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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How to do time evolution of operators in the Heisenberg Picture while staying in the Heisenberg Picture

Consider the time evolution of an operator in the Heisenberg picture: $$\tag{1}i\hbar \frac{d}{d t} \hat{A}_{H}(t) = \left([ \hat{A}_S(t), \hat H_S (t)] + i\hbar \frac{d}{d t} \hat{A}_S(t) ...
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Does Peskin & Schroeder Eq. (4.26), $U(t_1,t_2)U(t_2,t_3) = U(t_1,t_3)$ imply $[H_0,H_{int}] = 0$?

Peskin & Schroeder equation (4.17) define the operator, \begin{equation} U(t,t_{0})~=~e^{i(t-t_{0})H_{0}}e^{-i(t-t_{0})H} \tag{4.17} \end{equation} where $$H~=~H_0+H_{\text{int}}\tag{4.12}$$ is ...
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Why does the density matrix $\rho$ obey a wrong-signed Heisenberg equation of motion?

The density matrix is defined as $$ \rho_\psi ~:=~ \frac{\lvert\psi(t)\rangle \langle \psi(t)\vert}{ \langle \psi(t) |\psi(t)\rangle }$$ in the Schrödinger picture. $\rho_\psi$ is obviously a time ...
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Commuting with time evolution operator implies commuting with Hamiltonian

Consider a quantum system (finite dimensional) has overall Hamiltonian: $H_t = H_0 + w(t)H_c$ with $H_0, H_c$ constant in time and traceless and $w(t)$ a, not too badly behaved, function of time. ...
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The equivalence between Heisenberg and Schroedinger pictures

In quantum mechanics, the two pictures of Schroedinger and Heisenberg are taken as equivalent, where in the former wavefunctions are time variants and operators are not, and in the latter it is the ...
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321 views

How to find the time evolution for two-component spinor? [closed]

Taking the wave function I found here as the initial condition for the Schrodinger equation. I would like to find how the wave function will depend on time, at least for small times, and calculate ...
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Why isn't the Time-Independent Schrödinger Equation an equation of motion?

I thought an equation of motion was something where you are given a Lagrangian and, using the Euler-Lagrange equation, you then find the equations of motion for that system. Same basic idea for the ...
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59 views

How to solve the inverse square law equation of motion

From $$m\boldsymbol{\ddot{r}}=\boldsymbol{\hat{r}} f(r)$$ I can get $$r''-r \theta '^2=-\frac{k}{m r^2}$$ $$2 r' \theta '+r \theta ''=0$$ Now it seems that all the books tells me the method to solve ...
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Why is time evolution unitary

Is the reason why the time evolution operator is unitary based on purely physical arguments, i.e. that the physical processes that an isolated system undergoes shouldn't depend on any particular ...
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Why does $tr \ e^{-\frac{i}{h}\hat{H}t}= \int d^nr \left< \textbf{r}| e^{-\frac{i}{h}\hat{H}t} | \textbf{r} \right>$ hold?

I would like to consider the trace of the time evolution operator $e^{-\frac{i}{\hbar}\hat{H}t}$ Apparently in single-particle quantum mechanics is can be represented as $$ tr \ ...
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Can the chance of finding a particle diminish over time?

Let's assume we have a wave function described by a wave equation and it is a function of space and time $\psi : \mathbb{R}^4 \rightarrow \mathbb{C}$. This function needs to be normalized, so if I ...
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Why $E=mc^2$ formula does not include time?

For $E=mc^2$ formula, if an object of mass $m$ kg goes with speed of light (in theory), it transforms energy according to $E=mc^2$ but if there is no time complexity this will not happen. So I think ...
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About imaginary time evolution method

When I try to work out the ground state of a system, there is a method making use of imaginary time and splitting operators, I wan to know, when the ground state is fine and satisfying for my purpose? ...
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316 views

Who is doing the normalization of wave function in the time evolution of wave function?

In the Schrodinger equation, at any given time $t$ we should jointly add another sub equation, like $$||\psi_t(x)|| = 1$$ where $\psi_t(x) = \Psi(x,t)$, and then try to solve the two equations ...
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Boltzmann equation (Number density)

I'm trying to understand the Boltzmann equations use in the early Universe. The derivation is somewhat tedious, but in the end I end up with: $$a^{-3}\frac{d}{dt}\left(n_1a^3\right) = ...
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Why discrepancies in the Schrödinger equation? [duplicate]

Why is there seemingly two definitions of the Schrödinger equation? \begin{equation} i\hbar\frac{\partial}{\partial t}\Psi=\hat H\Psi. \end{equation} And \begin{equation} i\hbar ...
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115 views

What is the most agreed upon quantum mechanical equation of motion?

On multiple Wikipedia articles, it mentions several quantum mechanical equations of motion, namely those by Schrödinger and Heisenberg. Which one is the most accurate and agreed upon quantum ...
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Why is time evolution of wavefunctions non-trivial?

(Note: This post focuses on a single simple example, however I'm asking about the error in general in my logic). Consider the infinite potential well "particle in a box" system described by ...
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What really generates time evolution?

A fundamental principle of quantum mechanics, as far as I can tell, states that the Hamiltonian generates time evolution. A common result about generators are the following: let $\mathrm T$ be the ...
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Find Equation of Motion given Hamiltonian

So I am given a harmonic oscillator in an electric field. At $t=0$, we are given that the oscillator is in the ground state. The Hamiltonian is: $$H=\hbar \omega[a^{\dagger}a+\frac12+\kappa ...
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137 views

Time evolution of a quantum system

A quantum system has Hamiltonian $H$ with normalised eigenstates $\psi_n$ and corresponding energies $E_n$ ($n = 1,2,3...$). A linear operator $Q$ is defined by its action on these states: $$ ...
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444 views

What exactly does the Hamiltonian operator tell us?

I'm confused about how energy and time are linked. On the one hand, the Hamiltonian seems to describe the time evolution of the system because in the time dependent Schrodinger equation, $$ \hat H ...
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Creation and Annihilation Operators

Let $\widehat{a}^{+}_{i}$ and $\widehat{a}_{i}$ be the usual bosonic creation and annihilation operators. Consider $$\widehat{q}_{i} = \sqrt{\frac{\hbar}{2m_{i}w_{i}}}(\widehat{a}_{i}+ ...
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How do I enforce the no-slip boundary condition in time dependent incompressible pipe flow?

This is a technical problem which must have been solved already. It won't be in beginners textbooks but there should be a solution somewhere. I welcome reading suggestions. Maybe someone with ...
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316 views

How to get the time derivative of an expectation value in quantum mechanics?

The textbook computes the time derivative of an expectation value as follows: $$\frac{d}{dt}\langle Q\rangle=\frac{d}{dt}\langle \Psi|\hat Q\Psi\rangle=\langle \frac{\partial\Psi}{\partial t}|\hat ...
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Does Heisenberg equation of motion imply the Schrodinger equation for evolution operator?

Let us choose to postulate (e.g. considering the analogy of the Hamiltonian being a generator of time evolution in classical mechanics) $$ i\hbar \frac{d\hat{U}}{dt}=\hat{H}\hat{U}\tag{1} $$ where ...
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251 views

Solving the Schrödinger equation where the initial wave function is an energy eigenfunction

I was watching Allan Adams' lecture on energy eigenfunctions, and there's one part (around 43 minutes into the lecture) that confuses me. Suppose we have the initial wave function $\Psi (x,0)$ such ...
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1answer
138 views

Schrodinger basis kets with Time-dependent Hamiltonian

I was reading through the proof of the Adiabatic Theorem (in Sakurai) and I realised I'm not quite sure how Schrodinger Basis kets behave when we have a time-dependent Hamiltonian. I know that with a ...
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100 views

Hamiltonian mechanics and conservation of energy?

Can anyone explain to me Hamiltonian mechanics relation to conservation of energy? I'm not very good at mathematics, and I know it's important into understanding Hamiltonian mechanics. However, can ...
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129 views

Trace operation on dynamic equation: physical meaning?

Suppose we have Heisenberg equation of motion for some observable $A$, $$ i\hbar\frac{dA}{dt}= -[H,A] $$ since the trace of any finite dimensional commutator structure vanish(not something like ...
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What does the term 'equation of motion' refer to?

What does the term equation of motion refer to? If I am asked a question of the form 'What is the equation of motion of this object?', what should I write?
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Time evolution of states - Is total energy constant or not?

Suppose the state of the particle is given as follows: $$ |\psi_{(t)}\rangle = \frac{1}{\sqrt2} \left( e^{-\frac{i\omega t}{2}} |0\rangle + e^{-\frac{3i\omega t}{2}} |1\rangle \right) $$ Where the ...
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Time evolution of quantum states

Time evolution of a quantum state is fully described by a one parameter family of unitary operators. What I can't seem to understand is, given some unitary operator acting on some Hilbert space, can ...
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Is there a known equation for evolution of classical particle probability density?

Suppose we have some very imprecise knowledge of classical particle's coordinates and momentum: what we can only tell is the probability density to find it in some point of phase space. This is ...
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Question about equation 2.27 from Pachos's Introduction to topological quantum computing

http://quince.leeds.ac.uk/~phyjkp/Files/IntroTQC.pdf above is the PDF that is hosted on his website. The equation is on page 22 (pg 30 in the pdf). In chapter 2. It is the second equation of the ...
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The formal solution of the Schrodinger equation

Let's have Schrodinger equation (or some equation in Schrodinger form) $$ \tag 1 i \partial_{0} \Psi ~=~ \hat{H} \Psi . $$ One likes to write that it has formal solution $$ \tag 2 \Psi (t) ~=~ ...
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Time Evolution Operator in Interaction Picture (Harmonic Oscillator with Time Dependent Perturbation)

1. The problem statement, all variables and given/known data Consider a time-dependent harmonic oscillator with Hamiltonian $$\hat{H}(t)=\hat{H}_0+\hat{V}(t)$$ $$\hat{H}_0=\hbar \omega \left( ...
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When should one apply the unitary time evolution operator?

When is it appropriate to use $\hat U$, the unitary time evolution operator? For example, say I had a system in a certain potential that is changed to a different one at time $t = 0$. Would it be ...
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QFT Dyson series: why are we solving the Schrodinger equation?

In quantum field theory, the solution of the time evolution operator of the Schrodinger equation (in the interaction picture) is given by Dyson's series, which is used to calculate the S-matrix. Why ...
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Is it possible to derive Schrodinger equation in this way?

Let's have wave-function $\lvert \psi \rangle$. The full probability is equal to one: $$\langle \Psi\lvert\Psi \rangle = 1.\tag{1}$$ We need to introduce time evolution of $\Psi $; we know it in ...
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Is continuous evolution from one eigenstate of operator $O$ to another $O$-eigenstate possible?

Eigenvectors associated with distinct values of an observable are orthogonal, according to quantum mechanics. Does this entail that a quantum system cannot continuously evolve from one eigenstate ...
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Heisenberg picture of QM as a result of Hamilton formalism

Consider the formula for the total time-derivative of a physical value in Poisson's formalism: $$\tag{1} \frac{dA}{dt} = -\{H, A\}_{P.B.} + \frac{\partial A}{\partial t}, $$ where $\{A, B\}_{P.B.}$ is ...
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State collapse in the Heisenberg picture

I've been studying quantum mechanics and quantum field theory for a few years now and one question continues to bother me. The Schrödinger picture allows for an evolving state, which evolves through ...
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The unitary time-evolution in the interation picture

I'm currently consuming a course on QFT where we need to define the unitary time-evolution to get the time evolution of the wave function in the interaction picture: $\hat{U}(t_1,t_0) = ...
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Quantum Mechanical Operators in the argument of an exponential

In Quantum Optics and Quantum Mechanics, the time evolution operator $$U(t,t_i) = \exp\left[\frac{-i}{\hbar}H(t-t_i)\right]$$ is used quite a lot. Suppose $t_i =0$ for simplicity, and say the ...
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Relationship between a formal vector derivative and time evolution of an operator

I'm an undergraduate in physics, with all the lack of knowledge inherent in that. In two of my classes, my professors introduced two equations which look eerily similar. The first, from general ...
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Equation of motion in quantum system

The commutation relation between H and $L_i$ is $[H,L_i]$. I want to show the equation of motion. Is this right: $ [H,L_i] = -i \frac{dL_i}{dt}?$
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“Inverted” quantum oscillator

I'm trying to understand the problem of the "inverted" oscillator, which has the following Hamiltonian: $$ \hat{H}=\frac{\hat{p}^{2}}{2m}-\frac{k\hat{x}^{2}}{2} $$ Suppose that a particle at the ...
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Expectation value of time-dependent Hamiltonian

I'm trying to solve a problem in QM with a forced quantum oscillator. In this problem I have a quantum oscillator, which is in the ground state initially. At $t=0$, the force $F(t)=F_0 \sin(\Omega t)$ ...
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Time evolution of a reduced density matrix

For a bipartite quantum system evolving under some master equation, is the time derivative of the reduced density matrix equal to the partial trace of the time derivative of the matrix? In other ...