We study the time-dependent Schrödinger equation with matrix-valued potential presenting a generic crossing of type B, I, J or K in Hagedorn's classification. We use two-scale Wigner measures for describing the Landau–Zener energy transfer which occurs at the crossing. In particular, in the case of multiplicity 2 eigenvalues, we calculate precisely the change of polarization at the crossing. Our method provides a unified framework in which codimension 2, 3 or 5 crossings can be discussed. We recover Hagedorn's result for wave packets, from Wigner measure point of view, and extend them to any data uniformly bounded in L2. The proof is based on a normal form theorem which reduces the problem to an operator-valued Landau–Zener formula.
The visible spectral data
available for a range of bis(cyclopentadienyl)titanium(111)
derivatives (d1 systems) are discussed in relation to the proposed
one-electron energy level schemes suggested for molecules of C2v symmetry.
The interplay between two different discrete electron effects has been studied in quantum dots. These are the Coulomb blockade effect and a novel resonant tunnelling process that occurs within the dot itself. We demonstrate that the two processes are governed by two different energy scales. We find that electron–electron interactions (not the single electron energy level spectrum) are required to explain both processes.
Electron energy level calculations for semiconductor nanostructures