# Interaction (Intermediate) Representation

###### From A. Messiah’s Quantum Mechanics p.321 and 722 ff. , Dover (1999)

*Any unitary transformation of the vectors and the observables of the Schrödinger (or Heisenberg) representations defines a new representation.*

*Any problem of quantum mechanics essentially consists of the determination of the properties of the unitary operator . **

In the **intermediate representation**, we suppose that the Hamiltonian can be put in the form:

where is the Hamiltonian of a Schrödinger equation whose solution is know.

Let be the evolution operator corresponding to . We obtain vectors and observables in the interaction representation by applying the unitary tranformation to the Schrödinger* *ones.

The evolution operator for the states in the intermediate representation is:

The time dependence of is determined by:

The vectors in the intermediate representation *move* in time satisfying the Schrödinger* *equation:

On the other hand, the physical quantities are represented by *moving observables* that are subject to the Heisenberg equation of motion written with the “unperturbed” Hamiltonian :

…

The evolution operator in the interaction representation is:

→

This Integral equation can be solved by iteration.

To fit with the introduction of the C. Kittel’s book *Quantum Theory of Solids* and Abrikosov’s *Methods of Quantum Field Theory in Statistical Physics*, we note that, if is time independent and putting we have: