Chapter One - Charged particles in electromagnetic fields

Noether's theorem is based on two fundamental ideas. The first is the extremum of the action and the second is the invariance of the action under infinitesimal continuous transformations in space and time. The first gives Hamilton's principle of least action, which results in the Euler–Lagrange equations. The second gives the Rund–Trautman identity for the generators of infinitesimal transformations in space and time. We apply these ideas to a charged particle in an external electromagnetic field. A solution of the Rund–Trautman identity for the generators is obtained by solving generalized Killing equations. The Euler–Lagrange equations and the Rund–Trautman identity are combined to give Noether's theorem for a conserved quantity. When we use the Lagrangian and the generators of infinitesimal transformations for a charged particle in an external electromagnetic field, we obtain the work-energy theorem.

Lie algebraic approach to the time-dependent quantum general harmonic oscillator and the bi-dimensional charged particle in time-dependent electromagnetic fields

The first problem faced is that of finding a transition from the quantum description of a fermion by means of a 4-spinor satisfying Dirac's equation to a classical or ray optics limit describing a spinning particle in an electromagnetic field. The solution is obtained by first using quaternions as a natural tool to describe the orientation of a classical particle and then transforming Dirac's 4-spinor into a quaternion in a novel way. These two quaternions are shown to be closely related and facilitate the transition between the two pictures in the ray optics approximation. The same formalism is then applied to the second problem which is comparing the exact classical and Dirac solutions to the motion of a particle in a plane electromagnetic field. Here the correspondence is effectively one of identity. This work involves using an improved classical equation of motion for a classical spinning charged particle which could be of practical value in experimental situations where the spin of a fermion is relevant but interference and other quantum effects are not. Finally there is derived the unexpected consequence that the magnitude of the intrinsic angular velocity of the classical particle is twice the mass, in the absence of an electromagnetic field, or twice the action in the more general case.

It is shown that the charged symplectic form in Hamiltonian dynamics of classical charged particles in electromagnetic fields defines a generalized affine connection on an affine frame bundle associated with spacetime. Conversely, a generalized affine connection can be used to construct a symplectic 2-form if the associated linear connection is torsion-free and the antisymmetric part of theR 4* translational connection is locally derivable from a potential. Hamiltonian dynamics for classical charged particles in combined gravitational and electromagnetic fields can therefore be reformulated as aP(4)=O(1, 3)⊗R 4* geometric theory with phase space the affine cotangent bundleAT * M of spacetime. The sourcefree Maxwell equations are reformulated as a pair of geometrical conditions on the ℝ4* curvature that are exactly analogous to the source-free Einstein equations.

A canonical formalism for the relativistic classical mechanics of many particles is proposed. The evolution equations for a charged particle in an electromagnetic field are obtained and the relativistic two-body problem with an invariant interaction is treated. Along the same line a quantum formalism for the spinless relativistic particle is obtained by means of imprimitivity systems according to Mackey theory. A quantum formalism for the spin-1/2 particle is constructed and a new definition of spin1/2 in relativity is proposed. An evolution equation for the spin-1/2 particle in an external electromagnetic field is given. The Bargmann Michel, and Telegdi equation follows from this formalism as a quasiclassical approximation. Finally, a new relativistic model for hydrogenlike atoms is proposed. The spectrum predicted is in agreement with Dirac's when radiative corrections have been added.

This paper suggests a principle to find a unitary operator U which transforms non-physical quantity, zero-potential Hamiltonian H0, into true physical quantity UH0U† for a charged particle in classical electromagnetic field, and puts forward a unified form of constructing gauge-independent transition probabilities in this case. Different methods correspond to different unitary operators which satisfy the above-mentioned principle.

Polarizability and its generalization

We analyse the physical meaning of quantum phase effects for point-like charges and electric (magnetic) dipoles in an electromagnetic (EM) field. At present, there are known eight effects of such a kind: four of them (the magnetic and electric Aharonov – Bohm phases for electrons, the Aharonov – Casher phase for a moving magnetic dipole and the He – McKellar – Wilkens phase for a moving electric dipole) had been disclosed in 20th century, while four new quantum phases had recently been found by our team (A. L. Kholmetskii, O. V. Missevitch, T. Yarman). In our analysis of physical meaning of these phases, we adopt that a quantum phase for a dipole represents a superposition of quantum phases for each charge, composing the dipole. In this way, we demonstrate the failure of the Schrödinger equation for a charged particle in an EM field to describe new quantum phase effects, when the standard definition of the momentum operator is used. We further show that a consistent description of quantum phase effects for moving particles is achieved under appropriate re-definition of this operator, where the canonical momentum of particle in EM field is replaced by the interactional EM field momentum. Some implications of this result are discussed.

Представлен новый метод решения релятивистских уравнений движения заряженных частиц в электромагнитных полях, учитывающий условие постоянства их значений на каждом временном шаге. Проведено сравнение точности и эффективности вычислений при решении тестовых задач в дву- и трехмерной постановках на основе нового метода, метода Бориса и его модификаций. В каждом случае рассматривались варианты аналитически и дискретно заданных значений электрического и магнитного полей. At present, the Boris method is mostly common for the numerical solution of problems of the dynamics of charged particles in electromagnetic fields. In recent years, new modifications of the Boris method have appeared that are capable to simplify, refine, or speed up calculations. However, despite the rather large number of publications which consider algorithms for calculating the trajectories of charged particles, new, more accurate and high-performance algorithms are desired. The article presents a detailed analysis of some explicit methods for solving the problem of the motion of charged particles in electromagnetic fields, which differ in how they set the average values of velocities. A new scheme based on an analytical solution, which has not been studied before, is proposed. For the Boris, Vay, Higuera – Cary schemes and the new VD1 scheme, a comparative analysis of the accuracy, convergence and counting time for the relativistic and nonrelativistic cases is carried out. In the nonrelativistic case, the new scheme allows accurate solving of the equations for motion of charged particles, and in the relativistic case, the accuracy of the new scheme remains higher compared to other schemes. As the relativistic factor increases, the accuracy of all considered schemes for calculating the trajectory of particles in an electromagnetic field decreases. For large values of the relativistic parameter, the new scheme is more accurate than the other schemes and retains the second order. In practical calculations, when particles can have different speeds, the scheme automatically adjusts to the given speed of the particle. The presented new scheme for calculating trajectories can be important for solving a wide range of problems in astrophysics and thermonuclear fusion, where high accuracy in determining particle trajectories in inhomogeneous fields for large moments of time is required.

Chapter two - General relations for the motion of charged particles in electromagnetic fields

On progressive blast envelope evolution of charged particles in electromagnetic fields

In this work, we consider potential energy of recently conceptualized optimal (balanced) bianisotropic particles in electromagnetic fields. The case of non-resonant lossless particles is studied. Knowing the potential energy of optimal bianisotropic particles in the fields of the respective excitations, we find the acting force on the particle in inhomogeneous external fields. It is found that for optimal particles with the balanced values of the polarizabilities the potential energy and acting force are time independent.

In this section we derive the equations of motion of a single particle in an electromagnetic field, using the expressions for Lagrangians and actions obtained above [15].KeywordsMagnetic FieldDrift VelocityLorentz ForceTransverse MotionLongitudinal MotionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this chapter we consider some problems in solid-state physics and in the theory of motion of charged particles in electromagnetic fields, whose solutions lead to incomplete cylindrical functions.

Relativistic charged particles in electromagnetic fields follow paths that increase in complexity with an increasingly complex field. When traveling in these fields, the particles are acted upon by an electromagnetic force comprised of an electric component and a magnetic component. While solving for these paths in simple electromagnetic fields can be done analytically, the task becomes significantly more difficult when the fields get more complex. Thus, in these situations, numerical methods are required to find solutions. One of the most well-known such methods is the Boris method, which is explored in this paper. Before this method is applied to any complex situations, its accuracy must be ensured by testing on simple cases which are solvable by hand. These cases include a 1D electric field in the direction of the initial velocity of the particle, and a 1D magnetic field perpendicular to the particle's velocity. With the accuracy proven, the method was applied to the cases of a force-free field, a dipolar field, and a quadrupolar field. In the latter two cases the method produced very interesting results that could provide significant insight that would be very difficult to achieve analytically. In the case of the force-free field, however, the method shows some limitations, as a precise cancellation of the force produced by the electric and magnetic fields is required to produce a straight line and the Boris method has some difficulty achieving this, especially when using a large time step.