What is Time evolution: Definition and 123 Discussions
Time evolution is the change of state brought about by the passage of time, applicable to systems with internal state (also called stateful systems). In this formulation, time is not required to be a continuous parameter, but may be discrete or even finite. In classical physics, time evolution of a collection of rigid bodies is governed by the principles of classical mechanics. In their most rudimentary form, these principles express the relationship between forces acting on the bodies and their acceleration given by Newton's laws of motion. These principles can also be equivalently expressed more abstractly by Hamiltonian mechanics or Lagrangian mechanics.
The concept of time evolution may be applicable to other stateful systems as well. For instance, the operation of a Turing machine can be regarded as the time evolution of the machine's control state together with the state of the tape (or possibly multiple tapes) including the position of the machine's read-write head (or heads). In this case, time is discrete.
Stateful systems often have dual descriptions in terms of states or in terms of observable values. In such systems, time evolution can also refer to the change in observable values. This is particularly relevant in quantum mechanics where the Schrödinger picture and Heisenberg picture are (mostly) equivalent descriptions of time evolution.
The Maxwell wavefunction of a photon is given in [here] as follows:
Because the curl operation mixes 3 different components, this wavefunction only works for a minimum of 3 space dimensions, with each grid point having 6 component numbers ##{E^1, E^2, E^3, B^1, B^2, B^3}##, and with the...
I want to develop a 2D random field and its change with time with constant velocity. My process:
1. Define a 2D grid [x, y] with n \times n points
2. Define 1D time axis [t] with n_t elements
3. Find the lagrangian distance between the points in space with the velocity in x and y ...
I was working on plotting fidelity with time for two quantum states. First I used discrete time( t= 0,1,2,3...etc) to plot my fidelity. I got constant fidelity as 1 with continuous value of time. Next I used discrete set of values ( t=0 °,30 °,60 °,90 °). Here I saw my fidelity decreases and...
During time evolution of one photon with vacuum state with hamiltonian as a^†b+b^†a, the answer is cos(t/ℏ)|0,1⟩+isin(t/ℏ)|1,0⟩. But i don't know how to do calculation to get this answer. Can someone please help me?
I tried to do this calculation:
|0⟩|1⟩(t)=e−iHtℏ|0⟩|1⟩...
Since it asks for the time evolution of the wavefunction in the momentum space, I write : ##\tilde{\Psi}(k,t) = < p|U(t,t_{0})|\Psi> = < U^\dagger(t,t_{0})p|\Psi>##
Since ##U(t,t_{0})^\dagger = e^{\frac{i}{\hbar}\frac{\hat{p^2}t}{2m}}##, the above equation becomes
##\tilde{\Psi}(k,t) =...
Dear everybody,
Let me ask a question regarding the unitary time evolution of a given Hamiltonian.
Let's start by considering a Hamiltonian of the form ##H(t) = H_0 + V(t)##. Then, I move to the interaction picture where the Schrödinger equation is written as
$$ i\hbar \frac{d}{dt}...
Question
---
So I've done a calculation which seems to suggest if I combine the system of a measuring apparatus to say an experimenter who "reacts" to the outcome of the the measurement versus one who does not. Then the change in entropy in both these situations is bounded by:
$$ \Delta S_R...
Hi everyone!
I am studying spectral methods to solve PDEs having in mind to solve a heat equation in 2D, but now i am struggling with the time evolution with boundary conditions even in 1D. For example,
$$
u_t=k u_{xx},
$$
$$
u(t,-1)=\alpha,
$$
$$
u(t,1)=\beta,
$$
$$
u(0,x)=f(x),
$$
$$...
In the section 8-2 dealing with resolving the state vectors, we learn that
|\phi \rangle =\sum_i C_i | i \rangle
and the dual vector is defined as
\langle \chi | =\sum_j D^*_j \langle j |Then, the an inner product is defined as
\langle \chi | \phi \rangle =\sum_{ij} D^*_j C_i \langle j | i...
Watching Dr. Susskind show how to find the time evolution of the average of an observable K, he writes:
I can not for the life of me figure out he derived it, and he also did something which I found terribly annoying throughout which is set hbar to 1, so after steps you lose where the hbar...
In this paper ##J=\frac{\partial f_1(X_1)}{\partial X_1}\frac{\partial f_2(X_2)}{\partial X_2}\frac{\partial f_3(X_3)}{\partial X_3}## where ##f_2(X_2),f_1(X_1),f_3(X_3)## evolves with time.
Now using this ##\dot J=\frac{d}{dt}(\frac{\partial f_1(X_1)}{\partial X_1}\frac{\partial...
In QM, states evolve in time by action of the Time Evolution Unitary Operator, U(t,t0). Without the action of this operator, states do not move forward in time. Yet even stationary states, like an eigenstate of energy, still contain a time variable – they oscillate in time at a fixed...
For this problem at t=0
Ψ(x,0)=Ψ1-Ψ3
Where Ψ1 and Ψ3are the normalised eigenstates corresponding to energy level 1 and 3 of the infinite square well potential.
Now for it's time evolution it will be Ψ1exp(-iE1t/ħ)- Ψ3exp(-iE3t/ħ)
And taking the time given in the question the time part of the...
Homework Statement
Homework Equations
For this question my ans. is coming option (3) since the time part of the wave comes out to be same for both the energy states which is (-1)^(-1/8) and (-1)^(-9/8) respectively (using exp(-iEt/ħ)).
But the correct option is given option (4).
Am I right...
Hi all, I am rather confused about the following concept. Assistance is greatly appreciated!
A time-dependent probability amplitude can be written as
$$\langle a_k| e^{-\frac{i}{\hbar}\hat{H}t} |\psi\rangle$$
where ##a_k## is an eigenvalue. Suppose I want the x-representation of the ket, I can...
Homework Statement
Question attached here:
I am just stuck on the first bit. I have done the second bit and that is fine. This is a quantum field theory course question but from what I can see this is a question solely based on QM knowledge, which I've probably forgot some of.
Homework...
Homework Statement
The unitary time evolution of the density operator is given by
$$\rho(t)=\textrm{exp}(-\frac{i}{\hbar}Ht)\,\rho_0 \,\textrm{exp}(\frac{i}{\hbar}Ht)$$
General definition of entropy is
$$S=-k_B\,Tr\,\{\rho(t) ln \rho(t)\}$$
Proof: $$\frac{dS}{dt}=0$$
Homework Equations
I am not...
Suppose I have a 1-D harmonic oscilator with angular velocity ##\omega## and eigenstates ##|j>## and let the state at ##t=0## be given by ##|\Psi(0)>##. We write ##\Psi(t) = \hat{U}(t)\Psi(0)##. Write ##\hat{U}(t)## as sum over energy eigenstates.
I've previously shown that ##\hat{H} = \sum_j...
Homework Statement
Part e)
Homework Equations
I know that the time evolution of a system is governed by a complex exponential of the hamiltonian:
|psi(t)> = Exp(-iHt) |psi(0)>
I know that |psi(0)> = (0, -2/Δ)
The Attempt at a Solution
I'm stuck on part e.
I was told by my professor...
Suppose I want to find the ground states corresponding to several Hamiltonian operators ##\left\{ \hat{H}_i \right\}##, which are similar to each other. As an example, let's take the ##\hat{H}_i##:s to be anharmonic oscillator Hamiltonians, written in nondimensional form (##\hbar = m = 1##) as...
In quantum mechanics time evolution is defined via a unitary operator
$$U(t^\prime,t) = e^{-iH(t^\prime-t)}$$
Now let's forget about the fact that we know this exponential representation and that we know that the U's fulfill the group axioms, i.e. that we can multiply any two U's, regardless...
Homework Statement
[/B]
I'm trying to solve the following problem. (a) was easy but I am stuck at (b).
Homework Equations
[/B]
Since we are told that the Hamiltonian is conserved, and the answer is in terms of the uncertainty of H, I assume I have to use the conservation of uncertainty...
Why do the time-evolution operator in quantum mechanics ##\exp{iHt}## and the Gibbs-weight operator in statistical physics ##\exp{-H/T}## have the same functional form? – i.e. both exponentials of the Hamiltonian operator.
The Matsubara trick/method just takes this as a fact in thermal QFT; but...
Hello.
I am trying to prove that the uncertainty in energy for a normalized state limits the speed at which the state can become orthogonal to itself.
The problem is number 2 on https://ocw.mit.edu/courses/physics/8-05-quantum-physics-ii-fall-2013/assignments/MIT8_05F13_ps6.pdf
Having issues...
Homework Statement
I am not sure about (c) and (d). Firstly, I calculated the eigenvector of A :
|v_1> = ( |2 > - |1> )/ √(2) ,eigenvalue -2
|v_2> = ( |2> + |1>) / √(2) , eigenvalue 2
For (c), basically it follows from part (b) where the probability of a_1 is given by the formula | <v_1 | ψ...
Heads up, I only recently got into quantum mechanics and don't feel like I got a solid grasp on the material yet.
1. Homework Statement
Given is the wave function of a free particle in one dimension:
\begin{equation}
\psi(x,0) = \left( \frac{2}{\pi a^2} \right)^{1/4} e^{i k_0 x} e^{-x^2/a^2}...
Homework Statement
A particle in an infinite potential well ##V(x) = 0, -\frac{a}{2} \leq x \leq \frac{a}{2}##, and infinite elsewhere is in it's ground state. Subsequently, the potential is removed and the particle is free to move.
How does the probability distribution in x and p change...
Homework Statement
Hey guys,
I have a question that asks;
Assume an Electron Nuetrino (U1) is produced at t = 0. Find the state U(t) for later times t > 0.
To give some context the question is based on a two state system where U1 = Collumn vector (sin(theta) cos(theta)) and U2...
Homework Statement
The eigenstates of the momentum operator with eigenvalue k are denoted by |k>, and the state of the system at t = 0 is given by the vector
|{ψ}>=\int \frac {dk}{2π} g(k)|{k}>
Find the state of the system at t, |ψ(t)>.
Compute the expectation value of \hat{P}.
Homework...
In some texts of fundamental quantum mechanics, it introduces the wave packet by Fourier transformation of a momentum wave into a spatial version. This is easy to understand because, analogy to the optical wave, a typical beam could compose waves of more than one frequencies. The general form is...
hey,
this thread might be a bit longer, but I have already calculated everything and I am quiet sure that it is right :)
Just need someone who confirms me or not :)
1. Homework Statement
Time evolution of probabilities. An electron inside a quantum well of length L is at time t = 0 in the...
Can anybody give a natural interpretation of operators and states in the Heisenberg Picture? When I imagine particles flying through space, it seems that the properties of the particles are changing, rather than the position property itself. Is there any way I should be thinking about these...
How how can we calculate the future evolution of a particle after the infinite square well potential is (somehow) turned off, releasing it into a free state? Assuming that it was in the ground state before.
Hi,
I just completed my second year of my physics undergraduate degree. And recently did a course on Quantum Mechanics. I have a few questions regarding the basic theory and postulates, probably, because due to lack of full clarity.
So,
Consider a wave function ψ(x,o), which is well behaved and...
Homework Statement [/B]
For a general operator ## \hat{O}##, let ##\hat{O}_{mn}(t)## be defined as:
$$ \hat{O_{mn}}(t) = \int u^{*}_{m}(x,t) \hat{O} u_{n}(x,t) $$
and
$$ \hat{O_{mn}} = \int u^{*}_{m}(x) \hat{O} u_{n}(x) $$
##u_{m}## and ##u_{n}## are energy eigenstates with corresponding...
In "The Theoretical Minimum" of Susskind (p.98) it says that if we take any two basisvectors |i \rangle and |j \rangle of any orthonormal basis, and we take any linear time-development operator U, that the inner product between U(t)|i \rangle and U(t)|j \rangle should be 1 if |i \rangle=|j...
Hi, I am a bit confused about unitary time evolution, I understand that a closed quantum system can be explained by unitary time evolution which ensures that the probability of all possible outcomes is always 1. But for an open quantum system we can't in general explain it with a unitary time...
Homework Statement
I ended up solving the problem as I was typing it up, I am posting what I did anyway as it took so long to type and might be useful to someone else.
I am trying to figure out the position representation of a coherent state and it's time evolution. I should be getting a...
Is it possible to work out analytically how the mass density profile ρ(r) of a ball of gas (spherically symmetric) evolve with time given the initial profile ρ0(r)? The assumption here is that the particles collapse only under the influence of gravity. I thought of this question in the process...
How is (5.240b) derived? I get {U^{-1}}^\dagger(t, t_0)\,U^{-1}(t, t_0)=I instead.
My steps:
\begin{align}<\psi(t_0)\,|\,\psi(t_0)>&=\,<U(t_0, t)\,\psi(t)\,|\,U(t_0, t)\,\psi(t)>\\
&=\,<U^{-1}(t, t_0)\,\psi(t)\,|\,U^{-1}(t, t_0)\,\psi(t)>\\
&=\,<\psi(t)\,|\,{U^{-1}}^\dagger(t, t_0)\,U^{-1}(t...
From a physical perspective, is the reason why one requires that the norm of a state vector (of an isolated quantum system) is conserved under time evolution to do with the fact that the state vector contains all information about the state of the system at each given time (i.e. the...
Hey all,
I got some question referring to the interaction picture. For example:
I have the Hamiltonian ##H=sum_k w_k b_k^\dagger b_k + V(t)=H1+V(t)##
When I would now have a time evolution operator:
##T exp(-i * int(H+V))##.
(where T is the time ordering operator)
How can I transform it...
The equivalence between descriptions of time evolution in QM are rigorously defined and proved for conservative systems as explained for instance among many other sources in Jauch's "Foundations of quantum mechanics" in the chapter 10. However, and an exception is the cited reference, it is not...
Homework Statement
This is a problem from my Statistical Mechanics book by Pathria.
[/B]
At ##t=0##, the ground state wavefunction of a one-dimensional quantum harmonic oscillator with potential ##V(x)=\frac{1}{2}\omega_0^2 x^2## is given by,
\psi(x,0)=\frac{1}{\pi^{1/4}...
Hi PF there is one thing that i cannot understand here.
Please look at eqn 1068
I try to compute the first term (without ##V^\dagger##)
I get something like
##c_f (t) =-i/\hbar exp [i(\omega + \omega_{fi})t/2] \frac{sin(\omega + \omega_{fi})t/2}{(\omega + \omega_{fi})/2} \}##
Unlike eqn 1071...
Homework Statement
A quantum system with a ##C^3## state space and a orthonormal base ##\{|1\rangle, |2\rangle, |3\rangle\}## over which the Hamiltonian operator acts as follows:
##H|1\rangle = E_0|1\rangle+A|3\rangle##
##H|2\rangle = E_1|2\rangle##
##H|3\rangle = E_0|3\rangle+A|1\rangle##...