Donald H. Kobe

Gauge invariant formulation of the interaction of electromagnetic radiation and matter

The conventional perturbation theory in quantum mechanics for the interaction of electromagnetic radiation with matter, which is based on the time‐dependent vector and scalar potentials, is shown to be dependent on the gauge. The problem is reformulated, using the approach of Yang, in a gauge‐invariant way, in which the transitions between states are induced by the quantum‐mechanical power operator. In order to use an unperturbed Hamiltonian which does not involve the electromagnetic field, the gauge is chosen to give the electric dipole interaction. This choice can be made if the magnetic field is negligible and the electric field is slowly varying over atomic dimensions. For a comparison of the gauge‐invariant and conventional formulation, a two‐level atom in a radiation field is considered. For the probability that an atom originally in the lower state will make a transition to the upper state when the field is not in resonance, the gauge‐invariant formulation gives the correct result, whereas the gaug...

Helmholtz’s theorem revisited

Helmholtz’s theorem for three‐vectors is applicable to electrostatics and magnetostatics, but it must be generalized to antisymmetric second‐rank tensors to be applicable to electromagnetism. A proof of Helmholtz’s theorem for three‐vectors, which is shorter than the usual one, is given. Helmholtz’s theorem is shown to be a special case of the Hodge decomposition theorem in the theory of differential forms.

Gauge transformations and the electric dipole approximation

Some subtle aspects of the electric dipole approximation (EDA) are discussed, and some persistent criticisms answered. The usual form of the EDA is that the scalar potential is −E⋅r and the vector potential is zero. The EDA does not describe the electromagnetic wave, but only the effect of the electromagnetic wave in the long‐wavelength limit on a nonrelativistic electron bound in an atom. The EDA is the first term in the multipole expansion of the scalar potential in the multipolar gauge, in which the potentials are expressed as integrals over the electric and magnetic fields.

Derivation of Maxwell’s equations from the local gauge invariance of quantum mechanics

Maxwell’s equations are derived from the principle of form invariance of quantum mechanics under multiplication of the wave function by a space‐ and time‐dependent phase factor (local gauge transformations of the first kind). The principle leads to the introduction of the vector and scalar potentials, which are shown to transform under the usual gauge transformations of electromagnetism (gauge transformations of the second kind). The electric and magnetic fields are introduced in the usual way to obtain observable fields which are gauge independent. Faraday’s law and the condition of no magnetic monopoles are obtained from the gauge transformations of the potentials. Conservation of energy and the linearity of the field equations are assumed to obtain Gauss’ law and the Ampere‐Maxwell law.

Derivation of Maxwell’s equations from the gauge invariance of classical mechanics

The Lagrangian for a single classical charged particle is made form invariant under the addition of a total time derivative by adding an interaction Lagrangian which involves compensating fields. The compensating fields are the vector and scalar potentials of the electromagnetic field which couple to the current and charge densities, respectively. To insure form invariance of the Lagrangian, the vector and scalar potentials must undergo the usual gauge transformations of electromagnetism. The electric and magnetic fields, which are gauge invariant, are obtained by examining the equation of motion for the charged particle. Faraday’s law and the condition that there are no magnetic monopoles are obtained from the expressions for the electric and magnetic fields in terms of the potentials. The simplest possible gauge‐invariant Lagrangian which is quadratic in the electric and magnetic fields is constructed. From the principle of least action Gauss’ law and the Ampere–Maxwell law are obtained.

Synthesis of the Planck and Bohr formulations of the correspondence principle

Planck formulated the correspondence principle between quantum and classical mechanics as the limit in which the Planck constant h goes to zero. Bohr formulated the correspondence principle to be the limit of large quantum numbers. For three common quantum mechanical systems it is shown that in order for eigenvalues of quantum mechanical observables to have meaningful classical limits, it is necessary to take the double limit as both the Planck constant goes to zero and the quantum number goes to infinity, subject to the constraint that their product is equal to an appropriate classical action. This synthesis of the Bohr and Planck formulations of the correspondence principle is also used to show that the quantum mechanical transition frequency between adjacent levels approaches the corresponding classical frequency. The features these systems have in common in their classical limit are explained by general considerations of the classical limit of the Schrodinger equation.

Helmholtz theorem for antisymmetric second‐rank tensor fields and electromagnetism with magnetic monopoles

A generalized Helmholtz’s theorem is proved, which states that an antisymmetric second‐rank tensor field in 3+1 dimensional space‐time, which vanishes at spatial infinity, is determined by its divergence and the divergence of its dual. When the divergence of the antisymmetric electromagnetic field strength tensor is equal to the electric charge‐current density and the divergence of the dual of the electromagnetic field strength tensor is equal to the magnetic charge‐current density, the equations of electromagnetism are obtained. As a convenience, in the solution of the equations of electromagnetism two different four‐vector potentials can be used, one of which couples to the electric charge‐current density and the other to the magnetic charge‐current density.

Second Quantization as a Graded Hilbert Space Representation

A review of the second quantization formulation of quantum mechanics is given based on the idea of a graded Hilbert space whose elements are represented by column matrices containing zero, one, two, ···, particle functions. The creation and annihilation operators are defined in terms of intuitive ideas about adding a particle to or removing a particle from the system. The (anti)commutation relations are shown to follow from the (anti)symmetry of the wavefunctions. The field operators which describe the creation or annihilation of a particle at a point are then obtained and their properties discussed. These operators are then used to express general operators in second quantization. Finally, a basis is introduced and the equations of occupation-number space are obtained.

Lagrangians for dissipative systems

A Lagrangian and Hamiltonian formulation can be given for a damped harmonic oscillator with damping linear in the velocity. The canonical momentum is not equal to the kinetic momentum, and the Hamiltonian is not equal to the energy. On the other hand, a pendulum accreting mass has the same Lagrangian and equation of motion. However, in this case the canonical momentum is equal to the kinetic momentum, and the Hamiltonian is equal to the energy. No ambiguity arises if the physical situation is kept in mind.

Generalization of Coulomb’s law to Maxwell’s equations using special relativity

Maxwell’s equations are obtained by generalizing the laws of electrostatics, which follow from Coulomb’s law and the principle of superposition, so that they are consistent with special relativity. In addition, it is necessary to assume that electric charge is a conserved scalar. The Lorentz force on a charged particle and its energy conservation condition are obtained by making Newton’s second law for the particle in an electrostatic field consistent with special relativity. Magnetic monopoles can be introduced into Maxwell’s theory in a way consistent with special relativity.