heisenberg picture Definition and 36 Threads

In physics, the Heisenberg picture or Heisenberg representation is a formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which the operators (observables and others) incorporate a dependency on time, but the state vectors are time-independent, an arbitrary fixed basis rigidly underlying the theory.
It stands in contrast to the Schrödinger picture in which the operators are constant, instead, and the states evolve in time. The two pictures only differ by a basis change with respect to time-dependency, which corresponds to the difference between active and passive transformations. The Heisenberg picture is the formulation of matrix mechanics in an arbitrary basis, in which the Hamiltonian is not necessarily diagonal.
It further serves to define a third, hybrid, picture, the interaction picture.

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  1. Rayan

    The time-dependence of the expectation values of spin operators

    So first I derived the expressions for the dynamics of the spin operators and got: $$ \frac{d\hat{S}_y}{dt} = w\hat{S}_x^H $$ $$ \frac{d\hat{S}_x}{dt} = w\hat{S}_y^H $$ $$ \frac{d\hat{S}_z}{dt} = 0 $$ Now I want to calculate the time-dependence of the expectation values of the spin operators...
  2. T

    I Quantum Circuit Confusion On Time Evolution

    Hi all, When working in the Heisenberg picture, we can represent implementing time evolution on an operator via a Hamiltonian H through a quantum circuit type picture like the following: where time is on the vertical axis and increases going up and the block represents the unitary gate...
  3. T

    I Time dependence of operators in the Schrödinger picture

    I found in wikipedia following formula describing the derivative of operator ## A_H ## considered in Heisenberg picture, where ## A_S ## is it's representation in Schroedinger picture: ## \frac{d}{dt}A_\text{H}(t)=\frac{i}{\hbar}[H_\text{H},A_\text{H}(t)]+\left( \frac{\partial...
  4. O

    I Limit of quantum mechanics as h -> 0

    Starting from the Heisenberg equation of motion, we have $$ih \frac{\partial p}{\partial t} = [p, H]$$ which simplifies to $$ih \frac{\partial p}{\partial t} = -ih\frac{\partial V}{\partial x}$$ but this just results in ## \frac{\partial p}{\partial t} = -ih\frac{\partial V}{\partial x}## and...
  5. LittleSchwinger

    A Decoherence in the Heisenberg Picture

    When I'm teaching Advanced QM, I like to include how to describe some processes in the Heisenberg picture (e.g. double slit) so that a student's thinking isn't overly attached to the "dynamics of the quantum state", but they can also understand effects involving operator evolution. This is a...
  6. qft-El

    A Heisenberg picture and Path integrals (Zee QFT)

    Reading the introduction to path integrals given in the latest edition of Zee's "Quantum field theory in a nutshell", I have found a remark which I don't really understand. The author is evaluating the free particle propagator ##K(q_f, t; q_i, 0)## $$\langle q_f\lvert e^{-iHt}\lvert q_i...
  7. Bruno Cardin

    A Expectation value in Heisenberg picture: creation and annihilation

    So, I have a hamiltonian for screening effect, written like: $$ H=\sum_{k}^{}\epsilon_{k}c_{k}^{\dagger}c_{k}+ \frac{1}{\Omega}\sum_{k,q}^{}V(q,t)c_{k+q}^{\dagger}c_{k} $$ And I have to find an equation for the time evolution of the expected value of the operator ##c_{k-Q}^{\dagger}c_{k}##. I...
  8. Haorong Wu

    I Applying Heisenberg picture to density operator

    Suppose that a particle evolves from point A to point B. The state of the particle can be written as $$\rho=\sum \left | m\right >\rho_{mn}\left< n\right | .$$ Because the basis is evolving as the particle travels, I am considering applying the Heisenberg picture to the density operator. Let...
  9. Haorong Wu

    How to calculate an operator in the Heisenberg picture?

    I have some problems when calculating the operators in Heisenberg picture. First, ##\frac {dx} {dt} = \frac {1} {i \hbar} \left [ x, H \right ] = \frac {p} {m}##. Similarly, ##\frac {dp} {dt} = \frac {1} {i \hbar} \left [ p, H \right ] = - m \omega ^ 2 x##. These are coupled equations. I...
  10. LarryS

    I Quantum Measurement under Heisenberg Picture?

    When a quantum measurement occurs under the Schrodinger picture, the wave function collapses to one of the eigenvectors of the operator-observable and the value measured is the corresponding eigenvalue of that eigenvector. What happens during a quantum measurement under the Heisenberg picture...
  11. F

    I Interpretation of QM in the Heisenberg Picture

    This is in reference to a question, never fully resolved, posed here: https://www.physicsforums.com/threads/interpretation-of-the-heisenberg-picture-in-qm.816449/ The von Neumann postulates for Quantum Theory - Evolution (Schrödinger's equation) and Projection (Born's rule) are always framed...
  12. Haorong Wu

    The position and momentum operators for a free particle in Heisenberg picture

    Homework Statement From Griffiths GM 3rd p.266 Consider a free particle of mass ##m##. Show that the position and momentum operators in the Heisenberg picture are given by$$ {\hat x}_H \left( t \right) ={\hat x}_H \left( 0 \right) + \frac { {\hat p}_H \left( 0 \right) t} m $$ $$ {\hat p}_H...
  13. A

    Quantum mechanics - Heisenberg picture

    Homework Statement We consider the O2- molecule, with the Hamiltonian and position operator having matrix representations in terms of the Pauli matrices: In the Heisenberg picture, the position operator is: (1) Find the eigenvalues and eigenstates of x(t) at time t=pi*hbar/(4A) (2) The...
  14. Q

    I Interpret Heisenberg Picture: Operators & States

    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...
  15. atyy

    Can decoherence be formulated in the Heisenberg picture?

    In another thread, stevendaryl and I were trying to understand whether MWI can be formulated in the Heisenberg picture. Since neither of us really understands MWI, I tried to retreat to safer ground by asking: Can decoherence be formulated in the Heisenberg picture?
  16. zonde

    Photon energy in Heisenberg picture

    I would like to ask a quick (I suppose) question. Does a photon have definite energy in Heisenberg picture? My motivation for this question comes from reading that Hamitonian is generator of time evolution. But in Heisenberg picture time evolution is associated with operator not quantum system...
  17. D

    Instantaneous eigenstates in the Heisenberg picture

    I'm a bit confused as to what is meant by instantaneous eigenstates in the Heisenberg picture. Does it simply mean that if vectors in the corresponding Hilbert space are eigenstates of some operator, then they won't necessarily be so for all times ##t##, the eigenstates of the operator will...
  18. S

    Interpretation of the Heisenberg picture in QM

    I was always a bit puzzled by the Heisenberg picture (not mathematically, I'm fine with that, but conceptually) - if a "state" describes a system, how can it not be time-dependent, if the system changes? I just found an alternative way of looking at it which seems to make sense to me, but I'm...
  19. D

    Harmonic oscillator in Heisenberg picture

    For the harmonic oscillator in 1-D we get the 2nd time derivative of the x Heisenberg operator = -ω2 x. When that is integrated it gives xH (t) = Acos(ω t) +Bsin (ω t) where A and B are time independent operators. My question is why are the constants A and B incorporated into the terms as a...
  20. Breo

    Heisenberg Picture Formalism: Rewriting Operators in Covariance

    Why does Heisenberg picture formalism requires to rewrite operators in explicit covariance?
  21. H

    Some questions about the heisenberg picture

    I have some questions regarding the Heisenberg picture of QM. 1. How does one calculate probabilities for measuring eigenvalues? Are the eigenvectors simply time dependant? 2.Does this mean for initial value problems, the initial data of the system is contained in the operators? (Or...
  22. D

    How does Weinberg's Quantum Theory of Fields develop QFT in Heisenberg picture?

    Hello, Can anyone recommend me a book that develops QFT in Heisenberg picture? I have found Källen - Quantum Electrodynamics. Thanks in advance for the answers.
  23. I

    Books that emphasize Heisenberg picture

    Hi fellas, one friends that is Mathematician asked me to recommend some textbook that emphasizes Heisenberg picture and where this picture is rigorously explained. If anyone knows some good book for this I would be grateful :) Regards, Ivan
  24. Q

    Equation of Motion in Heisenberg Picture

    Homework Statement A particle of mass m is in a harmonic oscillator potential with spring constant k. An observable quantity is given in the Schrodinger picture by the operator: Z = a^{\dagger}a a^{\dagger} a a) Determine the equation of motion of the operator in the Heisenberg...
  25. C

    Heisenberg picture manifests Lorentz invariance?

    In several textbooks of QM I have read that Lorentz invariance is manifest in Heisnberg picture. How can we deduce that?
  26. K

    Schrödinger and Heisenberg picture

    Im sorry, I accidently edited my opening post instead of posting a new one.. The question was regarding the statement that the state ket is stationary in the Heisenberg picture when the basis kets are transforming in time. Because the state ket is a superposition of the base kets it should the...
  27. zonde

    Heisenberg Picture: Popularity & Photon Double-Slit Treatment

    I would like to find out how popular is Heisenberg picture. Is there someone who finds Heisenberg picture useful? And as well - are there any ideas how photon double-slit should be treated in Heisenberg picture?
  28. B

    Question about the Heisenberg Picture

    Homework Statement I've seen this example for using the Heisenberg equation of motion to solve the Simple Hamonic Oscillator. http://en.wikipedia.org/wiki/Heisenberg_picture#Commutator_relations" However, if you were only interested in one variable, let's say position, on how the the...
  29. N

    Heisenberg picture describes emission, Schroedinger picture does not

    Am I right in thinking that the Heisenberg matrix interpretation describes emission, while the Schroedinger interpretation does not?
  30. pellman

    Time derivative in the Heisenberg picture?

    On the Wikipedia page for http://en.wikipedia.org/wiki/Heisenberg_picture#Mathematical_details" we find this relation \frac{d}{dt}A(t)=\frac{i}{\hbar}[H,A(t)]+\left(\frac{\partial A}{\partial t}\right) I don't understand what the distinction between \frac{d}{dt}A(t) and...
  31. N

    Applying Heisenberg Picture Confusion

    I was trying to follow http://www.youtube.com/watch?v=dCua1R9VIiQ&p=EFD655A9E0B979B7&playnext=1&index=54" lecture at the 4:15 mark but am having a little difficulty. In particular, why doesn't he have to take the commutator of all four of the terms you get when you square (p-eA). Is he using...
  32. A

    Operator commutator (Heisenberg picture)

    Homework Statement It's a part of a bigger problem, but what I need help with is finding the commutator between x(t) and x(t) at a different time. So basically I need [x(t1),x(t2)] Homework Equations A(t)=e^{iHt/\hbar}Ae^{-iHt/\hbar} The Attempt at a Solution The farthest I can...
  33. pellman

    Time translation in the heisenberg picture?

    A fundamental quantity that we calculate with QM is \langle \Phi|\Psi\rangle-- the probability amplitude for observing a system to be in state |\Phi\rangle given that it is in state |\Psi\rangle. In the Schrodinger picture the states are time-dependent and we can ask, "What is the probability...
  34. L

    Did I Make a Mistake? A Sign Error in an Exercise Answer

    Did I do this right? An exercise in a book where my answer came out opposite from the author's. [PLAIN]"camoo.freeshell.org/22.9.pdf"[/URL] Laura
  35. M

    Usefulness of Heisenberg picture?

    I'm a little confused as to why anyone would want to use the Heisenberg picture of time evolution instead of the Schrodinger picture, beyond showing that the equations of motion are similar to those of classical mechanics. For example, consider a free particle. Using the Heisenberg equations of...
  36. B

    Finding expectation value using Heisenberg picture

    We have a particle in a harmonic oscillator potential. The eigenstates are denoted {|0>,|1>,...,|n>,...}. Initially the particle is in the state |s> = exp(-ipa)|0>, where p is the momentum operator. I need to find <x> as a function of time using the Heisenberg picture. The problem is, how do...
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