Related Experiment Video
Updated: May 9, 2026

Liquid-cell Transmission Electron Microscopy for Tracking Self-assembly of Nanoparticles
Published on: October 16, 2017
Semiclassical trace formula for the two-dimensional radial power-law potentials
A G Magner1, A A Vlasenko, K Arita
1Institute for Nuclear Research, 03680 Kiev, Ukraine. magner@kinr.kiev.ua
Abstract:
The trace formula for the density of single-particle levels in the two-dimensional radial power-law potentials, which nicely approximate up to a constant shift the radial dependence of the Woods-Saxon potential and its quantum spectra in a bound region, was derived by the improved stationary phase method. The specific analytical results are obtained for the powers α=4 and 6. The enhancement of periodic-orbit contribution to the level density near the bifurcations are found to be significant for the description of the fine shell structure. The semiclassical trace formulas for the shell corrections to the level density and the energy of many-fermion systems reproduce the quantum results with good accuracy through all the bifurcation (symmetry breaking) catastrophe points, where the standard stationary-phase method breaks down. Various limits (including the harmonic oscillator and the spherical billiard) are obtained from the same analytical trace formula.
Related Concept Videos
Velocity Potential
Gravitational Potential Energy for Extended Objects
Poisson's And Laplace's Equation
Calculations of Electric Potential I
The ring is divided into infinitesimal small arcs such that point M is equidistant from all the arcs. Here, the cylindrical coordinate system is used to calculate the electric potential at point M. A general element of the arc between angles θ and θ + dθ is of the length Rdθ and has a charge of λRdθ.
Orthogonal Trajectories
Differential Form of Maxwell's Equations

