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Chinese Journal of Physics

Spin-Flip Operator in Two Electron Hopping System

Dongnan Su

Published Jun 1, 1992• 1 citation
OpenAlex

Abstract

For a hvo electron hopping system we work the spin parts within the framework of quantum mechanics by taking the spin-flip operators as a part of the Hamiltonian. The system of two electrons hopping in a magnetic field with the dipole-dipole interaction each other and with a spin-flip perturbation is modelled. For the positive dipole-dipole interactions, we have derived a formula to determine the magnitude of the spin-flip operators. For free spin systems without the dipole-dipole interactions, we find also the solutions for the systems obeying the nowadays Pauli exclusion principle as recalled in this paper. When the spin-flips are tuned off mathematically, the system is not able to return back to usual no spin-flip case by possibly leaving with some residual effects or results in the wave function descriptions of the system. We find that these effects are even not satisfied by the so-called extended symmetrization principle as defined and proposed in this paper. This is a new consequence.

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For a hvo electron hopping system we work the spin parts within the framework of quantum mechanics by taking the spin-flip operators as a part of the Hamiltonian. The system of two electrons hopping in a magnetic field with the dipole-dipole interaction each other and with a spin-flip perturbation is modelled. For the positive dipole-dipole interactions, we have derived a formula to determine the magnitude of the spin-flip operators. For free spin systems without the dipole-dipole interactions, we find also the solutions for the systems obeying the nowadays Pauli exclusion principle as recalled in this paper. When the spin-flips are tuned off mathematically, the system is not able to return back to usual no spin-flip case by possibly leaving with some residual effects or results in the wave function descriptions of the system. We find that these effects are even not satisfied by the so-called extended symmetrization principle as defined and proposed in this paper. This is a new consequence.
Texts on Quantum Mechanics all cover the electron’s magnetic dipole, or spin. However, texts on Electromagnetic Field Theory never mention it, and electromagnetic machines such as electron microscopes are designed without any reference to a magnetic dipole. No experiment has ever shown the presence of a permanent magnetic dipole in free electrons. All we have is Pauli’s assertion that blurring masks the experimental results. This paper, based on Field Theory, considers the alternative possibility, that free electrons have an induced rather than a permanent magnetic dipole, and demonstrates that the consequences of that alternative approach are to create a framework for an electromagnetic model of atomic behavior. This is a framework only. It identifies three separate pieces of work needed to create the full working model.
I will discuss the spin dynamics of two magnetic atoms trapped in an external trap. Assuming that initially the atoms are in their orbital ground state I will discuss in details all possible elementary processes of spin flipping. Dipole-dipole interactions couple spin degrees of freedom of the two particles to their quantized orbital motion. Because of conservation of the total angular momentum if a magnetization of the system changes the system acquires an orbital angular momentum and atoms circulate around the center of the trap. I show that spin dynamics driven by dipolar interactions is possible only when the energies of the initial and the final sates are equal. In a case of weak magnetic moments of atoms this condition can be fulfilled utilizing a resonant external magnetic field, which tunes energies of involved states via the linear Zeeman effect. I will illustrate the resonant dynamics comparing the theoretical predictions to some results of experiments on Chromium atoms.
The standard theoretical treatment of the Kapitza-Dirac effect---that is, the scattering of an electron passing through a standing-wave laser field---is extended here through the use of the Pauli equation to account for the interaction of the electron spin with the magnetic field of the standing wave. Prescriptions for determining unitarity-preserving approximations for the transition probabilities for scattering both with and without rotation of the electron spin direction are provided. This formalism is used to develop a perturbation theory for the spin-flip probability which, in the strong-field limit of interest here, reduces to a fairly simple relation between SSS-matrix elements for scattering with and without change in spin orientation, each expressed in terms of a Bessel function. A similar perturbative procedure is applied to estimate corrections to the standard theory for scattering in the absence of spin-flip processes, in which interactions that change the net number of photons in the field are ignored.
The spin evolution of two spin-1/2 valence electrons, located at two separated ions, is investigated with the help of Breit equation. This evolution that arise solely depends on their dipole-dipole magnetic interaction. Special attention is paid to the effects beyond the leading order of the dipole-dipole magnetic interaction. In contrast to just including the leading order, it is found that the spin-spin interactions flips the spin of both electrons together, while the spin-orbit interactions typically changes the spin of just single electron. As a consequence, different behaviour is observed in the time evolution of two different entangled Bell states.
By a canonical transformation on the Dirac Hamiltonian for a free particle, a representation of the Dirac theory is obtained in which positive and negative energy states are separately represented by two-component wave functions. Playing an important role in the new representation are new operators for position and spin of the particle which are physically distinct from these operators in the conventional representation. The components of the time derivative of the new position operator all commute and have for eigenvalues all values between $\ensuremath{-}c$ and ccc. The new spin operator is a constant of the motion unlike the spin operator in the conventional representation. By a comparison of the new Hamiltonian with the non-relativistic Pauli-Hamiltonian for particles of spin \textonehalf{}, one finds that it is these new operators rather than the conventional ones which pass over into the position and spin operators in the Pauli theory in the non-relativistic limit. The transformation of the new representation is also made in the case of interaction of the particle with an external electromagnetic field. In this way the proper non-relativistic Hamiltonian (essentially the Pauli-Hamiltonian) is obtained in the non-relativistic limit. The same methods may be applied to a Dirac particle interacting with any type of external field (various meson fields, for example) and this allows one to find the proper non-relativistic Hamiltonian in each such case. Some light is cast on the question of why a Dirac electron shows some properties characteristic of a particle of finite extension by an examination of the relationship between the new and the conventional position operators.
Quantization of spin and orbital angular momenta in standard quantum mechanics is rooted in the fundamental rotational symmetry of space in conjunction with the results of experiments on atomic and subatomic particles conducted with the Stern-Gerlach apparatus. We explain some of the most important features of spin and orbital angular momenta as inescapable consequences of elementary Stern-Gerlach experiments viewed in light of the rotational symmetry of space. The rotation operator is first derived for spin-½ particles, then extended to spin-1 particles by treating the case of a hydrogen atom in its ground state, where the spin of the proton combines with that of the electron to create a spin-1 triplet state. Along the way, we examine the evolution of a particle’s spin state under an applied magnetic field, where the interaction energy arises from the existence of a magnetic dipole moment for each particle in proportion to its spin angular momentum. Whereas a constant magnetic field B0 causes the particle’s magnetic moment to gyrate around the direction of B0, it will be seen that the presence of a small oscillating magnetic field perpendicular to B0 brings about the so-called Rabi oscillations, which involve periodic flippings of the particle’s spin orientation. In the case of a hydrogen atom in its ground state, the interaction between the magnetic dipole moments of the proton and the electron splits the ground state between a lower energy singlet having a net angular momentum of zero (spin-0 state), and a higher energy triplet having a total angular momentum of ħ (spin-1 state). This hyperfine splitting is the source of the well-known 21-cm line observed in the spectrum of the hydrogen atom, which arises from transitions between the atom’s singlet and triplet states. In the final section, we put forward an argument (originally made by Richard Feynman) that ties the rotational symmetry of space to the conservation of angular momentum in quantum mechanics. The argument is, in fact, quite general, enabling one to deduce other conservation laws from various inherent symmetries of quantum mechanical systems.