time by discretising the ...
65 views

### Time-independent and time-dependent perturbation theory yield different results

First, here's the problem statement. Suppose you have an infinite square well of length $a$, where the box extend from $x=0$ to $x=a$. At $t=0$, you add a perturbation $H'$ of the form: \begin{...
82 views

### Is the Process of Projection of a Generic State Onto a Subspace Impossible? [closed]

I can define (in a standard way) the process of projection of a generic state onto the subspace $\mathcal{G}$ as a process that takes a generic state $|\psi\rangle$ of the Hilbert space $\mathcal{H}$ ...
184 views

### Fine Tuning of the Universe

I'm an A level student looking into the fine tuning of various constants. Physicists explain the extensive effects that would happen if these constants were to be changed/different and hence, how this ...
188 views

### Baker-Campbell-Hausdorff (BCH) Formula for the Time Evolution Operator

In following Prof. Toyer's Computational Quantum Physics lecture notes, I came across the following: In computing the Schrödinger equation in real space, one can make a "split operator" Ansatz, for ...