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 ...

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What approximation does Tamm-Dancoff approximation (CI singles) correspond to in real time Time-Dependent Density Functional Theory?

Starting from equations of motion for time-dependent density functional theory (in real time) $$ \frac{ {\rm d} \rho_{nn} }{ {\rm d} t} = i \left[ \rho_{nn}^{(1)}, h^{\rm KS} \right] ...
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51 views

Why is the energy operator special?

Only the energy operator controls the time dependence of a quantum system, but not the others, why is that?
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The $T\rightarrow \infty $ limit in quantum field theory

I am new to quantum field theory. Prior to this, I have been using quantum mechanics for a few years. I am reading the book by A. Zee, ''quantum field theory in a nutshell'', 2nd Ed.. On page 18, ...
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47 views

Writing operator evolution as a quantum dynamical map

In the Heisenberg picture we have the evolution of the operator in time given by: $$A(t)=U^+A(0)U$$ I was looking into the theory of open quantum systems where we introduce the concept of a quantum ...
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Magnus Expansion in Floquet theory

I wonder how to obtain the second equality as follows in Eq. (44) of http://www.tandfonline.com/doi/abs/10.1080/00018732.2015.1055918?journalCode=tadp20 \begin{eqnarray} ...
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time evolution of the state $|a'\rangle$ with Hamiltonian $H=|a'\rangle\delta\langle{a}''|+|a''\rangle\delta\langle{a}'|$ [closed]

Reference to Chapter2, Problem8.b, Modern Quantum Mechanics, Sakurai: $|a'\rangle$,$|a''\rangle$: eigenket of the hermitian operator $A$ and the Hamiltonian, $$ ...
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82 views

Expectation Value of Unitary Time Evolution Operator in Quantum Mechanics

Does the expression $\langle \Psi_i|U(t)|\Psi_i\rangle$ have a specific meaning, where $U(T)$ is the unitary time evolution operator of $\Psi$, and $\Psi_i$ is the initial state of $\Psi$? If so, ...
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Time-Evolution of a 3-State System [closed]

The Hamiltonian for a three-state system is, in some basis $|1\rangle ,|2\rangle,|3\rangle$ $$\hat{H}= \left( \begin{array}{ccc} E_0 & 0 & A \\ 0 & E_1 & 0 \\ A & 0 & E_0 ...
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3answers
241 views

Time derivative of a function in Phase Space

Consider a function $\mathcal{H}(q_i,p_i;t)$ such that it obeys the equation: $$ \frac{d\mathcal{H}}{dt}=\frac{\partial\mathcal{H}}{\partial t}$$ What does this equation imply (read: mean), ...
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114 views

Hamilton's Equations

The last step of this derivation of Hamilton's Equations is what's making me doubt it. It is as follows: Assuming the existence of a smooth function $\mathcal{H}(q_i,p_i)$ in $(q_i(t), \,p_i(t))$ ...
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2answers
102 views

Schroedinger Picture more general than Heisenberg Picture?

When thinking about the two pictures, what I found to be strange was: I can write the postulate of time-evolution in the Schroedinger picture by: \begin{align} i \hbar \frac{d}{dt} \lvert \Psi(t) ...
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Relationship between the Lindblad Equation and Redfield Equation

Both the Lindblad and Redfield Equation both model the open quantum system dynamics given a Hamiltonian and some operators. What is the relationship between the two equations? How can they transformed ...
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Wave packets in Dirac equation

Gaussian wave packets remain Gaussians after evolution in case of the Schrodinger equation. It is a very useful property of these wave packets. I don't think the same is true for a Gaussian wave ...
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Quantum transitions between energy states [closed]

When a quantum system is acted on by time dependent perturbation, the initial state evolves according to the new time-dependent Hamiltonian and grows to some superposition of states. During the time ...
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2answers
139 views

Are the eigenstates of an operator time independent?

In the Schrodinger picture, are the eigenstates of an operator time independent? Is it their expectation values that evolve in time rather than the actual eigenstates? For example, say I have an ...
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3answers
127 views

Prove time-dependent hamiltonian is hermitian from unitarity of time-evolution operator

When we solve the Schrodinger equation for the time-evolution operator: \begin{equation} i\hbar\frac{\partial}{\partial t}U(t,t_{0})=HU(t,t_{0}), \end{equation} We have three cases to be treated ...
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1answer
85 views

Confusion with time ordering

I am thinking about Proof of correlation function formula in quantum field theory and have realized there is a deeper confusion underpinning that. Consider: $$T\{U_I(T, t_2)\Phi_I(x_1)\}$$ where ...
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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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How long does it take to a local perturbation to propagate along a quantum system?

Imagine to have a one-dimensional system in its ground state, and to apply a local perturbation at one edge of the system. How does the system evolve after being perturbed? More specifically, how ...
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2answers
221 views

Time dependent and time independent Schrödinger equations

I'm trying to understand the relation between the time dependent and time dependent Schrödinger equations. In particular, we know that the TDSE is $$H\Psi=i\hbar \frac{\partial \Psi}{\partial t}$$ ...
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Picture-independence of quantum mechanics

I've been thinking about the equivalence of the Heisenberg and Schrödinger pictures of quantum mechanics in the following terms lately: a quantum system is a Hilbert space $\mathcal{H}$ equipped with ...
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74 views

Time evolution of wave function in QM

Recently I've been studying quantum dynamics with Sakurai's modern quantum mechanics, but I am confused with why the time evolution operator is written as ...
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1answer
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Transformations of states in quantum mechanics

In Classical Mechanics we usually describe the possible configurations of a system by points on a smooth manifold $M$ which is the configuration manifold of the system. In that case, when we talk ...
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3answers
417 views

Is the superposition of stationary states a stationary state? If not, then why not?

I am a beginner in Quantum mechanics and as I understand,the superposition of stationary states is also a solution of time-independent Schrödinger equation (TISE). The wave functions that are the ...
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1answer
64 views

Could you explain this flow of calculation?

I am reading this book, Quantum Optics by Walls and Milburn. I am working on Chapter 6 which is about the Stochastic Methods. I don't understand a calculation in this chapter. Let $w(t)$ be the ...
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1answer
68 views

What is the time evolution operator in quantum mechanics [duplicate]

I'm curious about what happen to a system when the configuration of the system changes. If we have a system in a state $|\psi_{\textrm{in}}\rangle$ and we change the configuration of the system, the ...
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How to describe time evolution in relativistic QFT?

I must confess that I'm still confused about the question of time evolution in relativistic quantum field theory (RQFT). From symmetry arguments, from the representation of the Poincare group through ...
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1answer
118 views

Time evolution operator acting on a non-eigenket

I'm taking a course in QM at my university, and I'm trying to work out an assignment given to the class by our professor. The setup is as follows: The problem is about a simplified description of ...
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1answer
171 views

How/Why did Feynman relate the element of Hamiltonian matrix $H_{12}$ to the amplitude to go from $|1\rangle$ to $| 2\rangle$?

$$ \newcommand{\bk}[2]{\left\langle #1 | #2 \right\rangle} \newcommand{\ket}[1]{\left| #1 \right\rangle} \newcommand{\bra}[1]{\left\langle #1 \right|} \newcommand{\biik}[3]{\left\langle #1 | #2| ...
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1answer
77 views

Probability amplitude for motion from $x_i$ to $x_f$ in Heisenberg picture

In M. Nakahara's book Geometry, Topology and Physics on page 19, the probability amplitude for a particle to move from $x_i$ at time $t_i$ to $x_f$ at time $t_f$ is given as $$ \tag{1} \langle x_f, ...
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2answers
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Time evolution in Quantum Mechanics abstract state space

As I've learned the first postulate from Quantum Mechanics can be stated as follows: The states of a quantum system are described by vectors in a complex Hilbert space $\mathcal{H}$. The book ...
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Completeness relations of eigenstates in the Heisenberg picture

I've been reading Srednicki's introduction to path integrals and I'm slightly unsure of the notation that he uses for the completeness relation of position eigenstates in the Heisenberg picture. In ...
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63 views

Time evolution of expectation value of an operator

I'm studying QM from Sakurai, and I have a doubt regarding the proof given that in the case of time independent Hamiltonian the expectation value of an observable doesn't change with time. The ...
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2answers
435 views

The curious case of the time derivative of the expectation value of the position

Having defined the expectation value of position as follows $$ \langle x \rangle = \int x {\lvert\Psi(x,t)\rvert}^2dx $$ The time derivative of the expectation value is derived in my literature in ...
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1answer
113 views

What is time evolution operator?

Could you explain to me (level 1 years undergrade) what is a time evolution operator? I read on Wikipedia, and it confuses me.
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Effective theories and unbounded operators

If you have two operators, one the true Hamiltonian $H$ and one we call an effective Hamiltonian $H_{eff}$ and say they agree on every eigenvector with eigenvalue up to $E_{eff}.$ Above that, they can ...
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293 views

Time-ordering and Dyson series

In Dyson series we use a time-ordered exponential by arguing that a Hamiltonian at two different instants of time does not commute. Why is it that so? Can anyone explain with example why should the ...
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0answers
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Evolution of a 'state' in the Heisenberg picture

Suppose that we have a Hamiltonian, $\hat{H}$, and an operator $\hat{A}$ which satisfies the Heisenberg equation$^{[a]}$ $$i \frac{d}{dt} \hat{A} = [\hat{A},\hat{H}].$$ Can we create a 'state' by ...
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How can a solution of the time-independent Schrödinger equation evolve in space?

I understand that if the Hamiltonian does not depend on the time, the Schrödinger Equation becomes separable, so you get $$ H \psi(x) = E \psi(x) $$ and $$ \Psi(x,t) = ...
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Lindblad equation solution

I have been trying to solve a Lindblad Equation and then thought about whether there is a closed form Lindblad Equation solution for most types. Googling hasn't lead me to anything useful. So, is ...
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3answers
245 views

Why does Hamiltonian follow the property $H^*_{ij} = H_{ji} $?

I was reading Feynman's Lectures III's Hamiltonian Matrix. There I found this property of Hamiltonian Matrix: The Hamiltonian has one property that can be deduced right away, namely, that ...
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Why does time evolution operator have the form $U(t) = e^{-itH}$?

Let's denote by $|\psi(t)\rangle$ some wavefunction at time $t$. Then let's define the time evolution operator $U(t_1,t_2)$ through $$ U(t_2,t_1) |\psi(t_1)\rangle = |\psi(t_2)\rangle \tag{1}$$ and ...
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1answer
146 views

How to get Heisenberg equation of motion? [closed]

A system Hamiltonian is given by $$ ...
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1answer
105 views

Time evolution of scalar field

Consider the quantized real scalar field acting on the vacuum state $\vert 0 \rangle $. We can interpret the state $\phi(\textbf{x})\vert 0 \rangle $ (defined in the Schrodinger picture at $t=0$) as a ...
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1answer
77 views

Entropy change in Heisenberg picture

If we stick with Heisenberg picture where density matrix $\rho$ is constant, how do we account for entropy increase? I've read the answer to State collapse in the Heisenberg picture but I don't see ...
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3answers
315 views

Constructing solutions to the time-dependent Schrödinger's equation

The following question is from David Griffiths' Introduction to Quantum Mechanics: Problem 2.13 A particle in the harmonic oscillator potential starts out in the state $$\Psi(x,0) = A[3 ...
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Does measurement change the evolution of wave function?

Basically any measurement is on wave function $|\psi\rangle$ is done by operator $X$ such that $X|\psi\rangle$ results observable $x$ with some probability. But what happens to $|\psi\rangle$? Does ...
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Field expansion in Peskin & Schroeder

Peskin and Schroeder state something which I'm not fully understanding. More specificially I think it's just phrased in a way I'm not understanding. In the Schrodinger picture we can expand the real ...
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1answer
83 views

A trace formula of two noncommutative operators

In many cases of quantum many-body problems, the Hamiltonian $H$ can always be divided into two parts, i.e. $H_0$ and $H'$. In this occasion, one can systemically calculate the partition function ...
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Time evolution of a classical system [closed]

For a harmonic oscillator the Liouville operator is given by $$L = p \partial_q- q \partial_p.$$ Now I have a phase space distribution $f(t,q,p)$ for which it holds (in general) $$f(t+\tau,q,p)= ...