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. P

    Time-dep Hamiltonian commute with time evolution?

    Hi all, I'm attempting to prove that i \frac{d \xi (t)}{dt}=[\xi(t),H(p,q ; t)] where the Hamiltonian is explicitly time-dependent, in general. We also have some unitary U(t) which generates time-evolution. I wrote up a quick proof but realized afterward that I had assumed that H and...
  2. Xenosum

    Time Evolution of the Complex Scalar Field

    Homework Statement Consider the Lagrangian, L, given by L = \partial_{\mu}\phi^{*}(x)\partial^{\mu}\phi(x) - m^2\phi^{*}(x)\phi(x) . The conjugate momenta to \phi(x) and \phi^{*}(x) are denoted, respectively, by \pi(x) and \pi^{*}(x) . Thus, \pi(x) = \frac{\partial...
  3. M

    How Does Time Evolution Affect Quantum Measurement Probabilities?

    Homework Statement A quantum system has Hamiltonian H with normalised eigenstates ψn and corresponding energies En (n = 1,2,3...). A linear operator Q is defined by its action on these states: Qψ1 = ψ2 Qψ2 = ψ1 Qψn = 0, n>2 Show that Q has eigenvalues 1 and -1 and find the...
  4. B

    How Does a Magnetic Field Influence Quantum State Evolution?

    Homework Statement A magnetic field pointing in ##\hat{x}##. The Hamiltonian for this is: ##H= \frac{eB}{mc}\begin{pmatrix} 0 & \frac{1}{2}\\ \frac{1}{2} & 0 \end{pmatrix}## where the columns and rows represent ##{|u_z\rangle, |d_z\rangle}##. (a) Write this out in Dirac...
  5. D

    A few conceptual questions on time evolution of quantum states

    Hi guys, Sorry if this isn't quite the right place to post this, but I have a few conceptual questions that I'd like to clear up about time evolution of a quantum state. Firstly, what is the exact argument for the evolution operator \hat{U}\left(t,t_{0}\right) being independent of the initial...
  6. D

    Unitary time evolution - products of consecutive evolutions

    Hi all, I was wondering if anyone could clarify my understanding of unitary time evolution of quantum states, in particular for products of time evolution's: Suppose we know state of a quantum system at t=t_{0}, given by \vert\psi\left(t_{0}\right)\rangle, then to determine its state at...
  7. V

    Homogeneous gravity field time evolution position and momentum operato

    Homework Statement I am trying to solve Problem 21 from this sheet: Homework Equations The equation describing the time evolution of operators is given in the problem. The Attempt at a Solution I have found the commutators of the position and momentum operator with the...
  8. I

    Finding the Time Evolution of a Particle in a Divided Box

    Homework Statement A box containing a particle is divided into a left and right compartment by a thin foil. The two orthonormal base kets |L> and |R> stand for the particle being in either the left or the right compartment, respectively. Hence, any state ket in our system can be...
  9. binbagsss

    Quantum Mechanics - Time evolution operator , bra ket states.

    The question is to calculate the time evoution of S_{x} wrt <\Psi(t)\pm l where <\Psi\pm (t) l= ( \frac{1}{\sqrt{2}}(exp(^{+iwt})< \uparrow l , \pm exp(^{-iwt})< \downarrow l ) [1] Sx=\frac{}{2}(^{0}_{1}^{1}_{0} ) Here is my attempt: - First of all from [1] I see that l \Psi\pm (t) > = (...
  10. X

    Time Evolution operator in Interaction Picture (Harmonic Oscillator)

    Homework Statement Consider a time-dependent harmonic oscillator with Hamiltonian \hat{H}(t)=\hat{H}_0+\hat{V}(t) \hat{H}_0=\hbar \omega \left( \hat{a}^{\dagger}\hat{a}+\frac{1}{2} \right) \hat{V}(t)=\lambda \left( e^{i\Omega t}\hat{a}^{\dagger}+e^{-i\Omega t}\hat{a} \right) (i)...
  11. X

    Time Evolution of Hydrogen Atom in a Magnetic Field

    Homework Statement A hydrogen atom is prepared in its ground state with spin up along the z-direction. At time t = 0 a constant magnetic field ##\vec{B}## (pointing in an arbitrary direction determined by ##\theta## and ##\phi##) is turned on. Neglecting the fine structure and terms...
  12. TrickyDicky

    What is the impact of nonperturbative effects on time-dependent quantum theory?

    If I haven't understood this tricky stuff very badly when the Hamiltonian is time independent, then Schrödinger’s equation implies that the time evolution of the quantum system is unitary, but for the time-dependent Hamiltonian one must add some mathematically "put by hand" assumptions (although...
  13. W

    Time evolution with time dependent Hamiltonians

    I understand that in general, it's not true that in the case of a time dependent hamiltonian, the exponential map of the Hamiltonian is not a unitary transformation/the time evolution operator? U(t) \ne e^{-i \frac{H(t)}{\hbar} t} Is this thing allegedly not unitary or is it just not time...
  14. C

    Time evolution of a quantum system

    Homework Statement Let the time evolution of a system be determined by the following Hamiltonian: $$\hat{H} = \gamma B \hat{L}_y$$ and let the system at t=0 be described by the wave function ##\psi(x,y,z) = D \exp(-r/a)x,## where ##r## is the distance from the origin in spherical polars. Find...
  15. C

    Expectation value of the time evolution operator

    This problem pertains to the perturbative expansion of correlation functions in QFT. Homework Statement Show that \langle0|T\left[exp\left(i\int_{-t}^{t}dt' H_{I}^{'}(t')\right)\right]|0\rangle = \left(\langle0|T\left[exp\left(-i\int_{-t}^{t}dt'...
  16. Q

    Time evolution operator versus propagate

    I am trying to understand the how the time evolution operator is used versus the Feynman propagate. My limited understanding is the following for which I am seeking clarity: 1. The time evolution operator is a unitary operator which enables us to calculate a probability amplitude of one...
  17. U

    TDSE, time evolution between states (Check my working please?)

    Homework Statement Hi guys, I've recently taken up quantum, so it's all very new to me, it would be greatly appreciated if someone could check my working!Let ψ1(x) and ψ2(x) be two orthonormal solutions of the TISE with corresponding energy eigenvalues E1 and E2. At time t = 0, the particle is...
  18. A

    Time Evolution of Measurables(Hamiltonian)

    Supposing a physical quantity f whose operator commutes with the Hamiltonian operator H, and supposing it has no explicit time dependence, then the result regarding the time derivative of the operators gives us that the quantity is conserved and its mean value does not change with time.The...
  19. H

    Neutron Spin Time Evolution in Interferometer

    Hey Everyone, I'm working on a question and can't quite get the answer out. QUESTION: Part (a) "\left|\alpha\right\rangle and \left|\beta\right\rangle are the eigenfunctions for neutrons polarized respectively along positive and negative z directions. If the neutron, initially in...
  20. O

    Sakurai page 181: Time evolution of ensembles

    From "Modern Quantum Mechanics, revised edition" by J. J. Sakurai, page 181. Equation (3.4.27), at some time t_0, the density operator is given by \rho(t_0) = \sum_i w_i \mid \alpha^{(i)} \rangle \langle \alpha^{(i)} \mid Equation (3.4.28), at a later time, the state ket changes from \mid...
  21. B

    Time evolution of density matrix

    Homework Statement Hi there. just working on a problem from sakurai's modern quantum mechanics. it is: A) Prove that the time evolution of the density operator ρ (in the Schrodinger picture) is given by ρ(t)=U(t,t_{0})ρ(t_{0})U^\dagger(t,t_{0}) B) Suppose that we have a pure ensemble at...
  22. R

    Time evolution of a particle in an ISW after the well

    Homework Statement How does the state of a particle in an ISW evolve with time after the width of the well doubles - from a to 2a If the particle starts in the ground state of the half width well, then immediately after the well doubles it will be undisturbed therefore the initial wave...
  23. J

    Energy of interacting particle and time evolution of p(x)

    In classical QM, using a photon to measure the location or momentum of an electron collapses the electron's wave function at the point of measurement, which then, over time, spreads out again (what I'll call "diffuses"). Fine. The question is: Does the energy of the measuring photon change the...
  24. L

    Time evolution of operators as projectors - confusion

    I have a confusion regarding expressing operators as projectors in Schrodinger and Heisenberg pictures. Please help. Consider a two-state system with |1> and |2> We know that e.g. a raising operator can be expressed as: \hat{\sigma}_+=|2><1| But here's my line of thought: In the...
  25. K

    Finding the time evolution of a state

    Homework Statement 1) Using energies and eigenstates that I've worked out, find time evolution ψ(t) of a state that has an initial condition ψ(0) = \begin{pmatrix} 1 \\ 0\\ \end{pmatrix} 2) Find the expectation values < Sy> and <Sz> as a function of time. Homework Equations...
  26. C

    Commutative free particle time evolution

    Homework Statement http://img853.imageshack.us/img853/2532/70224197.png Homework Equations i know schrödingher eq. and basic quantum formula The Attempt at a Solution i showed that the equality at the first question but i can not start from (a) part. how and where am i supposed to start...
  27. D

    Time evolution operator in terms of Hamiltonian

    Homework Statement "Show that if the Hamiltonian depends on time and [H(t_1),H(t_2)]=0, the time development operator is given by U(t)=\mathrm{exp}\left[-\frac{i}{\hbar}\int_0^t H(t')dt'\right]." Homework Equations i\hbar\frac{d}{dt}U=HU U(dt)=I-\frac{i}{\hbar}H(t)dt The Attempt at a...
  28. L

    Dirac's Quantum Mechanics - the definition of the time evolution operator

    Dirac's "Quantum Mechanics" - the definition of the time evolution operator I'm reading Dirac's "Principles Of Quantum Mechanics" to learn more about the formal side of the subject. I have a question about the way he defines the time evolution operator in the book. Either there's a mistake or...
  29. T

    Time Evolution Of A 1-D Gaussian Wave Packet Under The Gravitational Potential

    Hello Colleagues, I am curious about a problem in Quantum Mechanics that incorporates the evolution of a Gaussian Wave Packet under the Gravitational Potential. What I am interested in is equation (3) in the following paper: "On the quantum analogue of Galileo's leaning tower...
  30. K

    The time evolution operator (QM) Algebraic properties

    Homework Statement The hamiltonian for a given interaction is H=-\frac{\hbar \omega}{2} \hat{\sigma_y} where \sigma_y = \left( \begin{array}{cc} 0 & i \\ -i & 0 \end{array} \right) the pauli Y matrix Homework EquationsThe Attempt at a Solution So from the time dependant schrodinger...
  31. G

    How Does the Delta Emerge in the Time Evolution Operator's Exponential Form?

    If you have some Hamiltonian represented by a 2x2 matrix ## H = \left( \begin{array}{cc} 0 & \Delta \\ \Delta & 0 \end{array} \right) ## And you want to use the time evolution operator ## U = \exp ( - \frac{i}{\hbar} H t ) ## it says that ## U = \exp (- \frac{i \Delta}{\hbar} t) ## Why...
  32. D

    Time Evolution of Polarized Cavity

    Homework Statement Suppose that a particular x-polarized cavity mode is described, at time t = 0, by the state |ψ(0)> = (1/√2)(|n> + |n+1>) Find |ψ(t)> for t > 0. This is best done in the Schrodinger picture. Evaluate the expectation of the electric field operator Ex and the uncertainty...
  33. G

    Question: How do symmetries and time evolution interact in quantum mechanics?

    My question is the following: when in quantum mechanics one introduces symmetry, says that a states and observables transform both, in order to mantain mean values intact (kind of like a change of coordinate system), i.e.: |\psi>\rightarrow U|\psi> and O\rightarrow UOU^\dagger...
  34. B

    Commutation of Hamiltonian and time evolution operator

    Can anyone explain how the time evolution operator commutes with the Hamiltonian of a system ( given that the the Hamiltonian does not depend explicitly on t ) ?
  35. C

    Time evolution and Feynman Integrals

    Homework Statement With the time evolution time operator, where there is time dependent hamiltonian, show the new form of the feynman propagator between two states. Consider the Weyl Integral. 2. Equations from \newcommand{\mean}[1]{{<\!\!{#1}\!\!>}}...
  36. A

    Time Evolution of Wave Function

    I'm getting bogged down in what is probably a very basic subject and it's holding me back. I'm not really sure how to determine the wave function \psi(x,t) given a function \psi(x,t=0); and since this is pretty much the under-pinning of every homework problem I've seen so far it's a huge issue...
  37. K

    Time evolution operator - Confusion

    Hi everyone. I am given a somewhat common potential well V(x)=0 for ¦x¦<a and infinite elsewhere. I am told that at t = 0 my particle is in a state represented by the wavefunction \psi(x,0)= A(\sin{(\frac{\pi x}{a})}+ \sqrt{2} \cos{(\frac{3 \pi x}{2 a})}) where A is a constant use for...
  38. ChristinaJ

    Mathematica Mathematica: Plot time evolution data in 3D

    Hi all, I have 3 lists of data where lists; A={numbers}, B={numbers} and C={{y,m,d,h,m,s},{...}} (i.e. C is in the form DateList). I have no problem plotting using DateListPlot for the 2D case (C VS A) but can't seem to plot all 3 sets of data together. Mathematica seems not to be able...
  39. D

    Time Evolution of Schrodinger's Equation

    Hello! Here is my question: Consider a particle of mass m, whose initial state has wavefunction \psi(x), in an infinite potential box of width a. Show that the evolution under the Schrodinger equation will restore the initial state (possibly with a phase factor) after time...
  40. J

    Time evolution of a diffusion equation

    Hallo everyone, I have a 1-D diffusion equation with decay as dA/dt = d2A/dx2-L*A with initial condition C(x,0)=C0=exp(-ax) and boundary condition= -Ddc/dx = I0 where L= decay constant A = certain concentration the concentration A is not in equilibrium. We can solve the above...
  41. A. Neumaier

    Time evolution in quantum field theories

    It requires more than that: a well-defined, selfadjoint Hamiltonian. See http://arxiv.org/pdf/quant-ph/9907069 for a gentle introduction and counterexamples. An in depth discussion is given in Vol. 1 of the math physics treatise by Reed and Simon, or Vol.3 of the math physics treatise by...
  42. D

    Time Evolution of Gaussian Wave Packet in Anisotropic Harmonic Oscillator

    Homework Statement So this kinda incorporates my last questions. I have a particle described by a Gaussian wave packet. And it moves in 2D anisotropic h.o potential with commensurate frequencies (1:2). I've solved the x part (I was messing around with the nasty integral, which in the end...
  43. J

    About the Schrodinger equation, the Hamiltonian, time evolution?

    Forgive me if this is a poorly asked question but I am not yet completely fluent in quantum mechanics and was just looking at the energy eigenvalue equation H|\Psi\rangle = i\hbar \frac{\partial}{\partial t}|\Psi\rangle = E|\Psi\rangle . We've got the Hamiltonian operator H acting on the state...
  44. C

    Why does the Hamiltonian generate time evolution?

    The first part of QFT seems to be almost entirely mathematical formalism, without really requiring a whole lot of physical insight to proceed. For instance, we can start with the free-field scalar Lagrangian, minimize it using the Euler-Lagrange equation to arrive at the Klein-Gordon equation...
  45. B

    Time evolution of hydrogen wave function that is not an energy eigenfunction

    I recently had a probelm in QM to find the time evolution of a hydrogen prepared in a state with a wave function that is not an energy eigenfunction: specifically, psi = Y21*R2p where Y is then the D spherical harmonic. Of course, n=2 hydrogen doesn't have d oribtals. So the problem is I...
  46. B

    Explicit form of time evolution operator

    The time evolution operaton may be written formally as: This is an actual solution to: only in the case that [H(t1),H(t2)]=0 (that is: the hamiltonian commutes in different instants of time) Of course, this includes the case of a time independent hamiltonian. If this is not the case, the...
  47. P

    Time evolution operator on momentum basis

    Hey i was wondering how to express the time evolution operator U(t,to) to a momentum eigen state |p> for a particle moving in the xdirection under a zero potential, V= 0. The reason i need this is that iam told the only way to get the matrix element of the time evolution operator using position...
  48. P

    How do you find the matrix element using Dirac notation?

    I've been trying to solve some questions using dirac notation, and most seem to be pretty straight forward (once you set everything up) but i always seem to get stuck when i try to find the matrix element and i can't seem to find the proper way to express the eigenstates given... so for example...
  49. O

    How Does the Time Evolution of Expectation Shape Scientific Understanding?

  50. I

    Using the time evolution operator

    I hope someone can help me out here, I am confused with a line of text I read - it is an example of a 2D Hilbert space with orthonormal basis e1, e2. The Hamiltonian of the system is the Pauli matrix in the y-direction. Given by the matrix: \sigma_{y} = (\frac{0, -i}{i, 0}) The eigenvectors...
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