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THE APPLICATION OF THE TAMM-DANCOFF METHOD IN THE QUANTUM THEORY OF NON- LINEAR WAVE EQUATIONS

Károly Ladányi

Published Oct 1, 1958• 0 citations
OpenAlex

Abstract

No abstract. 13:005086A mathematical analysis is made of spontaneous photon decay into two photons in a uniform external field. (H.V.J.)

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The tunneling effect is the most popular phenomenon of quantum physics and is present in modern physical theories. Still, the most important features of this effect are already present in toy models - low dimensional quantum mechanics with high potential barriers. Even these simple systems cannot be solved analytically and approximations have to be made. Tunneling is closely related to splitting of the ground state energy which is degenerate in the classical limit. In this thesis we use two methods to estimate the energy splitting in one dimensional quantum mechanics. In semi-classical approximation the splitting is calculated in the weak coupling limit. The other approach, Tamm-Dancoff method gives very precise numerical results for finite coupling. Comparing these two approaches we are able to confirm results of the semi-classical approximation and determine range of its applicability. Several potentials are considered: anharmonic double well, cosine potential and cosine potential in periodic space with finite number of minima. We also address a non-generic problem of anharmonic triple well potential.
A connection is made between the exact eigen states of the BCS Hamiltonian and the predictions made by the Tamm-Dancoff Approximation. This connection is made by means of a parametrised algebra, which gives the exact quasi-spin algebra in one limit of the parameter and the Heisenberg-Weyl algebra in the other. Using this algebra to construct the Bethe Ansatz solution of the BCS Hamiltonian, we obtain parametrised Richardson-Gaudin equations, leading to the secular equation of the Tamm-Dancoff Approximation in the bosonic limit. An example is discussed in depth.