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

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

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

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

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

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

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

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

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

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