B.1 SCHRÖDINGER Picture

In the SCHRÖDINGER picture the operators $\hat{O}_\mathrm{S}$ are time-independent

$$\begin{array}{l} \displaystyle \hat{O}_\mathrm{S}(t)\ =\ \hat{O}_\mathrm{S}(t_0)\ = \ \hat{O}_\mathrm{S} \ , \end{array}$$ (B.2)

where $t_0$ is assumed to be the time reference point. The time dependence of the state vector $\Psi_\mathrm{S}(t)$ is obtained from the SCHRÖDINGER equation

$$\begin{array}{l} \displaystyle i\hbar \partial_t \vert\Psi_\m... ...e \ = \ \hat{H} \vert\Psi_\mathrm{S}(t) \rangle \ , \end{array}$$ (B.3)

which has the formal solution

$$\begin{array}{ll}\displaystyle \vert\Psi_\mathrm{S}(t)\rangle... ...at{H}(t-t_0)/\hbar}\vert\Psi_\mathrm{S}(t_0)\rangle \end{array}$$ (B.4)

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