Some questions about the heisenberg picture

In summary, the Heisenberg picture of quantum mechanics involves time-dependent eigenvectors and the calculation of probabilities for measuring eigenvalues. The initial data of the system is contained in the operators for initial value problems, and thorough worked examples can be found in lecture notes such as the one provided in the conversation.
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HomogenousCow
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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 equivalently the initial values of their time dependant eigenvectors)
3.Could someone link me to some thorough worked examples of problems in the Heisenberg picture, the ones readily available over google tend to be short and unsatisfying.
Thanks

Edit: Found some good lecture notes for #3
Here's the link of anyone wants them http://www.physics.usu.edu/torre/Quantum Mechanics/6210_Spring_2008/Lectures/15.pdf
 
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1. What is the Heisenberg picture?

The Heisenberg picture is one of the two formulations of quantum mechanics, the other being the Schrödinger picture. In this picture, the operators representing physical observables, such as position and momentum, evolve in time while the state vector remains constant.

2. How is the Heisenberg picture different from the Schrödinger picture?

In the Heisenberg picture, the operators evolve in time while the state vector remains constant, whereas in the Schrödinger picture, the state vector evolves in time while the operators remain constant. This leads to different equations of motion for the operators and different interpretations of physical observables.

3. What is the significance of the Heisenberg uncertainty principle in the Heisenberg picture?

The Heisenberg uncertainty principle, which states that the more precisely one property of a particle is measured, the less precisely the other property can be known, is a fundamental principle in the Heisenberg picture. This is because the operators representing these properties do not commute, and their uncertainties are related by the uncertainty principle.

4. How does the Heisenberg picture handle time-dependent Hamiltonians?

In the Heisenberg picture, the operators representing physical observables evolve in time according to the Heisenberg equations of motion, which are derived from the time-dependent Schrödinger equation. This allows for the handling of time-dependent Hamiltonians, which describe the energy of a system.

5. What are some practical applications of the Heisenberg picture?

The Heisenberg picture is commonly used in quantum field theory, where it allows for the calculation of time-dependent quantities such as scattering amplitudes. It is also used in the study of quantum systems, particularly in the analysis of quantum harmonic oscillators and quantum entanglement.

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