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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4
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2answers
825 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 ...
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1answer
133 views

If $| \alpha(t) \rangle = e^{-i\omega t} |\alpha_0 \rangle$, then why is there time dependence in expected values?

The time evolution of a coherent state $| \alpha(t) \rangle$ is given by: $$| \alpha(t) \rangle = e^{-i\omega t} |\alpha_0 \rangle$$ So then it seems to me that it should be $$\langle \alpha(t)| = ...
-1
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2answers
135 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 ...
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3answers
3k views

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

In the Schrödinger 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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1answer
939 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) = n_1^{(0)}n_2^{(...
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2answers
143 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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2answers
382 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 $$V(x)=\...
2
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1answer
354 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 ...
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1answer
851 views

Interaction pictures of Quantum Mechanics

I want to understand the Schrödinger, Heisenberg and interaction picture and have a few questions about them: So in general you have a time-dependent Hamiltonian $H$, as for example the potential may ...
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0answers
149 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 E_0\cos(...
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1answer
258 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: $$ Q\...
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4answers
901 views

Liouville's theorem and the preservation of topology

What might be a simple proof showing that the time evolution of the phase space volume can't lead to splitting off of the phase space volume? By Liouville's theorem, the total phase space volume is ...
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1answer
4k 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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1answer
1k 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}+ \widehat{a}^{+}...
13
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1answer
963 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 ...
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1answer
1k 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 Q\...
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4answers
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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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3answers
1k 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 ...
4
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1answer
753 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 ...
2
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3answers
4k 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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1answer
375 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 $[x,...
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3answers
147 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
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1answer
159 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 ...
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0answers
156 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 ...
4
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1answer
330 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 (...
6
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1answer
297 views

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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2answers
2k views

The formal solution of the time-dependent Schrödinger equation

Consider the time-dependent Schrödinger equation (or some equation in Schrödinger form) written down as $$ \tag 1 i \partial_{0} \Psi ~=~ \hat{ H}~ \Psi . $$ Usually, one likes to write that it has a ...
4
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1answer
2k 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( \...
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2answers
274 views

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 ...
5
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2answers
1k 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 ...
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1answer
1k views

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 ...
2
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1answer
160 views

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 ...
6
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2answers
901 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 ...
28
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6answers
1k 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 ...
5
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1answer
417 views

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) = \exp\left(\...
2
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1answer
336 views

Time evolution of a quantum state

I have another point in QM that I would like clarified. Suppose $$\{|n\rangle\}$$ is a set of eigenstates of both the Hamiltonian $H$ and another operator $\hat O$ corresponding to an observable also. ...
5
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2answers
410 views

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 ...
10
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4answers
10k views

How is the hamiltonian a hermitian operator?

My book about quantum mechanics states that the hamiltonian, defined as $$H=i\hbar\frac{\partial}{\partial t}$$ is a hermitian operator. But i don't really see how I have to interpret this. First of ...
2
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1answer
314 views

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 ...
0
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1answer
134 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}?$
3
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2answers
292 views

“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 ...
5
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2answers
2k views

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 $F(t)=F_0 \sin(\Omega t)$ ...
2
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2answers
1k views

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 ...
19
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1answer
17k 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 $$ i\...
4
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4answers
2k 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) = i\hbar\frac{\partial}{\...
2
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4answers
311 views

Why do we consider the evolution (usually in time) of a wave function?

Why do we consider evolution of a wave function and why is the evolution parameter taken as time, in QM. If we look at a simple wave function $\psi(x,t) = e^{kx - \omega t}$, $x$ is a point in ...
3
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3answers
385 views

Does the second law of thermodynamics tell me how the entropy changes?

In thermodynamics I can e.g. compute the properties of ideal gases with certain energies $U_1,U_2$ in boxes with certain volumes $V_1$ and $V_2$. Say I have two such boxes and they have some specific ...
2
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6answers
8k 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 ...
8
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2answers
2k views

Equation of motion for the reduced density matrix

The equation of motion for the density matrix of a many body isolated quantum system is the von Neumann's equation: $\dot{\rho }(t)=i[\rho (t),H]$. How about the equation of motion for the reduced ...
8
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4answers
2k views

Can the universe be described by a Markov chain?

This may be a fairly basic question as I don't have a strong background in physics. I intuitively thought that the universe must be able to be described by a Markov chain. That is, I thought you ...