# 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.

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1. ### I Time evolution of the electromagnetic wavefunction on a lattice

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...
2. ### A 2D space and 1D time evolution of a random field

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 ...
3. ### I Is time continuous or discrete in quantum physics?

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...
4. ### A Time evolution of coherent state with vacuum

I have attached my work here. I don't know how to proceed further?
5. ### A Time evolution of one photon

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⟩...
6. ### Time evolution of a particle in momentum space

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) =...

39. ### I Proof of unitarity of time evolution in Susskind's book

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...
40. ### I Unitary Time Evolution: Explaining Open Quantum Systems

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...
41. ### Position representation of coherent state and time evolution

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...
42. ### Time evolution of density profile

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...
43. ### Prove the time evolution operator is unitary

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...
44. ### Conceptual questions on unitarity and time evolution

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...
45. ### Interaction picture - time evolution operator

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...
46. ### Differential equations time evolution

Homework Statement Homework EquationsThe Attempt at a Solution Any help would be appreciated
47. ### Descriptions of time evolution: closed vs open systems

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...
48. ### Time Evolution for particle with potential suddenly removed

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}...
49. ### Time evolution of a state. (a missing t)

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...
50. ### Obtain Time evolution from Hamiltonian

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##...