Simple, but completely general, equations of change are derived for a property, χ†OΨ of a quantum mechanical system, where χ (r,s,t) and Ψ (r,s,t) are any two wavefunctions satisfying a time‐dependent Schrodinger equation with the same Hamiltonian H, and O is an arbitrary operator. For a Hamiltonian H0= (2m)−1 p2+V (r) the equations of change of angular momentum density and kinetic energy density are derived. The equations of change are developed for systems having Breit–Pauli type Hamiltonians which involve spin as well as external static electromagnetic fields and relativistic corrections. The velocity operator ṙ is shown to contain a spin‐dependent term. The relativistic correction to the kinetic energy (p2)2 term is not consistent with the equation of continuity. An expression is derived for the rate of ’’spin flipping’’ of a hydrogen atom in an external electromagnetic field. Methods are derived for projecting the equations of change onto real three‐dimensional space. Natural spin orbitals are useful for describing the equation of continuity in three dimensions; however, they cannot describe the equation of motion or the other equations of change.

Quantum mechanical equations of change. I

We study the orbits of particles (time-like geodesics) around two fixed black holes when the energy is elliptic, i. e. it does not allow the motion to extend to infinity. Most orbits are chaotic, but in many cases there are also ordered motions around stable periodic orbits. The orbits that fall into the first or the second black hole are separated by unstable periodic orbits. These are the satellite periodic orbits around the black holes when they exist. But for certain intervals of parameters there are no satellite orbits around the first or the second black hole. Then the limiting orbits are like arcs of hyperbolae, reaching the curve of zero velocity.

Periodic orbits and chaos around two fixed black holes. II

This book is an attempt to build a consistent relativistic quantum theory of interacting particles. In the first part of the book "Quantum electrodynamics" we follow rather traditional approach to particle physics. Our discussion proceeds systematically from the principle of relativity and postulates of quantum measurements to the renormalization in quantum electrodynamics. In the second part of the book "Quantum theory of particles" this traditional approach is reexamined. We find that formulas of special relativity should be modified to take into account particle interactions. We also suggest reinterpreting quantum field theory in the language of physical "dressed" particles. This formulation eliminates the need for renormalization and opens up a new way for studying dynamical and bound state properties of quantum interacting systems. The developed theory is applied to realistic physical objects and processes including the energy spectrum of the hydrogen atom, the decay law of moving unstable particles, and the electric field of relativistic electron beams. These results force us to take a fresh look at some core issues of modern particle theories, in particular, the Minkowski space-time unification, the role of quantum fields and renormalization as well as the alleged impossibility of action-at-a-distance. A new perspective on these issues is suggested. It can help to solve the old problem of theoretical physics -- a consistent unification of relativity and quantum mechanics.

Relativistic Quantum Dynamics: A non-traditional perspective on space, time, particles, fields, and action-at-a-distance

On the relation between momentum and velocity for elementary systems

The kind of operator algebra familiar in ordinary quantum mechanics is used to show formally that in an irreducible unitary representation of the Poincare group for positive mass, the Newton–Wigner position operator is the only Hermitian operator with commuting components that transforms as a position operator should for translations, rotations, and time reversal and does not behave in a singular way that contradicts what can be learned from Lorentz transformations in the nonrelativistic limit.

Simple derivation of the Newton–Wigner position operator

The significance of the energy-momentum conservation laws, as consequences of the general invariance principles, is much enhanced if the momentum, and hence the energy, can be coupled, still by means of the invariance principles, to some directly observable quantity. If the positions of particles can be measured in the quantum-mechanical sense, and if the corresponding operator satisfies the obvious invariance requirements, the velocity can also be ascertained and its relation to the momentum four-vector can easily be derived from the displacement invariances. Serious doubts have been cast, however, on the measurability of the position. Hence, a more primitive method of measuring the velocity, by means of « traps » is described; it is essentially the experimental method. The connection between momentum and velocity is then derived, for the velocity measured in this way, from the displacement invariance principle.

S. M. W. Ahmad

Papers

Invariant theoretic derivation of the connection between momentum and velocity