History and Some Aspects of the Lamb Shift

Abstract

Radiation is a process common to classical and quantum systems with very different effects in each regime. In a quantum system, the interaction of a bound electron with its own radiation field leads to complex shifts in the energy levels of the electron, with the real part of the shift corresponding to a shift in the energy level and the imaginary part to the width of the energy level. The most celebrated radiative shift is the Lamb shift between the 2 s 1 / 2 and the 2 p 1 / 2 levels of the hydrogen atom. The measurement of this shift in 1947 by Willis Lamb Jr. proved that the prediction by Dirac theory that the energy levels were degenerate was incorrect. Hans Bethe’s calculation of the shift showed how to deal with the divergences plaguing the existing theories and led to the understanding that interactions with the zero-point vacuum field, the lowest energy state of the quantized electromagnetic field, have measurable effects, not just resetting the zero of energy. This understanding led to the development of modern quantum electrodynamics (QED). This historical pedagogic paper explores the history of Bethe’s calculation and its significance. It explores radiative effects in classical and quantum systems from different perspectives, with the emphasis on understanding the fundamental physical phenomena. Illustrations are drawn from systems with central forces, the H atom, and the three-dimensional harmonic oscillator. A first-order QED calculation of the complex radiative shift for a spinless electron is explored using the equations of motion and the m a s s 2 operator, describing the fundamental phenomena involved, and relating the results to Feynman diagrams.

The Analytical Modeling of Nuclear Size Effect on Relativistic Electron in Hydrogen Atom

“…The simple distribution (1) gives the correct analytical behavior of the electronic wavefunctions at r = 0 and has been used in many early analyses. These expansions are also useful for a general understanding of the effects involved [32].…”

“…The hydrogen atom is the fundamental two-body system and perhaps the most important tool of atomic physics and the challenge is to calculate its properties to the highest accuracy possible. The study of the hydrogen atom has been at the heart of the development of modern physics [1]. The hydrogen atom has been the testing ground for theoretical atomic physics for over a hundred years.…”

Dynamical Symmetries of the H Atom, One of the Most Important Tools of Modern Physics: SO(4) to SO(4,2), Background, Theory, and Use in Calculating Radiative Shifts

“…Understanding the atomic spectra of the hydrogen atom drove the discovery of quantum mechanics in the 1920's. The measurement of the Lamb shift in 1947 and its explanation by Bethe in terms of atom's interaction with the quantum vacuum fluctuations ushered in a revolution in quantum electrodynamics [5][6][7]. Exploring the symmetries of the hydrogen atom has been an essential part of this progress.…”

“…This radiative shift accounted for about 96% of the measured shift. The insight that one needed to include the interaction of the atom with the vacuum fluctuations and how one could actually do it ushered in the modern world of quantum electrodynamics [7]. Here, we compute in the non-relativistic dipole approximation and to first order in the radiation field, as did Bethe, the radiative shift, but we use group theoretical methods based on the SO(4,2) symmetry of the non-relativistic hydrogen atom as developed in this paper.…”

“…The imaginary part of the shift gives the width of the level [7]. The matrix element Ω NL can be converted to a matrix element of a function of the generators Γ A taken between eigenstates |nlm) of (Zα) −1 .…”