Relations Between Low-lying Quantum Wave Functions and Solutions of the Hamilton-Jacobi Equation

We discuss a new relation between the low lying Schroedinger wave function of a particle in a one-dimentional potential V and the solution of the corresponding Hamilton-Jacobi equation with -V as its potential. The function V is $\geq 0$, and can have several minina (V=0). We assume the problem to be characterized by a small anhamornicity parameter $g^{-1}$ and a much smaller quantum tunneling parameter $\epsilon$ between these different minima. Expanding either the wave function or its energy as a formal double power series in $g^{-1}$ and $\epsilon$, we show how the coefficients of $g^{-m}\epsilon^n$ in such an expansion can be expressed in terms of definite integrals, with leading order term determined by the classical solution of the Hamilton-Jacobi equation. A detailed analysis is given for the particular example of quartic potential $V={1/2}g^2(x^2-a^2)^2$.Comment: LaTex, 48 pages, no figur

Spin Exciton in quantum dot with spin orbit coupling in high magnetic field

Coulomb interactions of few ($N$) electrons confined in a disk shaped quantum dot, with a large magnetic field $B=B^*$ applied in the z-direction (orthogonal to the dot), produce a fully spin polarized ground state. We numerically study the splitting of the levels corresponding to the multiplet of total spin $S=N/2$ (each labeled by a different total angular momentum $J_z$) in presence of an electric field parallel to $B$, coupled to $S$ by a Rashba term. We find that the first excited state is a spin exciton with a reversed spin at the origin. This is reminiscent of the Quantum Hall Ferromagnet at filling one which has the skyrmion-like state as its first excited state. The spin exciton level can be tuned with the electric field and infrared radiation can provide energy and angular momentum to excite it.Comment: 9 pages, 9 figures. submitted to Phys.Rev.