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Interaction (Intermediate) Representation

December 11, 2008
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 U(t,t_0). *

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

\displaystyle H(t)=H_0(t)+V(t)

where H_0(t) is the Hamiltonian of a Schrödinger equation whose solution is know.

Let U_0(t,t_0) be the evolution operator corresponding to H_0. We obtain vectors and observables in the interaction representation by applying the unitary tranformation U_0^{\dagger}(t,t_0) to the Schrödinger ones.

\displaystyle \left|\psi_{\mathrm{I}}(t)\right>=U_0^{\dagger}(t,t_0)\left|\psi_S(t)\right>

\displaystyle A_{\mathrm{I}}(t)=U_0^{\dagger}(t,t_0)A_S U_0(t,t_0)

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

\displaystyle U_{\mathrm{I}}(t, t_0)=U_0^{\dagger}(t,t_0)U(t,t_0) 

The time dependence of U_{\mathrm{I}} is determined by:

\displaystyle V_{\mathrm{I}}(t)=U_0^{\dagger}(t,t_0) V(t) U_0(t,t_0)

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

\displaystyle i\hbar\frac{\mathrm{d}}{\mathrm{d}t}\left|\psi_{\mathrm{I}}(t)\right>=V_{\mathrm{I}}(t)\left|\psi_{\mathrm{I}}(t)\right>

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 H^{0}_{\mathrm{I}}(t):

The evolution operator in the interaction representation is:

\displaystyle U_{\mathrm{I}}(t,t_0)=1-i \hbar^{-1} \int_{t_0}^t V_{\mathrm{I}}(t') U_{\mathrm{I}}(t',t_0) dt'

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 H_0 is time independent and putting t_0=0 we have:

\displaystyle U_0(0,t)=e^{-iH_0/\hbar}

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