The central problem in this chapter is to solve for the wave function for a particle of charge q and rest mass m propagating in a general electromagnetic potential. It explores a few of the important implications for wave optics. The chapter begins with a review of basic quantum mechanics governing particle motion. It confines the discussion to only those topics which are relevant to the motion of a fast (unbound) charged particle in a general electromagnetic potential. The chapter seeks a dynamical equation to describe the motion of a single particle of charge q and rest mass m in the presence of a general electromagnetic potential. Physically, a close analogy exists with the wave optics of light propagation. In quantum mechanical wave optics, one starts with a surface of constant phase called a wave front. In wave optics, the aberrations are manifest as displacements in the surfaces of constant phase.
The central problem in this chapter is to solve for the wave function for a particle of charge q and rest mass m propagating in a general electromagnetic potential. It explores a few of the important implications for wave optics. The chapter begins with a review of basic quantum mechanics governing particle motion. It confines the discussion to only those topics which are relevant to the motion of a fast (unbound) charged particle in a general electromagnetic potential. The chapter seeks a dynamical equation to describe the motion of a single particle of charge q and rest mass m in the presence of a general electromagnetic potential. Physically, a close analogy exists with the wave optics of light propagation. In quantum mechanical wave optics, one starts with a surface of constant phase called a wave front. In wave optics, the aberrations are manifest as displacements in the surfaces of constant phase.
Let us return from optics to mechanics and explore the analogy to its fullest extent. In optics the old system of mechanics corresponds to intellectually operating with isolated mutually independent light rays. The new undulatory mechanics corresponds to the wave theory of light . Erwin Schrödinger, Nobel lecture , 1933 In this chapter we shall see that electromagnetic waves and electrons feature a number of common properties under conditions of spatial confinement. Simple and familiar problems from introductory quantum mechanics and textbook wave optics are recalled in this chapter to emphasize the basic features of waves in spatially inhomogeneous media. Herewith we make a first step towards understanding the properties of electrons and electromagnetic waves in nanostructures and notice that these properties in many instances are counterparts. Different formulas and statements of this chapter can be found in handbooks on quantum mechanics and wave optics. A few textbooks on quantum mechanics do consider analogies of propagation and reflection phenomena in wave optics with those in wave mechanics. Isomorphism of the Schrödinger and Helmholtz equations In Chapter 2 we discussed that an electron in quantum mechanics is described by the wave function, the square of its absolute value giving the probability of finding an electron at a specific point in space. This function satisfies the Schrödinger equation (2.56) which is the second-order differential equation with respect to space and the first-order differential equation with respect to time.
Essentially, we solve the particle-field equations obtained in the matter distribution quantum mechanics. In the framework of this theory, a particle is described by a density amplitude function, we call wave function, as a packet of waves with the time dependent phases proportional to the relativistic Lagrangian. The mass quantization of such a particle, as a distribution of matter, is obtained from the equality of the spatial integral of the density with the mass as a characteristic of the relativistic dynamics of this matter. The particle dynamics, as a motion of a mass spatial distribution, is obtained from the equality of the wave propagation velocities with the distribution coordinate velocities. This description is a physical one, compared to the Schrödinger-Heisenberg description leading to unphysical results as a null velocity for the kinetic energy as a constant of motion, or Zitterbewegung. The description of a particle interaction with an electromagnetic field by additional Lagrangian terms, with a vector potential conjugated to coordinates, as in the Aharonov-Bohm effect, and an electric potential conjugated to time, leads to the Lorentz force, and the Maxwell equations. In the two conjugated spaces, of the coordinates and of the momentum, the dynamics of a quantum particle is described by propagation operators, depending on the coordinate-momentum product, applied to time dependent wave functions satisfying matter-field equations. Unlike the standard Schrödinger or Schrödinger-Dirac equations, depending only on the Hamiltonian operator, instead of the Hamiltonian these equations contain the Lagrangian, which, besides the Hamiltonian, include additional terms with an explicit dependence on velocity, as it is expected in a relativistic theory. We obtain the wave function of a particle-antiparticle system interacting with an electromagnetic field, as a wave packet of spinors propagating in space with finite dimensions, given by the finite definition domain of the density function, in the space of the matter-field momentum, as a constant of motion. We show that an electrically charged quantum particle, rotating in a central electric field, generates a vector potential with a magnetic field perpendicular to the rotational plane and an electric field in this plane. By a quantum transition, a photon is emitted in the particle rotational plane, with the magnetic field perpendicular to this plane, and the electric field contained in this plane. In this framework, the vector potential rotation is correlated to a particle momentum rotation, for the canonical matter-field momentum as a constant of motion, without any electromagnetic field radiated outside.
Wave mechanics, geometrical optics, classical Hamilton--Jacobi dynamics, electron optics, and gauge-coupled quantum mechanics are usually introduced as conceptually distinct frameworks. Yet they share a common mathematical structure: an environment modifies a local phase relation, the phase defines a local propagation momentum, and the resulting momentum field determines characteristic curves along which propagation occurs. This paper develops a propagation-first synthesis of these structures for particles with nonzero rest mass. For a nonrelativistic spinless particle with wave functionΨ(r,t)=R(r,t)eiS(r,t)/ℏ,the Schrödinger equation in scalar and electromagnetic potentials is exactly equivalent, wherever the amplitude is sufficiently regular, to a continuity equation and a quantum Hamilton--Jacobi equation,∂t∂R2+∇⋅[R2m∇S−qA]=0,∂t∂S+2m(∇S−qA)2+qϕ+U+Q=0,whereQ=−2mℏ2R∇2R. The gauge-invariant kinetic propagation momentum is thereforeΠ=∇S−qA,and the associated local propagation velocity isv=mΠ. We call this coupled amplitude--phase representation the \textbf{Unified Propagation System}. The name denotes an organizational framework rather than a new fundamental equation. Within this framework, external interactions can be interpreted as modifications of a local phase--dispersion structure. For static scalar potentials, spatial variation of the local propagation momentum defines an effective matter-wave refractive index and reproduces the optical--mechanical analogy familiar from Hamiltonian and electron optics. Electromagnetic vector potentials generalize this picture: they enter directly into the gauge-invariant phase gradient and therefore modify propagation direction in a manner that cannot in general be represented by a scalar refractive index. This motivates a generalized definition of refraction: \textbf{refraction is the bending of propagation characteristics caused by spatial or temporal variation of the local phase--dispersion relation}. In the semiclassical limit this definition reduces directly to geometrical ray bending. Beyond that limit, the full amplitude-dependent term Q provides an additional modification of the propagation structure. The same equations can be written in a second-order form,mdtdv=q(E+v×B)−∇U−∇Q,showing that force-based and propagation-based descriptions can represent the same dynamics at different levels. We therefore propose that electromagnetic deflection, potential scattering, and matter-wave refraction may be regarded as different manifestations of a common propagation principle rather than as fundamentally unrelated mechanisms. The paper deliberately avoids a deeper ontological claim. It does not assert that the wave function, electromagnetic field, and particle arise from a common physical substrate. It instead identifies the propagation structure that any such deeper account would have to reproduce. A relativistic extension based on the gauge-covariant Hamilton--Jacobi equation suggests that the same architecture survives beyond the nonrelativistic Schrödinger regime.
In this chapter, we consider a quantum particle wave function with a bound spectrum of velocity c, and obtain the relativistic momentum based on the group velocity of this wave function. With a space-time isometry condition, the Lorentz transformation and the relativistic dynamics were obtained. Considering a field interacting with a quantum particle as a four-vector conjugated to the space-time vector in the time-dependent phase, we obtain the Lorenz force and the Maxwell equations. It is interesting that only the Ampère-Maxwell equation of a magnetic circuit is specific to the electromagnetic field, while the other equations are general for a field interacting with a charged quantum particle. Considering the time-dependent phase of a quantum particle interacting with an electromagnetic field with a space-time homogeneity condition, we obtain Lorentz transformations for this field. For a quantum particle at a non-relativistic velocity, we obtain a wave function with a very rapidly-varying factor, of a frequency proportional to the rest energy of this particle. From the Schrödinger equation of a particle with a relativistic Hamiltonian, we obtain a split of the wave function into four components, describing a proper rotation of this particle with an angular momentum called spin (Dirac’s relativistic electron theory). Moreover, we also calculate electron potential in the magnetic field, and two-electron interaction potential.
We set up the Maxwell's equations and subsequently the classical wave equations for the electromagnetic waves which together with their generating source, an oscillatory charge of zero rest mass in general travelling, make up a particle travelling similarly as the source at velocity ν in the field of an external scalar and vector potentials. The direct solutions in constant external field are Doppler-displaced plane waves propagating at the velocity of light c ; at the de Broglie wavelength scale and expressed in terms of the dynamically equivalent and appropriate geometric mean wave variables, these render as functions identical to the space-time functions of a corresponding Dirac spinor, and in turn identical to de Broglie phase waves previously obtained from explicit superposition. For two spin-half particles of a common set of space-time functions constrained with antisymmetric spin functions as follows the Pauli principle for same charges and as separately indirectly induced based on experiment for opposite charges, the complete wave functions are identical to the Dirac spinor. The back-substitution of the so explicitly determined complete wave functions in the corresponding classical wave equations of the two particles, subjected further to reductions appropriate for the stationary-state particle motion and to rotation invariance when in three dimensions, give a Dirac equation set; the procedure and conclusion are directly extendible to arbitrarily varying potentials by use of the Furious theorem and to particle motions in three dimensions by virtue of the characteristics of de Broglie particle motion. Through the derivation of the Dirac equation, the study hopes to lend insight into the connections between the Dirac wave functions and the electrodynamic components of simple particles under the government by the well established basic laws of electrodynamics.
Based on an observation that the basic mode of a common microwave waveguide is a solution to the Klein-Gordon equation, quantum mechanics is modeled as the wave-function propagated inside a waveguide. The guide width is determined by the rest-energy and the potential energy. Moreover, a geometrical optics model of the propagation illustrates how the width as a function of distance determines the velocity and accelerations of the particle. The model entails the disappearance of action-at-a-distance problem, and that interactions are conveyed via the potential, i.e. by photons.