What is heisenberg picture: Definition and 36 Discussions

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. 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. 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. 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. 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. 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...

13. 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. 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. 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. 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. 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. 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. 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. Heisenberg Picture Formalism: Rewriting Operators in Covariance

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