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# Kinetic Energy Partition Method for Quantum Systems with Competing Potentials

Date Issued

2015

Date

2015

Author(s)

Chen, Yu-Hsin

Abstract

For quantum systems with competing potentials, the conventional perturbation theory often yields an asymptotic series and the subsequent numerical outcome becomes uncertain. To tackle such kind of problems, we develop a general solution scheme based on a new energy dissection idea. Instead of dividing the potential energy into ""unperturbed"" and ""perturbed"" terms, a partition of the kinetic energy is performed. By distributing the kinetic energy term in part into each individual potential, the Hamiltonian can be expressed as the sum of the subsystem Hamiltonians with respective competing potentials. The total wavefunction is expanded by using a linear combination of the basis sets of respective subsystem Hamiltonians. We first illustrate the solution procedure using a simple system consisting of a particle under the action of double delta -function potentials and triple delta -function potentials . Next, this method is applied to harmonic oscillator in the infinite potential box , the hydrogen atom and 1D Stark effect. Compared with the exact solution, this new scheme converges much faster to the exact solutions for both eigenvalues and eigenfunctions. When properly extended, this new solution scheme can be very useful for dealing with strongly coupling quantum systems. In the mathematical allowed, we can shall dividing mass with ""negative"" term and ""positive"" term. Then the total Hamiltonian is conserved. Some kind of Hamiltonian system didn''t have discrete energy solution. But introduce with negative mass, we can regard as the potential energy inverse of x-axes. So the Hamiltonian system can be solved. We have 4 case to illustrate this method. The first case is antisymmetry delta -function potentials, this is a simply case to explain how to apply ""negative"" mass in the quantum system. The second case is ""positive"" $delta -function potential in the infinite potential box. The third case is Helium atom, we used negative mass to solved the electron repulsive potential energy. Optimize Helium system position we get -78.965eV Helium ground state energy, just 0.02% error with experimental value. The last case is 1D helium atom, it is a mathematical model. This model can reduce the double system integration, so there is a very persuasive example.

Subjects

Kinetic Energy Partition

Negative mass

Stark effect

Hydrogen atom

Helium ground state energy

Type

thesis

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