Interaction picture

The interaction picture (also called interaction representation, Dirac representation or Dirac representation called ) of quantum mechanics is a model for dealing with time-dependent problems, taking into account interactions.

It was introduced in 1926 by Paul Dirac to quantum mechanics. In the context of quantum electrodynamics the interaction picture by Tomonaga, Dirac and ( in an unpublished work as a student at City College of New York) was introduced by Julian Schwinger (1934 ). The treatment of relativistic quantum field theory in the interaction picture with second quantization was then input into the standard textbooks.

In the interaction picture, the following assumptions apply:

  • The system of the Hamiltonian given by where the time-dependent Hamiltonian of the unperturbed system, and the disturbance caused by the interaction described, it can be time-dependent. However, it can also be useful without an interaction is present, to hold such a formal splitting of the Hamiltonian.
  • States are time-dependent:
  • Operators are also time-dependent:
  • The dynamics of states is described by the adjusted Schrödinger equation, while the dynamics of the operators is given by the adjusted Heisenberg equation of motion.
  • Only certain calculations are easier to perform in the Dirac picture. The best example here is the derivation of the time-dependent perturbation theory.

To indicate that one uses the interaction picture, are states and operators occasionally with the index " I" or " D" marked ( as Dirac picture) ( as engl interaction. ): Or

The sense of the image is to insert the time evolution of the system, caused by the temporal dependence of the operators, while the time dependence of induced received in the development of the condition. This is just another picture and describes " the same physics ", ie all physically relevant quantities ( scalar, eigenvalues ​​, etc.) remain the same.

It defines two time evolution operators:

  • The "normal" which, as explained in the time evolution operator is defined by:
  • The time evolution operator generated only by:

The expectation value a of the operator must be the same in all pictures:

The time-dependent operator ( as in the Heisenberg picture ) is given by:

The time-dependent state can only indirectly - fully describing the dynamics of the state to share his time caused development on the reduction of the ( in the Schrödinger picture) - are defined:

This makes it possible to define the operator:

The time- independent portion of the Hamiltonian in the interaction picture is identical to that in the Schrödinger picture:

The dynamics of states is (similar to the Schrödinger picture) described by the equation:

The dynamics of the operators (such as in the Heisenberg picture) described by the Heisenberg equation of motion, with the time-independent Hamiltonian which describes the unperturbed system:

With the Dirac picture goes on in the Heisenberg picture.

At the time tune all three images match:

Derivation of the equations of motion

In preparation, the time derivatives are determined by and:

Equation of motion for the states:

Equation of motion for the operators:

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