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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24
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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 ...
6
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2answers
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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 ...
19
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1answer
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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\...
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3answers
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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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5answers
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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} ...
10
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6answers
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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 ...
10
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1answer
603 views

What's the relation between path integral and Dyson series?

If one solves the Schrodinger equation $$i\hbar\partial_tU(t,0) = H U(t,0)$$ for time evolution operator $U(t,0)$, one can get the following Dyson series $$U(t,0) = \sum_n(\dfrac{-i}{\hbar})^n\...
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3answers
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What is meant by unitary time evolution?

According to the time evolution the system changes its state the with the passage of time. Is there any difference between time evolution and unitary time evolution?
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3answers
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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 ...
9
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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
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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 ...
6
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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 ...
4
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1answer
930 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 ...
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6answers
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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 ...
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3answers
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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 ...
40
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3answers
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Why do excited states decay if they are eigenstates of Hamiltonian and should not change in time?

Quantum mechanics says that if a system is in an eigenstate of the Hamiltonian, then the state ket representing the system will not evolve with time. So if the electron is in, say, the first excited ...
28
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6answers
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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 ...
6
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3answers
617 views

What is a “picture” in quantum mechanics?

One of the basic ingredients of quantum mechanics is the possibility of working in different "pictures". Thus, while we normally work in the Schrödinger picture, in which states evolve according to ...
4
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3answers
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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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2answers
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QM Continuity Equation: Many-Body Version for Density Operator?

I am trying to brush up my rusty intuition on second quantization and many-particle systems and i came across the following problem: In 1-particle QM we have the continuity equation $$ \frac{\...
5
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1answer
366 views

Why is time-evolution unitary (the sequel)?

One foundational postulate of QM is that a closed physical system at one instant of time, say $t$, is completely described by a wavefunction $\psi \in S^1\subset H$ (where $H$ is a Hilbert space and $...
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2answers
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Heisenberg equation with time-dependent Hamiltonian

It is the root of quantum mechanics that Heisenberg picture and Schrödinger picture are equivalent? In most textbooks and wikipedia, the equivalence is proved with a time-independent Hamiltonian. ...
0
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2answers
475 views

General solution of states of time dependent Hamiltonian

Given a time dependent Hamiltonian which commutes at different times, we have the time evolution operator given by $$\mathcal{U}(t,0) = \text{exp}\bigg[-\bigg(\frac{i}{h}\bigg)\int_{0}^{t}dt' H(t')\...
5
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4answers
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Heisenberg Picture with a time-dependent Schrödinger Hamiltonian

So when the Hamiltonian is time-independent, we can define the Heisenberg state vectors by evolving the Schrödinger state vectors back in time: $$ | \psi \rangle_H = \hat{U}^\dagger (t)|\psi(t) \...
10
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1answer
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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 ...
12
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2answers
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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 ...
5
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1answer
328 views

Does Haags Theorem forbid Time-Evolution?

I didn't quite grasp the essence of Haags Theorem in the the way it is presented (for example on wikipedia), but the issue seems to be that if one wants to represent infinitely degrees of freedom ...
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 ...
3
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4answers
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Schrodinger equation for a Hamiltonian with explicit time-dependence?

Can I write a Schrodinger equation for time-dependent Hamiltonian like this: $$i\hbar\frac{d}{dt}\psi(t) = H(t)\psi(t)$$ and then perform Euler integration like this: $$\psi(t+\Delta t) = (1-\frac{...
6
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3answers
695 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 ...
3
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2answers
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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 $$H\Psi=i\hbar \frac{\partial \Psi}{\partial t}$$ ...
0
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1answer
942 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 ...
3
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3answers
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Why is time-evolution operator unitary?

When we shift the system's time from $t=0$ to $t = t$, we can define the following operator $\hat{U}$. $$\hat{U} = e^{- i \hat{H} t / \hbar} \, .\tag{1}$$ So many (as far as I read, almost all of) ...
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 ...
0
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2answers
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Time-ordering and Dyson series

In Dyson series we use a time-ordered exponential by arguing that Hamiltonians at two different instants of time do not commute. Why is that so? Can anyone explain with an example why should the same ...
5
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1answer
208 views

Time-dependence of free and interacting Hamiltonians

Consider an interacting field theory with Hamiltonian $$H=H_0+V$$ where $H_0$ is the Hamiltonian of the free theory and $V$ is the added interaction. Now, I know the full Hamiltonian $H$ should be ...
3
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2answers
217 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 $e^{-iEt/h}\psi(x)$.
3
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2answers
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Heisenberg equation of motion

In the Heisenberg picture (using natural dimensions): $$ O_H = e^{iHt}O_se^{-iHt}. \tag{1} $$ If the Hamiltonian is independent of time then we can take a partial derivative of both sides with respect ...
2
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2answers
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Are $\hat x$ and $\hat p$ assumed to be time-independent operators?

In the book Quantum Mechanics by Cohen-Tannoudji, at $G_{III}$, it is given that and then in the comment section, it is also given that so I'm pretty confused in here, because in one side, they say ...
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1answer
706 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 ...
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0answers
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Time evolution (spin-1 particle)

The state ket of a spin-1 particle with orbital angular momentum is given by $$\left|\alpha,0\right\rangle=\frac{f\left(r\right)}{4\sqrt2}\left[\sqrt2\left|1-1\right\rangle_L \left|11\right\...
2
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2answers
416 views

Two Level Atom Rotating Frame

Introduction I am trying to work out the Rabi problem. In particular I am trying to work it out from a perspective focusing on the operators rather than the kets. Think Heisenberg picture rather than ...
2
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2answers
150 views

Time dependence of a function of an operator

Suppose I know the time evolution of an operator is given by $\dot{\hat{O}} = \frac{i}{\hbar}[\hat{H}(t), \hat{O}(t)]$. Now I want to look at a function $\hat{f}(\hat{O}$, and I want to know the time ...
2
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4answers
183 views

Why does the time-evolution operator $U(t)$ depend explicitly on time in the Schrodinger picture?

Schrodinger's picture is that operators are time-independent. But time evolution operator $U(t)$ is time-dependent. Why is that?
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1answer
133 views

Why is time evolution unitary in $PT$-symmetry?

I have a question on the time evolution for a $PT$-symmetric Hamiltonian. So far I have only read that time evolution was unitary because $H$ commutes with $PT$ and the newly constructed operator $C$ ...
1
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1answer
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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\...
0
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1answer
89 views

Time resulting of a thermodynamics effect?

I heard in a documentary that : Irreversibility of time is linked to Heat Dissipation in thermodynamics. The theorist says "passage of time" is only a macro sensation, the underlying effect ...
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1answer
140 views

Operator Transformation and Time Evolution

UPDATE: See update at bottom of post Short Version: $\sigma^-_H$ in the Heisenberg picture. $U = e^{+i\omega (|e\rangle \langle e)_H}$. Prove that $$ \frac{d \sigma^-_H}{dt} e^{+i\omega t} = U^{\...