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 hydrogen atom has been at the heart of modern physics. Exploring the symmetry of the most fundamental two body system has led to advances in atomic physics, quantum mechanics, quantum electrodynamics, and elementary particle physics. In this pedagogic review we present an integrated treatment of the symmetries of the Schrodinger hydrogen atom, including the classical atom, the SO(4) degeneracy group, the non-invariance group or spectrum generating group SO(4,1) and the expanded group SO(4,2). After giving a brief history of these discoveries, most of which took place from 1935-1975, we focus on the physics of the hydrogen atom, providing a background discussion of the symmetries, providing explicit expressions for all the manifestly Hermitian generators in terms of position and momenta operators in a Cartesian space, explaining the action of the generators on the basis states, and giving a unified treatment of the bound and continuum states in terms of eigenfunctions that have the same quantum numbers as the ordinary bound states. We present some new results from SO(4,2) group theory that are useful in a practical application, the computation of the first order Lamb shift in the hydrogen atom. By using SO(4,2) methods, we are able to obtain a generating function for the radiative shift for all levels. Students, non-experts and the new generation of scientists may find the clearer, integrated presentation of the symmetries of the hydrogen atom helpful and illuminating. Experts will find new perspectives, even some surprises.Comment: 68 pages, 8 figures, has corrections to the typographical errors in the version published in Symmetr

A Gedanken spacecraft that operates using the quantum vacuum (Dynamic Casimir effect)

Conventional rockets are not a suitable technology for deep space missions. Chemical rockets require a very large weight of propellant, travel very slowly compared to light speed, and require significant energy to maintain operation over periods of years. For example, the 722 kg Voyager spacecraft required 13,600 kg of propellant to launch and would take about 80,000 years to reach the nearest star, Proxima Centauri, about 4.3 light years away. There have been various attempts at developing ideas on which one might base a spacecraft that would permit deep space travel, such as spacewarps. In this paper we consider another suggestion from science fiction and explore how the quantum vacuum might be utilized in the creation of a novel spacecraft. The spacecraft is based on the dynamic Casimir effect, in which electromagnetic radiation is emitted when an uncharged mirror is properly accelerated in the vacuum. The radiative reaction produces a dissipative force on the mirror that tends to resist the acceleration of the mirror. This force can be used to accelerate a spacecraft attached to the mirror. We also show that, in principal, one could obtain the power to operate the accelerated mirror in such a spacecraft using energy extracted from the quantum vacuum using the standard Casimir effect witha parallel plate geometry. Unfortunately the method as currently conceived generates a miniscule thrust, and is no more practical than a spacewarp, yet it does provide an interesting demonstration of our current understanding of the physics of the quantized electromagnetic field in vacuum.Comment: 18 pages, 3 figure

New Insights into the Lamb Shift: The Spectral density of the Shift

In an atom, the interaction of a bound electron with the vacuum fluctuations of the electromagnetic 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 $2S_{1/2}$ and the $2P_{1/2}$ 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 demonstrated the renormalization process required to deal with the divergences plaguing the existing theories and led to the understanding that it was essential for theory to include interactions with the zero-point quantum vacuum field. This was the birth of modern quantum electrodynamics (QED). Other calculations of the Lamb shift followed by Welton and Power in an effort to clarify the physical mechanisms leading to the shift. We have done a calculation of the shift using a group theoretical approach which gives the shift as an integral over frequency of a function, which we call the spectral density of the shift. The spectral density reveals how different frequencies contribute to the total energy shift. We find, for example, that half the radiative shift for the ground state 1S level in H comes from photon energies below 9700 eV, and that the expressions by Power and Welton do not have the correct low frequency behavior, although they do give approximately the correct value for the total shift.Comment: 24 pages, 11 figures, version of paper published in Physics in 2022 with corrections to typographical error

Model for Entangled States with Spin-Spin Interaction

A system consisting of two neutral spin 1/2 particles is analyzed for two magnetic field perturbations: 1) an inhomogeneous magnetic field over all space, and 2) external fields over a half space containing only one of the particles. The field is chosen to point from one particle to the other, which results in essentially a one-dimensional problem. A number of interesting features are revealed for the first case: the singlet, which has zero potential energy in the unperturbed case, remains unstable in the perturbing field. The spin zero component of the triplet evolves into a bound state with a double well potential, with the possibility of tunneling. Superposition states can be constructed which oscillate between entangled and unentangled states. For the second case, we show that changes in the magnetic field around one particle affect measurements of the spin of the entangled particle not in the magnetic field nonlocally. By using protective measurements, we show it is possible in principle to establish a nonlocal interaction using the two particles, provided the dipole-dipole potential energy does not vanish and is comparable to the potential energy of the particle in the external field

Effect of quantum and thermal jitter on the feasibility of Bekenstein’s proposed experiment to search for Planck-scale signals

A proposed experiment to test whether space is discretized [J. D. Bekenstein, Phys. Rev. D 86, 124040 (2012); Found. Phys. 44, 452 (2014)] is based on the supposed impossibility of an incident photon causing a displacement of a transparent block by less than the Planck length. An analysis of the quantum and thermal jitter of the block shows that it greatly diminishes the possibility that the experiment could reveal Planck-scale signals

Gedanken experiments with Casimir forces, vacuum energy, and gravity

Gedanken experiments are used to explore properties of quantum vacuum energy that are currently challenging to explore experimentally. A constant lateral Casimir force is predicted to exist between two overlapping finite parallel plates at 0 K, otherwise it would be possible to extract an arbitrary amount of energy from the quantum vacuum. A rigid unpowered object cannot be accelerated by the quantum vacuum because of the translational symmetry of space. By considering systems in which vacuum energy and other forms of energy are exchanged, we demonstrate that a change {\Delta}E in vacuum energy, whether positive or negative with respect to the free field, corresponds to an equivalent inertial mass and equivalent gravitational mass {\Delta}M={\Delta}E/c^2. We consider the possibility of a gravitational shield, and show that, if it exists, the energy to operate it would have to cancel the net energy extracted from the gravitational field, otherwise we could extract an arbitrary amount of energy from the field.Comment: 24 pages, 3 figure

Of Some Theoretical Significance: Implications of Casimir Effects

In his autobiography Casimir barely mentioned the Casimir effect, but remarked that it is "of some theortical significance." We will describe some aspects of Casimir effects that appear to be of particular significance now, more than half a century after Casimir's famous paper

The Role of Vacuum Fluctuations and Symmetry in the Hydrogen Atom in Quantum Mechanics and Stochastic Electrodynamics

Stochastic Electrodynamics (SED) has had success modeling black body radiation, the harmonic oscillator, the Casimir effect, van der Waals forces, diamagnetism, and uniform acceleration of electrodynamic systems using the stochastic zero-point fluctuations of the electromagnetic field with classical mechanics. However the hydrogen atom, with its 1/r potential remains a critical challenge. Numerical calculations have shown that the SED field prevents the electron orbit from collapsing into the proton, but, eventually the atom becames ionized. We look at the issues of the H atom and SED from the perspective of symmetry of the quantum mechanical Hamiltonian, used to obtain the quantum mechanical results, and the Abraham-Lorentz equation, which is a force equation that includes the effects of radiation reaction, and is used to obtain the SED simulations. We contrast the physical computed effects of the quantized electromagnetic vacuum fluctuations with the role of the real stochastic electromagnetic field

History and Some Aspects of the Lamb Shift

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