Use of Lambert’s theorem for then-dimensional Coulomb problem

Abstract

We present the analytical solution in closed form for the semiclassical limit of the quantum mechanical Coulomb Green function in position space in n dimensions. We utilize a projection method which has its roots in Lambert's theorem and which allows us to treat the system as an essentially one dimensional problem. The semiclassical result assumes a simple analytical form and is well suited for a numerical evaluation. The method can also be extended to classically forbidden space regions. Already for moderately large principal quantum numbers ν ≥ 5, the semiclassical Green function is found to be an excellent approximation to the quantum mechanical Green function.

Trilobites, butterflies, and other exotic specimens of long-range Rydberg molecules

“…6) calls to mind a trilobite fossil or a butterfly with spread wings, respectively 28 . Their distinctive nodal patterns have attracted interest due to their deep connections to periodic orbit theory and to a near separability of the diatomic Hamiltonian in elliptic coordinates [134,135]. One underappreciated point is the extent to which these wave functions localize the wave function near the perturber.…”

“…Their appearance in cylindrical coordinates (plotted as 'shadows' in figure 5 and more explicitly as surface plots in the insets of figure 6) calls to mind a trilobite fossil or a butterfly with spread wings, respectively 30 . Their distinctive nodal patterns have attracted interest due to their deep connections to periodic orbit theory and to a near separability of the diatomic Hamiltonian in elliptic coordinates [134,135]. One underappreciated point is 29 We eliminate second and third derivatives using the radial Schrödinger equation.…”

Ultralong-range Rydberg molecules

“…1 can be understood semiclassically. In a path integral picture ψ(r; R) results from interference of electron states associated to elliptical Keppler trajectories that pass simultaneously through the points r and R. Minima in the PECs arise when these tractories satisfy semiclassical Einstein-Brillouin-Keller quantization conditions which imply that the wave function has a certain integer number of nodes n ξ and n η along two different elliptical directions [78,79]. Interestingly, trilobite wave functions can also be realized by applying a sequence of electric and magnetic field pulses even when the groundstate atom is not present [80].…”

Transient capture of electrons in magnetic fields, or: comets in the restricted three-body problem

“…The semiclassical method for quantization provides the link between classical and quantum mechanics and highlights the role of closed orbits for the determination of eigenvalues and eigenfunctions. For systems where the electrons are not bound photodetachment and photoionization processes are amendable to a semiclassical treatment [1,2,3,4,5,6,7,8]. A spectacular experimental realization of the macroscopic extension of the photodetachment wavefunction was shown by Blondel et al [9,10].…”