On the “new Tamm-Dancoff” approximation treatment of the superconductivity Hamiltonian

H. Mitter, K. Yamazaki

A. Friederich, W. Gerling, K. Bleuler

B. P. Nigam

The Tamm-Dancoff approximation is applied to the interaction Hamiltonian used by Bogoliubov in his theory of superconductivity. It is shown that (i) the requirement that there is no virtual production of a single pair of electron and hole without phonons, or (ii) the electron-phonon coupling parameter tends to zero, leads to essentially the same compensating equation as obtained by Bogoliubov.

N. N. Bogolyubov

D. Ya. Petrina

Stijn De Baerdemacker

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.