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 ...

learn more… | top users | synonyms

1
vote
1answer
60 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, ...
-1
votes
0answers
49 views

Transforming operators and minus signs

If I have an operator $A_H$ in the Heisenberg picture, then it obeys the equation $-i \frac{\partial}{\partial t}A_H=[H,A_H]$. However, if I plug in the expression $H=i\frac{\partial}{\partial t}$ I ...
1
vote
3answers
211 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), ...
0
votes
1answer
110 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 $(q_i(t), \,p_i(t))$ ...
0
votes
2answers
78 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) ...
0
votes
1answer
31 views

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 ...
1
vote
0answers
38 views

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 ...
1
vote
0answers
49 views

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 ...
1
vote
2answers
129 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 ...
1
vote
3answers
88 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 ...
0
votes
1answer
71 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 ...
6
votes
2answers
301 views

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 ...
4
votes
0answers
56 views

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 ...
0
votes
2answers
98 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 $$H\Psi=i\hbar \frac{\partial \Psi}{\partial t}$$ ...
3
votes
0answers
77 views

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 ...
0
votes
0answers
31 views

Constants of motion of a Hamiltonian matrix

Given a Hamiltonian $H$ on $\mathbb{C}^n$ represented by some $n \times n$ matrix, I would like to find all constants of motion in the Heisenberg's picture. I know that in principle the Heisenberg ...
0
votes
1answer
65 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 ...
0
votes
1answer
63 views

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 ...
0
votes
3answers
242 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 ...
0
votes
1answer
63 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 ...
0
votes
1answer
60 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 ...
8
votes
2answers
281 views

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 ...
1
vote
1answer
109 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 ...
1
vote
1answer
129 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| ...
2
votes
1answer
74 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, ...
2
votes
2answers
115 views

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 ...
1
vote
1answer
109 views

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 ...
0
votes
1answer
50 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 ...
4
votes
2answers
339 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 ...
0
votes
1answer
80 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.
3
votes
1answer
62 views

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 ...
0
votes
2answers
232 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 ...
0
votes
0answers
66 views

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 ...
0
votes
2answers
86 views

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) = ...
1
vote
0answers
73 views

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 ...
2
votes
3answers
208 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 ...
5
votes
6answers
418 views

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 ...
1
vote
1answer
136 views

How to get Heisenberg equation of motion? [closed]

A system Hamiltonian is given by $$ ...
1
vote
1answer
95 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 ...
0
votes
1answer
62 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 ...
3
votes
3answers
284 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 ...
0
votes
5answers
161 views

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 ...
1
vote
1answer
126 views

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 ...
1
vote
1answer
80 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 ...
3
votes
0answers
51 views

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)= ...
3
votes
2answers
140 views

Time evolution of a wavepacket

I do not understand why if $H\psi = E\psi$, then the time-evolution of the wavefunction is given by $e^{-iEt/h}\psi(x)$.
0
votes
0answers
42 views

Does the time Evolution operator commute with the both free and interaction Hamiltonian?

Consider a quantum system (finite dimensional) which has overall Hamiltonian: $H_s = H_0 + w(s)H_c$ with $H_0, H_c$ constant in time and traceless and $w(t)$ a, not too badly behaved, function of ...
1
vote
1answer
114 views

Kraus operators + path integrals = Lindblad equation?

The other day our professor was talking about Kraus representation of density operator and the derivation of Lindblad equation. He told that this was related to the Feynman path integrals and that we ...
3
votes
1answer
113 views

Time Evolution of Position Operator

I am trying to understand why $$e^{-it\triangle}xe^{it\triangle}=x-2it\nabla$$ where $x$ is just multiplication operator by $x$. In particular, the text says this can be seen by differentiating with ...
0
votes
0answers
77 views

Time evolution of interaction Hamiltonian in the Heisenberg picture

How does the interaction Hamiltonian of a (finite dim) quantum system with Hamiltonian: $H(t) = H_0 + w(t) H_I$ evolve in time in the Heisenberg picture. Is there anything special about the way ...