B.2 Interaction Picture

In the interaction representation both the state vectors and the operators are time-dependent. The state vector in the interaction representation is given by

\begin{displaymath}\begin{array}{l} \displaystyle \vert\Psi_\mathrm{I}(t)\rangle...
...at{H}_{0}t/\hbar}\vert\Psi_\mathrm{S}(t)\rangle \ , \end{array}\end{displaymath} (B.5)

which is merely a unitary transformation at the time $ t$. The equation of motion of this state vector is found by taking the time derivative

\begin{displaymath}\begin{array}{ll} \displaystyle i\hbar\partial_t \vert\Psi_\m...
...at{H}_{0}t/\hbar}\vert\Psi_\mathrm{I}(t)\rangle \ . \end{array}\end{displaymath} (B.6)

Therefore, one obtains the following set of equations in the interaction picture

\begin{displaymath}\begin{array}{l} \displaystyle i\hbar\partial_t\vert\Psi_\mat...
...r} \hat{H}^\mathrm{int}e^{-i\hat{H}_{0}t/\hbar} \ . \end{array}\end{displaymath} (B.7)

An arbitrary matrix element in the SCHRÖDINGER picture can be written as

\begin{displaymath}\begin{array}{l}\displaystyle \langle\Psi_\mathrm{S}^{'}(t) \...
...at{H}_{0}t/\hbar}\vert\Psi_\mathrm{I}(t)\rangle \ , \end{array}\end{displaymath} (B.8)

which suggests the following definition of an operator in the interaction picture

\begin{displaymath}\begin{array}{l}\displaystyle \hat{O}_\mathrm{I}(t) \ = \ e^{...
...bar} \hat{O}_\mathrm{S}e^{-i\hat{H}_{0}t/\hbar} \ . \end{array}\end{displaymath} (B.9)

M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors