## 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:

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## Rabi’s Oscillations

Rabi oscillations: *quantum oscillations resulting from the coherent absorption and emission of photons driven by an electromagnetic wave.*

The Rabi oscillations demostrate the existence of quantum superposition states in which the system is simultaneously in two or more macroscopically different quantum states!

I report the link to a good introduction to Stimulated emission and Rabi’s oscillations made by Complex Photonic Systems (COPS) group at the University of Twente. Click > here <

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In this post I report the article “Questioni di geografia” about OECD SCIENCE, TECHNOLOGY AND INDUSTRY OUTLOOK 2008 , written by Vincenzo Moretti in his Blog. Read more…

## The Heisenberg Representation

An observable in the Schrödinger representation transforms into:

In general is not stationary even when doesn’t depend explicilty upon time.

## Evolution Operator

On the scale of quantum phenomena the evolution of a quantum system **ceases*** to be strictly causal* (There exist no net separation between the system itself and the observing instrument).

*Postulate (1): an isolated quantum system evolves in an exactly predictable manner.*

If the system was not subjected to any observation during the time interval , the ket vector representing the dynamical state of the system at the later time is exactly determined by specifying the ket vector at the previous initial time .

*Postulate (2): the linear superposition of states is preserved in the course of time.*

Consequently the correspondence between and is linear and defines a certain linear operator called the *evolution operator*.

.

For a *conservative *system the Hamiltonian does not depend explicitly upon the time. So can be deduced by the *Einstein’s postulate*: . We write the Schrödinger equation:

We postulate that the eigenvector evolves in time according to the law:

or

Consequently:

→

Differentiating we obtain:

satisfying the initial condition:

For a *non conservative *system the Hamiltonian depends explicitly upon time so the Einstein’s postulate losses all meaning. We postulate that is solution of the previous differential equation even when the system is not conservative, but is defined by the integral equation (Fredholm eq.):

→

In order that the norm of remain constant in time, it is necessary and sufficient that be Hermitian. If is Hermitian, is an *unitary operator*: