[NOTES/QM-09007] Interaction Picture of Quantum Mechanics

Pictures in quantum mechanics

We recall that the time evolution of quantum states is given by the time dependent Schr\"{o}dinger equation. \begin{equation}\label{eq01} i\hbar\pp[\ket{\psi}]{t} = H \ket{\psi}.\end{equation} and the operators do not carry any time dependence. The dynamical variables, represented by operators are specified once for all and do not evolve with time. This 'picture' of time evolution is called Schrodinger picture. Recall that the state vector by itself is not measurable quantity. Only the average values and the probabilities are measurable quantities. This allows the possibility of describing the time dependence in several ways.Heisenberg \cite{hpic} and Dirac pictures are two important alternate descriptions of time evolution.


We shall now discuss the interaction picture, also known as Dirac picture. We shall denoting the Schrodinger picture kets and operators by \(\ket{\psi}_S, X_S\) etc. and \(\ket{\psi}_I, X_I\) etc will denote the corresponding quantities in the interaction picture. Let the Hamiltonian of the system be written as sum of two parts \begin{equation}\label{eq02} H = H_0 + H'. \end{equation} \(H_0, H'\) will be called free part and the interaction part of the Hamiltonian \(H\), respectively. While \(H_0\) is assumed to be independent of time, the interaction Hamiltonian may or may not depend on time. The state of a system in the interaction picture is defined by \begin{equation}\label{eq03} \ket{\psi t}_I = e^{iH_0t/\hbar}\ket{\psi t}_S. \end{equation}

Dynamical variables in interaction picture

The dynamical variables of the interaction picture are defined by demanding that the average values in the interaction and Schr\"{o}dinger pictures coincide at all times: \begin{equation}\label{eq04} _I\average{\psi}{X_I}_I\equiv _S\average{\psi}{X_I}_S. \end{equation} Substituting \begin{equation}\label{eq05} \ket{\psi t}_S= e^{-iH_0t/\hbar}\ket{\psi t}_I, \end{equation} in \eqref{eq03}, we get \begin{equation}\label{eq06} _I\average{\psi}{X_I}_I= {_I\hspace{-0.5mm}\average{\psi}{e^{iH_0t/\hbar}X_Ie^{-iH_0t/\hbar}}}_I. \end{equation} Therefore, we use \begin{equation}\label{eq07} X_I = e^{iH_0t/\hbar} X_S e^{-iH_0t/\hbar} \end{equation} to define the an interaction picture dynamical variables. The time dependence of the interaction picture operators is very simple and is governed by the free Hamiltonian \(H_0\): \begin{equation}\label{eq08} i\hbar \dd[X_I(t)]{t} = \big[X_I, H_0\big]. \end{equation} As an example, it should be obvious that, the free particle Hamiltonian \(H_0\) in the interaction picture remains identical with \(H_0\): \begin{equation}\label{eq09} (H_0)_I = e^{iH_0t/\hbar}H_0 e^{-iH_0t/\hbar} = H_0. \end{equation} On the other hand, even though the interaction part of the Hamiltonian, \(H'\), may be independent of time, the interaction picture Hamiltoinan \(H'_I\) \begin{equation}\label{eq10} H'_I(t) = e^{iH_0t/\hbar}H'e^{-iH_0t/\hbar} \end{equation} is different from \(H'\) and depends explicitly on time.

Time Evolution of States

We will now derive the differential equation which gives the evolution of the state vectors. For this purpose we begin with \eqref{eq03} \begin{eqnarray}\nonumber i\hbar\dd{t} \ket{\psi t}_I &=& i\hbar \dd{t} \Big( e^{iH_0t/\hbar}\ket{\psi t}_S\Big) \\ &=&i\hbar\, \Big(\dd{t}e^{iH_0t/\hbar}\Big)\ket{\psi t}_S + e^{iH_0t/\hbar}\Big(i\hbar\dd{t}\ket{\psi t}_S\Big) + \\ &=& -H_0e^{iH_0t/\hbar}\ket{\psi_0}_S + e^{iH_0t/\hbar}\big(H_0+H'\big)\ket{\psi t}_S\nonumber\\ &=& e^{iH_0t/\hbar}(-H_0)\ket{\psi_0}_S + e^{iH_0t/\hbar}\big(H_0+H'\big)\ket{\psi t}_S\nonumber\\ &=& e^{iH_0t/\hbar}\big(H'\big)\ket{\psi t}_S \end{eqnarray} Next, we need to express the Schr\"{o}dinger picture state vector,\(\ket{\psi t}_S\), in the right hand side, in terms of the interaction picture state vector \(\ket{\psi t}_I\). Thus we get \begin{eqnarray} i\hbar\dd{t} \ket{\psi t}_I &=& e^{iH_0t/\hbar}\big(H'\big)e^{-iH_0t/\hbar}\ket{\psi t}_I. \end{eqnarray} Thus we get the desired equation for time evolution of the state vectors in the interaction picture in the final form \begin{eqnarray} \boxed{ i\hbar\dd{t} \ket{\psi t}_I = H'_I \ket{\psi t}_I} \end{eqnarray} where \(H'_I\) is given by \eqref{eq10}. Thus we see that the time dependence of the state vector in the interaction picture is governed by the operator \(H'_I\), the interaction part of Hamiltonian,\(H'\), transformed to the interaction picture.