This chapter is titled: “Construction of classical Hamiltonian function of a charged particle in an electric and a magnetic field.” Here we first obtain expression for Lagrangian function for a charged particle in an electric field and a magnetic field. Then we obtain Hamiltonian function from Legendre transformation of the Lagrangian function. To solve the problem of the book at non-zero magnetic field, we need Hamiltonian operator for an electron in presence of electric and a magnetic field. To get the operator, we first need classical Hamiltonian function of a charged particle in an electric and a magnetic field. We can replace operators for observables in the Hamiltonian function to get Hamiltonian operator.
This chapter is titled: “Construction of classical Hamiltonian function of a charged particle in an electric and a magnetic field.” Here we first obtain expression for Lagrangian function for a charged particle in an electric field and a magnetic field. Then we obtain Hamiltonian function from Legendre transformation of the Lagrangian function. To solve the problem of the book at non-zero magnetic field, we need Hamiltonian operator for an electron in presence of electric and a magnetic field. To get the operator, we first need classical Hamiltonian function of a charged particle in an electric and a magnetic field. We can replace operators for observables in the Hamiltonian function to get Hamiltonian operator.
The motion of a particle with charge q and mass m in a magnetic field given by B = kB0 + B1 [icos(omegat) + jsin(omegat)] and an electric field which obeys Ñ ×E = - ¶ B/ ¶ t is analysed classically and quantum-mechanically. The use of a rotating coordinate system allows the analytical derivation of the particle classical trajectory and its laboratory wavefunction. The motion exhibits two resonances, one at omega = omegac = -qBo/m, the cyclotron frequency, and the other at omega = omegaL = -qBo/2m, the Larmor frequency. For omega at the first resonance frequency, the particle acquires a simple closed trajectory, and the effective hamiltonian can be interpreted as that of a particle in a static magnetic field. In the second case a term corresponding to an effective static electric field remains, and the particle orbit is an open line. The particle wave function and eigenenergies are calculated.
Spinor wave functions; classical Lagrangian and Hamiltonian of a charged particle in an electromagnetic field, and its quantum Hamiltonian; gauge invariance; spin magnetic moment in a uniform magnetic field; magnetic resonance; the Stern--Gerlach experiment; neutron interferometry and rotations of spinor wave functions; treatment of a particle in a uniform magnetic field with and without the inclusion of spin degrees of freedom; Ahronov--Bohm effect for a charged spinless particle confined in a cylindrical shell.
Abstract In Chapter 2, a path integral representation of the quantum operator e-β H in the case of Hamiltonians H of the separable form p2/2m + V(q) has been constructed. Here, the construction is extended to Hamiltonians that are more general functions of phase space variables. This results in integrals over paths in phase space involving the action expressed in terms of the classical Hamiltonian H(p,q). However, it is shown that, in the general case, the path integral is not completely defined, and this reflects the problem that the classical Hamiltonian does not specify completely the quantum Hamiltonian, due to the problem of ordering quantum operators in products. When the Hamiltonian is a quadratic function of the momentum variables, the integral over momenta is Gaussian and can be performed. In the separable example, the path integral of Chapter 2 is recovered. In the case of the charged particle in a magnetic field a more general form is found, which is ambiguous, since a problem of operator ordering arises, and the ambiguity must be fixed. Hamiltonians that are general quadratic functions provide other important examples, which are analysed thoroughly. Such Hamiltonians appear in the quantization of the motion on Riemannian manifolds. There, the problem of ambiguities is even more severe. The problem is illustrated by the analysis of the quantization of the free motion on the sphere SN−1.
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.
Abstract The interaction between atoms or molecules and electromagnetic (EM) fields underlies all spectroscopic techniques and a great variety of desirable molecular properties. EM fields and EM waves are introduced via the famous Maxwell equations. The scalar and vector potential are defined, and the gauge freedom is outlined. The Coulomb gauge is adopted. The classical Hamiltonian for a charged particle in an EM field is derived, and from this the ‘minimal substitution’ rules for incorporating the fields in the quantum Hamiltonian are obtained. The operators describing the interaction of an electron with static electric and magnetic fields, including the magnetic fields from nuclear spins, are derived, followed by the derivation of the interaction between an electron and an EM wave.
Using the Landau and symmetric gauges for the vector potential of a constant magnetic field and the quantum problem of a charged particle moving on a flat surface, we show the classical electromagnetic gauge transformation does not correspond to a one-dimensional unitary group transformation U(1) of the wave function for the quantum case. In addition, with the re-examination of the relation between the magnetic field B and its vector potential A, we found that, in order to have a consistent formulation of the dynamics of the charged particle with both expressions, we must have that B=∇×A if and only if B≠0.