The interaction picture

Consider a quantum system described by a time-dependent Hamiltonian of the form

$${\rm H}(t) = {\rm H}_0 + {\rm H}_1(t)$$

In the language of perturbation theory, ${\rm H}_0$ is known as the unperturbed Hamiltonian and describes a system of interest such as a molecule or a condensed-phase sample such as a pure liquid or solid or a solution. ${\rm H}_1(t)$ is known as the perturbation, and it often describes an external system, such as a laser field, that will be used to probe the energy levels and other properties of ${\rm H}_0$.

We now seek a solution to the time-dependent Schrödinger equation

$${\rm H}(t)\vert\Psi(t)\rangle = \left({\rm H}_0 + {\rm H}_1(... ...\rangle = i\hbar{\partial \over \partial t}\vert\Psi(t)\rangle$$ (1)

subject to an initial state vector $\vert\Psi(t_0)\rangle$. In order to solve the equation, we introduce a new state vector $\vert\Phi(t)\rangle$ related to $\vert\Psi(t)\rangle$ by

$$\vert\Psi(t)\rangle = e^{-i{\rm H}_0(t-t_0)}\vert\Phi(t)\rangle$$ (2)

The new state vector $\vert\Phi(t)\rangle$ is an equally valid representation of the state of the system. In Chapter 10, we introduced the concept of pictures in quantum mechanics and discussed the difference between the Schrödinger and Heisenberg pictures. Eqn. (2) represents yet another picture of quantum mechanics, namely the interaction picture. Like the Schrödinger and Heisenberg pictures, the interaction picture is a perfectly valid way of representing a quantum mechanical system. The interaction picture can be considered as ``intermediate'' between the Schröginer picture, where the state evolves in time and the operators are static, and the Heisenberg picture, where the state vector is static and the operators evolve. However, as we will see shortly, in the interaction picture, both the state vector and the operators evolve in time, however, the time-evolution is determined by the perturbation ${\rm H}_1(t)$. Eqn. (2) specifies how to transform between the Schrödinger and interaction picture state vectors. The transformation of operators proceeds in an analogous fashion. If ${\rm A}$ denotes an operator in the Schrödinger picture, its representation in the interaction picture is given by

$${\rm A}_I(t) = e^{i{\rm H}_0(t-t_0)/\hbar}{\rm A}e^{-i{\rm H}_0(t-t_0)/\hbar}$$ (3)

which is equivalent to an equation of motion of the form

$${d{\rm A}_I(t) \over dt} = {1 \over i\hbar}[{\rm A}_I(t),{\rm H}_0]$$ (4)

Substitution of Eqn. (2) into the time-dependent Schrödinger equation yields

$$\left({\rm H}_0+{\rm H}_1(t)\right)e^{-i{\rm H}_0(t-t_0)/\hbar}\vert\Phi(t)\rangle = {\rm H}_0e^{-i{\rm H}_0(t-t_0)/\hbar}\vert\Phi(t)\rangle + e^{-i{\rm H}_0(t-t_0)/\hbar} i\hbar{\partial \over \partial t}\vert\Phi(t)\rangle$$

$${\rm H}_1(t)e^{-iH_0(t-t_0)/\hbar}\vert\Phi(t)\rangle = e^{-iH_0(t-t_0)/\hbar}i\hbar{\partial \over \partial t}\vert\Phi(t)\rangle$$

$$e^{iH_0(t-t_0)/\hbar}{\rm H}_1(t)e^{-iH_0(t-t_0)/\hbar}\vert\Phi(t)\rangle = i\hbar{\partial \over \partial t}\vert\Phi(t)\rangle$$ (5)

According to Eqn. (3), the $\exp[i{\rm H}_0(t-t_0)/\hbar]{\rm H}_1(t)\exp[-i{\rm H}_0(t-t_0)/\hbar]$ is the interaction-picture representation of the perturbation Hamiltonian, and we will denote this operator as $H_I(t)$. Thus, the time-evolution of the state vector in the interaction picture is given a Schrödinger equation of the form

$$H_I(t)\vert\Phi(t)\rangle = i\hbar{\partial \over \partial t}\vert\Phi(t)\rangle$$ (6)

The initial condition to Eqn. (6), $\vert\Phi(t_0)\rangle$ is, according to Eqn. (2), also $\vert\Psi(t_0)\rangle$. In the next section, we will develop an iterative solution to Eqn. (6), which will reveal a rich structure of the propagator for time-dependent systems.