In Quantum Physics, the spin and angular momentum operators are magnitudes introduced by means of a vector transformation law. However, interpreting the eigenvalues of its Z "components" as projections on said axis leads to certain contradictions supposedly avoided by a mandatory (presented as a freely selected) Z's orientation. It is shown that an oriented physical space almost forces us to project the angular momentum's and spin's eigenvalues onto its orientation's 3-form, which sidesteps entering into inconsistencies. The final conclusion is that this "rare" magnitude called spin, downright naturally comes in and plays thanks to the orientation of our three-dimensional space.
In Quantum Physics, the spin and angular momentum operators are magnitudes introduced by means of a vector transformation law. However, interpreting the eigenvalues of its Z "components" as projections on said axis leads to certain contradictions supposedly avoided by a mandatory (presented as a freely selected) Z's orientation. It is shown that an oriented physical space almost forces us to project the angular momentum's and spin's eigenvalues onto its orientation's 3-form, which sidesteps entering into inconsistencies. The final conclusion is that this "rare" magnitude called spin, downright naturally comes in and plays thanks to the orientation of our three-dimensional space.
The theory of angular momentum and spin in quantum mechanics seems to defy common-sense intuition.We render the theory intelligible again by pointing out that this apparent impenetrability merely stems from an {\em undue} parallel interpretation of the algebraic expressions for the angular-momentum and spin operators in the group representation theory of SO(3) and SU(2). E.g. the correct meaning of the quantumoperator L^z=ℏ(x∂y∂−y∂x∂) is not that it is the operatorfor the z-component Lz of the angular momentum L, but rather the expression of the operator for the angular momentum L when it is aligned with the z-axis. Hence what we are used to note (erroneously) as L^z is not a scalar but a vector operator.The same applies{\em mutatis mutandis} for the spin operators. In the correct interpretation, the wholealgebraic formalism is just the group representation theory for the rotationsof three-dimensional Euclidean geometry. It is thus mere,elementary high-school mathematics (in a less usual, more technical guise) and as such totally exempt of any physics, let alone quantum mysteries.The change of interpretation has no impact on the algebraic results, such that they remain in agreement with experimental data and the algebra quantum mechanics remains correct. It is all only a matter of the correct geometrical meaning of the algebra.All these statements are proved within the framework of the group representation theory for SO(3) and SU(2) which is the basic tool used to describe rotational motion in quantum mechanics.
As in classical mechanics, rotation in quantum mechanics is a transformation which deals with angular momentum. The difference with classical mechanics comes from the fact that angular momentum is a vector operator and not a usual vector and its components do not commute. As for any transformation in quantum mechanics, to each rotation we can associate an operator which acts in state space. The expression of this operator depends on whether the rotation is passive, that is we do a rotation of the coordinate axes and the physical system is left unchanged, or active, in which case the coordinate axes are unchanged and the rotation is performed on the physical system. In the first part (Chaps. 1 and 2) of this book, details concerning both aspects are given. Following the definition of the geometrical transformation associated with the most general rotation, we give the expression of the rotation operator for specific cases. Transformation laws for scalar fields, vector fields and spinor fields are given as well as transformation laws for scalar operators, vector operators and more generally, for operators of any rank. The second part (Chaps. 3 and 4) deals with angular momentum algebra. We define the coupling coefficients of 2, 3 and 4 angular momenta as well as the recoupling coefficients. The definition of the irreductible tensor operator, which is a generalisation of scalar and vector operators, is given as well as the Wigner-Eckart theorem. The application of this theorem to more complex cases is studied.
Abstract The concept of angular momentum in quantum mechanics has many counterintuitive aspects, such as assuming only certain discrete values and only certain orientations in space in the presence of external fields. Spin is more esoteric yet; it is hard to imagine how point‐particles with no spatial extent can have an intrinsic angular momentum. Nevertheless, spin is the most fundamental concept in magnetic resonance. Appreciation of the spin concept requires some understanding of the relativistic quantum mechanics developed by P. A. M. Dirac. The history of the evolution of the spin concept is traced—from the turn‐of‐the‐century observation of the Zeeman effect, through the old and new quantum mechanics, and on to Dirac's theory of the electron, which was published in the late 1920s. The special relativity theory is discussed, and there is a brief explanation of how spin emerges naturally in a relativistically correct formulation of quantum mechanics.
Most students of science and engineering know that every elementary particle, such as electrons, neutrons, photons, neutrinos, etc., has a quantum mechanical property called “spin” which can be measured (perhaps not easily, but at least in principle) and has a quantized value, including zero. The vast majority of these students mentally visualize spin as the angular momentum associated with the elementary particle spinning or rotating about its own axis (like a top or a planetary object). This mental picture, although convenient and comforting, is actually somewhat crude and certainly incomplete. Landau and Lifshitz, in their classic textbook on quantum mechanics [1], wrote “[the spin] property of elementary particles is peculiar to quantum theory. [It] has no classical interpretation... It would be wholly meaningless to imagine the ‘intrinsic’ angular momentum of an elementary particle as being the result of its rotation about its own axis.” The simplistic notion of self-rotation about an axis, shown in Fig. 1.1, can- not explain many features of spin, such as why its magnitude cannot assume continuous values and why it is quantized to certain specific values. It also causes serious problems if taken too literally. As we will see later (Problem 1.2), if we think of an electron as a solid sphere of radius equal to the Lorentz radius e2/(4πǫ0m0c 2) (where e is the electron’s charge, m0 is the mass, c is the speed of light in vacuum, and ǫ0 is the dielectric constant of vacuum), then the velocity on the surface of a rotating electron would have to be many times the velocity of light in vacuum if such a rotation were to generate an angular momentum equal to the electron’s spin. Obviously that would not be permitted by the theory of relativity. Indeed, a deep understanding of quantum mechanics is required to understand how the spin property comes about. Its origin is in relativistic quantum mechanics and really was first appreciated by Paul Andrew Maurice Dirac when he derived the Dirac equation which is the cornerstone of relativistic quantum mechanics. Richard Feynman, noted for his distaste for mysticism in physics, wrote about the notion of “spin”: “It appears to be one of the few places in physics where there is a rule which can be stated very simply, but for which no one has found a simple and easy explanation. The explanation is down deep in relativistic quantum mechanics.
The standard Bloch sphere representation has been recently generalized to describe not only systems of arbitrary dimension, but also their measurements, in what has been called the extended Bloch representation of quantum mechanics . This model, which offers a solution to the longstanding measurement problem, is based on the hidden-measurement interpretation of quantum mechanics , according to which the Born rule results from our lack of knowledge of the measurement interaction that each time is actualized between the measuring apparatus and the measured entity. In this article, we present the extended Bloch model and use it to investigate, more specifically, the nature of the quantum spin entities and of their relation to our three-dimensional Euclidean theater. Our analysis shows that spin eigenstates cannot generally be associated with directions in the Euclidean space, but only with generalized directions in the Blochean space, which apart from the special case of spin one-half entities, is a space of higher dimensionality. Accordingly, spin entities have to be considered as genuine non-spatial entities. We also show, however, that specific vectors can be identified in the Blochean theater that are isomorphic to the Euclidean space directions, and therefore representative of them, and that spin eigenstates always have a predetermined orientation with respect to them. We use the details of our results to put forward a new view of realism, that we call multiplex realism , providing a specific framework with which to interpret the human observations and understanding of the component parts of the world. Elements of reality can be represented in different theaters, one being our customary Euclidean space, and another one the quantum realm, revealed to us through our sophisticated experiments, whose elements of reality, in the quantum jargon, are the eigenvalues and eigenstates. Our understanding of the component parts of the world can then be guided by looking for the possible connections, in the form of partial morphisms, between the different representations, which is precisely what we do in this article with regard to spin entities.
For pedagogical purposes it is demonstrated that with little effort the customary discussion of rotations in introductory courses can be extended to naturally produce a number of angular momentum operator properties, if coordinate-free rotation operators are used in their exponential form. This form gives rise to a particularly useful coordinate-free representation of the generators of rotations in three-space, from which the student can derive the usual angular momentum formalism for the case j=1 by elementary vector algebra alone. Since in this approach the j=1 formalism appears as a set of relations among simple geometric objects in ordinary space, it admits of a geometric interpretation and will leave the student better prepared for a more profound appreciation of the abstract concepts of the full theory.