To be more specific, in this paper, we aim at extending the Ehrenfest theorem in one dimension to two dimensions.
The rest of the paper is organized as follows: Section 2 presents the methodology of the Ehrenfest theorem in two dimensions; in Section 3, we apply this methodology to a linear and quadratic coupled quantum dot to harmonic potential and present the numerical study of the semiquantum equations' motion.
In this paper, semiquantum chaos in two GaAs quantum dots coupled linearly and quadratically by two harmonic potentials was investigated using the Ehrenfest theorem. This work connected two distant concepts in physics: the quite popular double-dot potential in semiconductor structures proposed by Loss and DiVincenzo for qubits in a quantum computer and quantum chaos.
In Born-Oppenheimer molecular dynamics [6, 10, 11], it is assumed that the adiabatic and the Born-Oppenheimer approximations are valid and that the nuclei follow a semi-classical Newton equation whose potential is determined by the Ehrenfest theorem. It is further assumed that the electronic wave function is in its ground state (lowest energy).
The first term of the Lagrangian corresponds to the kinetic energy of the nuclei, the second term corresponds to the nuclear potential as obtained from the Ehrenfest theorem, and the last term is a Lagrange function which ensures that the orbitals remain orthonormal at all time.