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arXiv:1909.03931

Statistical Assemblies of Particles with Spin

G. Ramachandran

Published Sep 9, 2019• 0 citations
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Abstract

Spin, $s$ in quantum theory can assume only half odd integer or integer values. For a given $s$, there exist $n=2s+1$ states $|s,m\rangle$, $m=s,s-1,........,-s$. A statistical assembly of particles (like a beam or target employed in experiments in physics) with the lowest value of spin $s=\frac {1}{2}$ can be described in terms of probabilities $p_m$ assigned to the two states $m=\pm \frac {1}{2}$. A generalization of this concept to higher spins $s>\frac {1}{2}$ leads only to a particularly simple category of statistical assemblies known as `Oriented systems'. To provide a comprehensive description of all realizable categories of statistical assemblies in experiments, it is advantageous to employ the generators of the Lie group $SU(n)$. The probability domain then gets identified to the interior of regular polyhedra in $\Re^{n-1},$ where the centre corresponds to an unpolarized assembly and the vertices represent `pure' states. All the other interior points correspond to `mixed' states. The higher spin system has embedded within itself a set of $s(2s+1)$ independent axes, which are determinable empirically. Only when all these axes turn out to be collinear, the simple category of `Oriented systems' is realized, where probabilities $p_m$ are assigned to the states $|s,m\rangle$. The simplest case of higher spin $s=1$ provides an illustrative example, where additional features of `aligned' and more general `non oriented' categories are displayed.

DOI
10.48550/arxiv.1909.03931

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The recently established existence of spherical harmonic functions, Yℓm(θ,ϕ)Y_\ell^{m}(θ,ϕ)Yℓm​(θ,ϕ) for half-odd-integer values of ℓ\ellℓ and mmm, allows for the introduction into quantum chemistry of explicit electron spin-coordinates; i.e. spherical polar angles θs,ϕsθ_s, ϕ_sθs​,ϕs​, that specify the orientation of the spin angular momentum vector in space. In this coordinate representation the spin angular momentum operators, S2,SzS^2, S_zS2,Sz​, are represented by the usual differential operators in spherical polar coordinates (commonly used for L2,LzL^2, L_zL2,Lz​), and their electron-spin eigenfunctions are sin⁡θsexp⁡(±ϕs/2)\sqrt{\sinθ_s} \exp(\pmϕ_s/2)sinθs​​exp(±ϕs​/2). This eigenfunction representation has the pedagogical advantage over the abstract spin eigenfunctions, α,β,α, β,α,β, that ``integration over spin coordinates'' is a true integration (over the angles θs,ϕsθ_s, ϕ_sθs​,ϕs​). In addition they facilitate construction of many electron wavefunctions in which the electron spins are neither parallel nor antiparallel, but inclined at an intermediate angle. In particular this may have application to the description of EPR correlation experiments.
In this second paper in a series, we show that the the general statistical approach to nonrelativistic quantum mechanics developed in the first paper yields a representation of quantum spin and magnetic moments based on classical nonrelativistic spinning top models, using Euler angle coordinates. The models allow half-odd-integer spin and predict supraluminal speeds only for electrons and other leptons, which must be treated relativistically. The spin operators in the space-fixed frame satisfy the usual commutation rules, while those in the rotating body-fixed frame satisfy "left-handed" rules. The commutation rules are independent of the structure of the top, so all nonrelativistic rigidly rotating objects must have integer or odd-half-integer spin. Physical boundary conditions restrict all mixed spin states to involve only half-odd-integer or only integer spin eigenstates. For spin 1/2, the theory automatically yields a modified Pauli-Schrödinger equation. The Hamiltonian operator in this equation contains a rigid rotator term and a term involving the square of the magmetic field, as well as an interaction term having the usual form in spherically symmetric and some cylindrically symmetric models, valid for any magnetogyric ratio.
Chapter 1: On the existence of quantum representations for two dichotomic measurements. Under which conditions do outcome probabilities of measurements possess a quantum-mechanical model? This kind of problem is solved here for the case of two dichotomic von Neumann measurements which can be applied repeatedly to a quantum system with trivial dynamics. The solution uses methods from the theory of operator algebras and the theory of moment problems. The ensuing conditions reveal surprisingly simple relations between certain quantum-mechanical probabilities. It also shown that generally, none of these relations holds in general probabilistic models. This result might facilitate further experimental discrimination between quantum mechanics and other general probabilistic theories. Chapter 2: Possibilistic Physics. I try to outline a framework for fundamental physics where the concept of probability gets replaced by the concept of possibility. Whereas a probabilistic theory assigns a state-dependent probability value to each outcome of each measurement, a possibilistic theory merely assigns one of the state-dependent labels "possible to occur" or "impossible to occur" to each outcome of each measurement. It is argued that Spekkens' combinatorial toy theory of quantum mechanics is inconsistent in a probabilistic framework, but can be regarded as possibilistic. Then, I introduce the concept of possibilistic local hidden variable models and derive a class of possibilistic Bell inequalities which are violated for the possibilistic Popescu-Rohrlich boxes. The chapter ends with a philosophical discussion on possibilistic vs. probabilistic. It can be argued that, due to better falsifiability properties, a possibilistic theory has higher predictive power than a probabilistic one. Chapter 3: The quantum region for von Neumann measurements with postselection. It is determined under which conditions a probability distribution on a finite set can occur as the outcome distribution of a quantum-mechanical von Neumann measurement with hyphenation given that the scalar product between the initial and the final state is known as well as the success probability of the postselection. An intermediate von Neumann measurement can enhance transition probabilities between states such that the error probability shrinks by a factor of up to 2. Chapter 4: A presentation of the category of stochastic matrices. This chapter gives generators and relations for the strict monoidal category of probabilistic maps on finite cardinals (i.e., stochastic matrices). Chapter 5: Convex Spaces: Definition and Examples. We try to promote convex spaces as an abstract concept of convexity which was introduced by Stone as "barycentric calculus''. A convex space is a set where one can take convex combinations in a consistent way. By identifying the corresponding Lawvere theory as the category from chapter 4 and using the results obtained there, we give a different proof of a result of Swirszcz which shows that convex spaces can be identified with algebras of a finitary version of the Giry monad. After giving an extensive list of examples of convex sets as they appear throughout mathematics and theoretical physics, we note that there also exist convex spaces that cannot be embedded into a vector space: semilattices are a class of examples of purely combinatorial type. In an information-theoretic interpretation, convex subsets of vector spaces are probabilistic, while semilattices are possibilistic. Convex spaces unify these two concepts.
A thorough study of regular and quasi-regular polyhedra shows that the symmetries of these polyhedra identically describe the quantization of orbital angular momentum, of spin, and of total angular momentum, a fact which permits one to assign quantum states at the vertices of these polyhedra assumed as the average particle positions. Furthermore, if the particles are fermions, their wave function is anti-symmetric and its maxima are identically the same as those of repulsive particles, e.g., on a sphere like the spherical shape of closed shells, which implies equilibrium of these particles having average positions at the aforementioned maxima. Such equilibria on a sphere are solely satisfied at the vertices of regular and quasi-regular polyhedra which can be associated with the most probable forms of shells both in Nuclear Physics and in Atomic Cluster Physics when the constituent atoms possess half integer spins. If the average sizes of the constituent particles are known, then the average sizes of the resulting shells become known as well. This association of Symmetry with Quantum Mechanics leads to many applications and excellent results.