Scattering Phase Correction for Semiclassical Quantization Rules in Multi-Dimensional Quantum Systems

While the scattering phase for several one-dimensional potentials can be exactly derived, less is known in multi-dimensional quantum systems. This work provides a method to extend the one-dimensional phase knowledge to multi-dimensional quantization rules. The extension is illustrated in the example of Bogomolny＇s transfer operator method applied in two quantum wells bounded by step potentials of different heights. This generalized semiclassical method accurately determines the energy spectrum of the systems, which indicates the substantial role of the proposed phase correction. Theoretically, the result can be extended to other semiclassical methods, such as Gutzwiller trace formula, dynamical zeta functions, and semielassical Landauer Buttiker formula. In practice, this recipe enhances the applicability of semiclassical methods to multi-dimensional quantum systems bounded by general soft potentials.

Quantization rules for low dimensional quantum dots

This paper applies the analytical transfer matrix method （ATMM） to calculate energy eigenvalues of a particle in low dimensional sharp confining potential for the first time, and deduces the quantization rules of this system. It presents three cases in which the applied method works very well. In the first quantum dot, the energy eigenvalues and eigenfunction are obtained, and compared with those acquired from the exact numerical analysis and the WKB （Wentzel, Kramers and Brillouin） method; in the second or the third case, we get the energy eigenvalues by the ATMM, and compare them with the EBK （Einstein, Brillouin and Keller） results or the wavefunction outcomes. From the comparisons, we find that the semiclassical method （WKB, EBK or wavefunction） is inexact in such systems.

Notes on Application of WKB Method

The notes here presented are of the modifications introduced in the application of WKB method.Theproblems of two-and three-dimensional harmonic oscillator potential are revisited by WKB and the new formulationof quantization rule respectively.It is found that the energy spectrum of the radial harmonic oscillator,which isreproduced exactly by the standard WKB method with the Langer modification,is also reproduced exactly without theLanger modification via the new quantization rule approach.An alternative way to obtain the non-integral Maslov indexfor three-dimensional harmonic oscillator is proposed.

Approximate analytical solutions and mean energies of stationary Schrodinger equation for general molecular potential

The Schrodinger equation is solved with general molecular potential via the improved quantization rule.Expression for bound state energy eigenvalues,radial eigenfunctions,mean kinetic energy,and potential energy are obtained in compact form.In modeling the centrifugal term of the effective potential,a Pekeris-like approximation scheme is applied.Also,we use the Hellmann–Feynman theorem to derive the relation for expectation values.Bound state energy eigenvalues,wave functions and meanenergies of Woods–Saxon potential,Morse potential,Mobius squared and Tietz–Hua oscillators are deduced from the general molecular potential.In addition,we use our equations to compute the bound state energy eigenvalues and expectation values for four diatomic molecules viz.H_(2),CO,HF,and O_(2).Results obtained are in perfect agreement with the data available from the literature for the potentials and molecules.Studies also show that as the vibrational quantum number increases,the mean kinetic energy for the system in a Tietz–Hua potential increases slowly to a threshold value and then decreases.But in a Morse potential,the mean kinetic energy increases linearly with vibrational quantum number increasing.