B.3 HEISENBERG Picture

In the HEISENBERG representation state vectors are defined as

$$\begin{array}{l}\displaystyle \displaystyle \vert\Psi_\mathrm... ...{i\hat{H}t/\hbar} \vert\Psi_\mathrm{S}(t)\rangle \ .\end{array}$$ (B.10)

Its time derivative may be combined with (B.3) to yield $i\hbar\partial_t\vert\Psi_\mathrm{H}(t)\rangle = 0$, which shows that $\vert\Psi_\mathrm{H}(t)\rangle$ is time-independent. Since an arbitrary matrix element in the SCHRÖDINGER picture can be written as

$$\begin{array}{l}\displaystyle \langle\Psi_\mathrm{S}^{'}(t) \... ...-i\hat{H}t/\hbar}\vert\Psi_\mathrm{H}(t)\rangle \ , \end{array}$$ (B.11)

a general operator in the HEISENBERG picture is given by

$$\begin{array}{l}\displaystyle \hat{O}_\mathrm{H}(t) \ = \ e^{... ...H}t/\hbar}\hat{O}_\mathrm{S}e^{-i\hat{H}t/\hbar}\ . \end{array}$$ (B.12)

Equation (B.12) can be rewritten in terms of the interaction picture operators

$$\begin{array}{l} \displaystyle \hat{O}_\mathrm{H}(t) \display... ...(t) e^{i\hat{H}_0 t/\hbar} e^{-i\hat{H}t/\hbar} \ , \end{array}$$ (B.13)

or in terms of the operator $\hat{S}$ derived in the next section

$$\begin{array}{l}\displaystyle \hat{O}_\mathrm{H}(t) \ = \ \hat{S}(0,t) \hat{O}_\mathrm{I}(t) \hat{S}(t,0)\ . \end{array}$$ (B.14)

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