The Lorentz transformation can be explained in two ways i.e. the special Lorentz transformation and most general Lorentz transformation. The length contraction, time dilation and velocity addition formulas for the special and most general Lorentz transformations are clearly explained. The variation of mass due to the relativistic velocity has also explained. Using above mentioned parameters we have derived the formulas of the transformation of orbital angular momentum and spin angular momentum for special Lorentz transformation. We have calculated the values of orbital angular momentum for rest and moving observers. The values of orbital angular momentum have plotted for the different velocities of the moving observers.
The Lorentz transformation can be explained in two ways i.e. the special Lorentz transformation and most general Lorentz transformation. The length contraction, time dilation and velocity addition formulas for the special and most general Lorentz transformations are clearly explained. The variation of mass due to the relativistic velocity has also explained. Using above mentioned parameters we have derived the formulas of the transformation of orbital angular momentum and spin angular momentum for special Lorentz transformation. We have calculated the values of orbital angular momentum for rest and moving observers. The values of orbital angular momentum have plotted for the different velocities of the moving observers.
This chapter covers various aspects of orbital, total, and spin angular momenta and the Clebsch-Gordon coefficients. An approach that is not found in many books, is used to derive eigenvalues of orbital angular momentum. The Pauli theory of spin half systems, addition of angular momenta, eigenfunctions of orbital angular momentum, computation of Clebsch-Gordon coefficients, solved examples, and unsolved exercises are included in the chapter. The basic purpose of the chapter has been to bring out the quantum mechanical importance of the theory of angular momenta with the use of essential mathematical derivations, which are self-consistent, and to adopt an elegant modern approach. Special mention on why orbital angular momentum cannot have non-integer values is made. The concept of spin angular momentum, which is purely a quantum mechanical quantity, is introduced in a simple and most convincing manner.
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.
Abstract This chapter is seeking relativistic wave equations, which are invariant under Lorentz transformations, in an attempt to obtain a quantum mechanical description of relativistic particles. First, the chapter starts with the Klein–Gordon equation and then it discusses the Dirac equation. It also takes the non-relativistic limit of the Dirac equation to derive the Schrödinger equation with two additional terms, the Zeeman term and the spin–orbit coupling term. These two terms emerge naturally from the Dirac equation, and thus the spin, as an internal quantum number which behaves like angular momentum, is clearly identified. Finally, the existence of antimatter is clearly supported by the nature of the solutions to the Dirac equation.
Lorentz transformations of spin density matrices for a particle with positive mass and spin 1∕2 are described by maps of the kind used in open quantum dynamics. They show how the Lorentz transformations of the spin depend on the momentum. Since the spin and momentum generally are not independent, the maps generally are not completely positive and act in limited domains. States with two momentum values are considered, so the maps are for the spin qubit correlated with the qubit made from the two momentum values, and results from the open quantum dynamics of two coupled qubits can be applied. Inverses are used to show that every Lorentz transformation completely removes the spin polarization, and so completely removes the information, from a number of spin density matrices. The size of the spin polarization that is removed is calculated for particular cases.
In four-dimensional or higher-dimensional space, spin angular momentum is not known well. For instance, we are not sure how many components it has in four-dimensional space. In this paper, the spin properties are theoretically derived and investigated in general-dimensional space. Before studying spin, we review the orbital angular momentum properties first, because they are theoretically well known even in higher-dimensional space. Then, for the orbital angular momentum, we make a special but natural assumption, and see that it leads a quantity similar to spin or Pauli matrices. It is remarkable that all the spin properties are derived from the one and only one assumption. Consistency of our consideration is checked by calculating the four-dimensional version of spin spherical harmonics and its eigenvalues. This theory may enable us to understand the double degeneracy of spin-1/2 particles without bringing in something called spin.
Spin is well known to be an intrinsically relativistic property of particles. Nonetheless its effects are seen in many physical situations which are not obviously relativistic, perhaps the most obvious examples being magnets and the quantum mechanically permitted electronic configurations of the elements in the periodic table. The view of spin adopted in these and other problems is that the electron has a quantized spin ( s = 1/2) and that is a fundamental tenet of the theory, rather than something that has to be explained. In this chapter we are going to discuss the behaviour of spin-1/2 particles (electrons, protons and neutrons for example) without much direct reference to relativistic quantum theory and the origins of spin. This chapter should be instructive in its own right and as a guide to understanding spin when we come to discuss it in a fully relativistic context at various stages throughout this book. Unless otherwise specified, we will refer to electrons, but it should be borne in mind that the theory is equally applicable to any spin-1/2 particle. Students of quantum theory cannot delve very deeply into the subject without coming across the quantization of angular momentum. The orbital angular momentum usually surfaces in the theory leading up to the quantum description of the hydrogen atom.