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

3
votes
1answer
67 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 ...
2
votes
3answers
359 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 ...
0
votes
0answers
37 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 ...
1
vote
1answer
68 views

Why is probabilty conserved under time evolution of a system in quantum mechanics?

I've studied quantum mechanics to a certain degree, but one question that I've never been able to get a fully satisfactory answer to is why probability is conserved (by this I mean that it has either ...
1
vote
1answer
347 views

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

I would like to find the time evolution for the given Hamiltonian, the initial state of the system we choose two spinor wavefunction $\psi_{+}(t=0)$ and $\psi_{-}(t=0)$ as given below: The effective ...
0
votes
0answers
50 views

Unitary evolution operator

Assume we have a system in a state $\psi$ that is a superposition of eigenvectors of some observable $A$. Now we make a measurement of the observable $A$; the state after the measurement $\phi$ is a ...
5
votes
1answer
135 views

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 ...
17
votes
4answers
610 views

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 ...
0
votes
2answers
117 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 ...
0
votes
1answer
101 views

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}?$
2
votes
0answers
54 views

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) ...
5
votes
2answers
445 views

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 ...
7
votes
4answers
1k views

Which derivative with respect to time is which in the Heisenberg picture of quantum mechanics?

For an observable $A$ and a Hamiltonian $H$, Wikipedia gives the time evolution equation for $A(t) = e^{iHt/\hbar} A e^{-iHt/\hbar}$ in the Heisenberg picture as $$\frac{d}{dt} A(t) = \frac{i}{\hbar} ...
5
votes
3answers
111 views

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 ...
1
vote
3answers
131 views

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 ...
0
votes
1answer
52 views

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. ...
8
votes
1answer
5k views

Evolution operator for time-dependent Hamiltonian

When I studied QM I'm only working with time independent Hamiltonians. In this case the unitary evolution operator has the form $$\hat{U}=e^{-\frac{i}{\hbar}Ht}$$ that follows from this equation $$ ...
2
votes
2answers
136 views

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 ...
0
votes
2answers
78 views

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 ...
2
votes
2answers
68 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 ...
1
vote
1answer
194 views

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 ...
2
votes
1answer
69 views

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 \ ...
2
votes
4answers
1k views

Which Schrödinger equation is correct?

In the coordinate representation, in 1D, the wave function depends on space and time, $\Psi(x,t)$, accordingly the time dependent Schrödinger equation is $$H\Psi(x,t) = ...
4
votes
2answers
732 views

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 ...
-1
votes
2answers
91 views

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 ...
0
votes
0answers
20 views

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? ...
0
votes
0answers
38 views

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) = ...
1
vote
0answers
239 views

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 ...
3
votes
1answer
661 views

Time evolution in quantum mechanics

We know that an operator A in quantum mechanics has time evolution given by Heisenberg equation: $$ \frac{i}{\hbar}[H,A]+\frac{\partial A}{\partial t}=\frac{d A}{d t} $$ Can we derive from this ...
1
vote
2answers
137 views

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 ...
1
vote
1answer
77 views

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 ...
5
votes
0answers
188 views

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 ...
1
vote
1answer
138 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: $$ ...
1
vote
0answers
79 views

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 ...
3
votes
2answers
568 views

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 ...
11
votes
1answer
450 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 ...
1
vote
1answer
146 views

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}+ ...
1
vote
2answers
102 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 ...
1
vote
1answer
333 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 ...
7
votes
4answers
544 views

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 ...
0
votes
3answers
270 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 ...
3
votes
1answer
139 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 ...
1
vote
1answer
135 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 ...
0
votes
3answers
105 views

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?
2
votes
1answer
58 views

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 ...
1
vote
0answers
95 views

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 ...
3
votes
1answer
116 views

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 ...
12
votes
2answers
352 views

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) ~=~ ...
3
votes
0answers
387 views

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( ...
2
votes
6answers
2k views

What does a unitary transformation mean in the context of an evolution equation?

Let be the unitary evolution operator of a quantum system be $U(t)=\exp(itH)$ for $t >0$. Then what is the meaning of the equation $$\det\bigl(I-U(t)e^{itE}\bigr)=0$$ where $E$ is a real ...