Interaction Hamiltonian in quantum physics
Hsiang Shun Chou1*
1Institute of optoelectronic sciences, National Taiwan Ocean University, Keelung, Taiwan
* presenting author:Hsiang Shun Chou, email:hschou@mail.ntou.edu.tw
In the conventional formulation of quantum transitions, the Coulomb and the Lamb gauges lead to the velocity-form and the length-form interaction Hamiltonians, respectively. The matrix elements of the two interaction Hamiltonians differ by the ratio of the transition frequency over the field frequency. In the literatures [1-3], it was asserted that the conventional formulation requires the length-form interaction Hamiltonian in order to obtain the correct transition amplitudes. In this paper, we show that it is misleading. We propose a gauge-invariant procedure for calculating the transition ampltudes. We demonstrate that in the conventional formulation, the velocity-form interaction Hamiltonian is correct in any gauge. In the electric-dipole approximation, the gauge-invariant procedure yields the modified length-form interaction Hamiltinian.

[1] W. E. Lamb, Jr. , Phys. Rev. 85, 259 (1952).
[2] K. H. Yang, Ann. Phys. (N. Y.) 101, 62 (1976).
[3] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms-Introduction to Quantum Electrodynamics (John Wiley & Sons, 1989), pp. 269-271.


Keywords: Interaction Hamiltonian, Gauge invariance