What is Time dependence: Definition and 32 Discussions
A time-invariant (TIV) system has a time-dependent system function that is not a direct function of time. Such systems are regarded as a class of systems in the field of system analysis. The time-dependent system function is a function of the time-dependent input function. If this function depends only indirectly on the time-domain (via the input function, for example), then that is a system that would be considered time-invariant. Conversely, any direct dependence on the time-domain of the system function could be considered as a "time-varying system".
Mathematically speaking, "time-invariance" of a system is the following property:
Given a system with a time-dependent output function
y
(
t
)
,
{\displaystyle y(t),}
and a time-dependent input function
x
(
t
)
;
{\displaystyle x(t);}
the system will be considered time-invariant if a time-delay on the input
x
(
t
+
δ
)
{\displaystyle x(t+\delta )}
directly equates to a time-delay of the output
y
(
t
+
δ
)
{\displaystyle y(t+\delta )}
function. For example, if time
t
{\displaystyle t}
is "elapsed time", then "time-invariance" implies that the relationship between the input function
x
(
t
)
{\displaystyle x(t)}
and the output function
y
(
t
)
{\displaystyle y(t)}
is constant with respect to time
t
{\displaystyle t}
:
y
(
t
)
=
f
(
x
(
t
)
,
t
)
=
f
(
x
(
t
)
)
.
{\displaystyle y(t)=f(x(t),t)=f(x(t)).}
In the language of signal processing, this property can be satisfied if the transfer function of the system is not a direct function of time except as expressed by the input and output.
In the context of a system schematic, this property can also be stated as follows:
If a system is time-invariant then the system block commutes with an arbitrary delay.If a time-invariant system is also linear, it is the subject of linear time-invariant theory (linear time-invariant) with direct applications in NMR spectroscopy, seismology, circuits, signal processing, control theory, and other technical areas. Nonlinear time-invariant systems lack a comprehensive, governing theory. Discrete time-invariant systems are known as shift-invariant systems. Systems which lack the time-invariant property are studied as time-variant systems.
Consider the interaction of a two level atom and an electric field (semiclassically, we treat the field as 'external' i.e. not influenced by the atom; the full quantum treats the change in the field as well)
Electric field in semiclassical Hamiltonian: plane wave
##H_{int,~semiclassical}=-\mu...
Hello,
So about two weeks ago in class we looked at RLC circuits in our E&M course, and short story short... we compared the exchange of energy between the Capacitor and the Inductor (both ideal) to simple harmonic motion. Once the capacitor and inductor are not ideal anymore, we said it's...
Consider instead a thermally insulated container of volume V with a
small hole of area A, containing a gas with molecular mass m. At time t = 0, the density is ##n_0## and temperature is ##T_0##. As gas effuses out through a small hole, both density and temperature inside the container will...
Suppose I had a Lagrangian
$$L = q+ \dot{q}^2 + t.$$
This has explicit time dependence. Now consider another Lagrangian:
$$L = q+ \dot{q}^2 .$$
Which has no explicit time dependence. But after solving for the equations of motion, I get $$\dot{q} = t/2 + C.$$
So I could now write my Lagrangian...
It seems that the entanglement of two particles does not change with time and can cross long distanced as long an neither particle decoheres with the environment. This makes me wonder if the wave function for that entanglement can have any time or space dependence? I only did a brief search for...
Homework Statement
The answer is as follows: [/B]
However they said that time t=0 so I am confused how the exponent has a t in it surely it should be zero. Thanks
Homework Statement
I am not looking for a solution to the problem, as much as I need a clarification on what it's asking for. The problem:
"A particle of mass m slides down an inclined plane under the influence of gravity. The particle is starting its motion from rest. Find the time dependence...
A particle is in its ground state of an infinite square well of width a <xl i>=√2/a*sin(πx/a) and since it's an eigenstate of the Hamiltonian it will evolve as <xlα(t)>=√2/a*sin(πx/a)e^(-iE1t/ħ) where E=π2ħ2/2ma2
If the well now suddenly expands to witdh 2a
If the well suddenly expands to 2a...
It would be really appreciated if somebody could clarify something for me:
I know that stationary states are states of definite energy. But are all states of definite energy also stationary state?
This question occurred to me when I considered the free particle(plane wave, not a Gaussian...
In field theory we most of deal with theories whose Lagrangian densities are of the form (sticking to scalar fields for simplicity) $$\mathcal{L}= -\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi - \frac{1}{2}m_{\phi}^{2}\phi^{2} + \cdots$$ where ##\partial := \frac{\partial}{\partial x^{\mu}}##...
Let's assume a universe like ours which after inflation expands decelerated and accelerated thereafter.
How will the ratio Hubble length ##1/H## to scalefactor ##a## evolve over time? And how could one calculate the time dependence of this ratio.
Any help appreciated.
Hi. Say we have found a hamiltonian ##H## for some system.
So I know that if ##\frac{\partial H }{\partial t} \neq 0## then obviously the energy of the system is not conserved.
But if ##\frac{\partial H }{\partial t} = 0##, is the energy always conserved? Or do we need to find that ##\frac{d H...
I'm trying to elucidate certain concepts about time dependence and perturbation theory in quantum mechanics and QFT.
I get the impression that most of the important results that in principle can be considered to have a time dependence can actually be calculated in terms of time-independent...
Homework Statement
A uniform magnetic field B is perpendicular to the plane of a circular wire loop of radius R. The magnitude of the field varies with time according to B=B0exp(-t/τ) where B0 and τ are constants. The time dependence of the induced emf in the loop is
a) exp(-t2/τ2)
b)...
Hi folks, originally I read Peskin & Schroeder but then I realized it was too concise for me.
So I switched to Srednicki and am reading up to Chapter 5.
(referring to the textbook online edition on Srednicki's website)
Two questions:
1. In the free real scalar field theory, the creation...
A = \langle q_f(t) \mid q_i(t) \rangle = \langle q_{f,H} \mid e^{iH(t_0-t)} e^{-iH(t-t_0)} \mid q_{i,H} \rangle = \langle q_{f,H} \mid q_{i,H} \rangle
This means that A is time-independent, and depends only on the reference point ##t_0##. How is it possibly? From Schoedinger picture it...
I have a question about time dependence in the different pictures of QM. In the Schrodinger Picture, I've read that the time dependence is in the state vector, but one can construct Hamiltonians and other operators that are time dependent in the Schrodinger Picture. For instance, one can...
Hi all,
I was having a bit difficulty understanding the term scleronomic constraint.
From what I have read, it is a type of holonomic system(which means there is time dependence). However, the difference between the two types(scleronomic and rheonomic), is that although scleronomic...
What does "Solve for the time dependence of" mean?
Homework Statement
Use the Heisenberg equation of motion to solve for the time dependence of x(t) given the Hamiltonian
H = \frac{p^{2}(t)}{2m} + mgx(t)
Homework Equations
The Heisenberg equation of motion is
\frac{dA(t)}{dt} =...
Homework Statement
Find the wavefunction for an infinite well, walls are at x=0 and x=L(include the time dependence)
The Attempt at a Solution
I don't understand what it's meant by include the time dependence. Can I just find the time-independent wavefunction and then multiply it by...
How do I show that the scalar product is time independent?
I have: \frac{d}{dt}\int\Psi^{*}_{1}(x,t)\Psi_{2}(x,t)dx = 0
And have proceeded to take the derivatives inside the integral and using the time dependent Schrodinger eq. ending up with...
I have a few questions about the time dependence of energy and probability etc. of systems.
Firstly...
Say I have a particle in an infinite 1-D well.
I can work out the general wave function as a Fourier sum of the orthogonal sine functions.
Hence, the average/expectation value of the...
I am trying to analyse an RL circuit, particulartly the dependence of current in 2 resistors with time, in an RL circuit. This is what I think but I would really love some people to tell me if I am getting it wrong or if there is anything they want to add to it.
***Just after the the switched...
Homework Statement
The question comes straight from Intro to QM by Griffiths (pg 29, Q 2.6).
A wave equation is given representing an even mixture of the first two energy levels of the infinite square well. The task is to normalize the wave function, state it explicitly and then derive...
I am really puzzled. I have several questions about how Omega_M and Omega_Lambda evolve with time. Ultimately I want to reconstruct figure 1 one Sean Carroll's Website http://nedwww.ipac.caltech.edu/level5/March01/Carroll/Carroll1.html.
First of all, I will assume a flat universe throughout...
I would be very eager to have someone explain to me why it is justified to assume harmonic time dependence when seeking solutions to a wave equation. This is done many times in Jackson or Kittel. Isn't assuming harmonic time dependence in solving the wave equation not using part of the...
I'm trying to understand something that's coming from my Marion & Thornton (4th edition 1995 on p. 264 in a section titled "Conservation Theorems Revisited"). The topic is conservation of energy and introduction of the Hamiltonian from Lagrange's equations.
We're told that the Lagrangian...
Say I have a funciton sin(theta - phi)
theta and phi are both time-dependant. How do I take the time derivative of this function? Is there a general notation or do I have to assume theta = w1t and phi = w2t and go with that?
Homework Statement
A magnetic field with the time dependence shown in the figure below is at right angles to a N=197 turn circular coil with a diameter of d=4.2 cm.
What is the induced EMF in the coil at t=7.5ms in mV?
See attached graph
Homework Equations
|EMF|= N*|change...