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InteractionRepresentation

< The density matrix | Index | Addition of angular momenta >

Let us now consider a Hamiltonian composed of two parts, the main one, {$H_0$}, describing a simple solved problem, and a perturbation {$H_1$}: {$H=H_0+H_1$} If the perturbation commutes with the main part they share the same basis of eigenvectors and the solution is trivial. It they do not commute it is useful to introduce the Interaction representation.

Let us first consider the formal time dependent density matrix for {$H$}. If the mixture of states is represented by {$\rho_0$} at time t=0,its evolution is given by

{$ (1) \qquad\qquad \frac {d\rho} {dt} = \frac i \hbar ([H_0,\rho]+ [H_1,\rho])$}

The formal solution given in the previous section would be {$\rho(t)=e^{i(H_0+H_1)t/\hbar}$}, but we would know how to write this operator only if we knew already its eigenstates. It is useful to consider another density matrix, given by an intermediate step, the so-called Interaction representation:

{$ (2) \qquad\qquad \rho_I = e^{-i H_0 t /\hbar} \rho(t) e^{i H_0 t /\hbar} $}

where we recognize the unperturbed time evolution operator {$U(t)=e^{-iH_0t/\hbar}$}. We can multiply both members on the right by {$U(t)$} and on the left by {$U(-t)=U^\dagger(t)$}, to obtain

{$ (3) \qquad\qquad U^\dagger \rho_I U= \rho(t)$}

Taking the time derivative of this last equation we may obtain the equation of motion for operators in this Interaction representation:

{$\frac i \hbar H_0 \rho + U^\dagger \frac {d\rho_I} {dt} U + \frac i \hbar \rho H_0 = \frac {d\rho} {dt}=\frac i \hbar (H_0 \rho-\rho H_0 + H_1 \rho -\rho H_1)$}

where we have used eq. (1). The result may be read after multiplying back on the right by {$U^\dagger$} and on the left by {$U$}

{$ (4) \qquad\qquad \frac {d\rho_I} {dt} = \frac i \hbar U ( H_1 \rho -\rho H_1) U^\dagger = \frac i \hbar [H_{1I},\rho_I ] $}

where we have inserted {$U^\dagger U=1$} between {$H_1$} and {$\rho$} in the middle expression.

< The density matrix | Index | Addition of angular momenta >