[Paper Review] Derivation of Standard Treatment of Spin Addition From Probability Amplitudes

Habatwa Vincent Mweene|ArXiv.org|Mar 15, 2000

Scientific Measurement and Uncertainty Evaluation5 references3 citations

TL;DR

This paper derives the standard matrix treatment of spin addition from fundamental probability amplitudes using the Landé interpretation of quantum mechanics. By generalizing probability amplitude transformations, it shows that the conventional spin matrices for two spin-1/2 particles emerge as a special case in the limit of aligned quantization axes, revealing that standard results are a restricted form of a broader, more general framework for angular momentum coupling.

ABSTRACT

In a recent paper, we introduced a new way of treating systems of compounded angular momentum. We obtained the probability amplitudes for measurements on the systems and used these to derive the matrix treatment of compounded spin. However, the matrix forms are in 3- and 4- dimensional space and are therefore entirely different from the standard forms. This raises the question of the connection between these forms and the standard forms. In this paper, we answer this question. We not only derive the standard matrix treatment of spin addition - we discover a more generalized form of the theory. We apply the new generalized theory to the singlet and triplet states arising from the addition of the spins of two systems of spin 1/2 each. We obtain new generalized forms of the vectors and operators for these cases, and show that they reduce to the standard forms in the appropriate limit.

Motivation & Objective

• To establish a rigorous derivation of the standard matrix treatment of compounded spin from first principles using probability amplitudes.
• To demonstrate that the standard spin matrices for two spin-1/2 particles are not fundamental but emerge as a limiting case of a more general formalism.
• To resolve the conceptual gap between the new generalized probability amplitude approach and the conventional matrix formulation in angular momentum coupling.
• To show that the standard treatment is insufficient for non-factorizable observables, especially in entangled systems.
• To lay the foundation for a more complete, generalized angular momentum theory beyond the standard formalism.

Proposed method

• Utilizes the Landé interpretation, where quantum states are probability amplitudes connecting initial and final measurement outcomes.
• Applies the Hermiticity and orthogonality conditions on probability amplitudes to ensure consistency with quantum mechanics.
• Derives the transformation law between probability amplitudes (Eq. 3) as the foundation for transitioning from wave mechanics to matrix mechanics.
• Constructs generalized spin operators [r^(1)]₁ and [r^(2)]₂ in arbitrary quantization directions using spherical angles θ and φ.
• Applies the transformation to the singlet and triplet states of two spin-1/2 particles, deriving 4×4 and 3×3 matrix representations.
• Takes the limit where quantization axes align (θ_d = θ₁, φ_d = φ₁) to recover the standard Pauli matrix forms.

Experimental results

Research questions

• RQ1 How can the standard matrix treatment of spin addition be derived from probability amplitudes rather than postulated?
• RQ2 What is the relationship between the generalized probability amplitude formalism and the conventional matrix representation of spin states?
• RQ3 Why do standard spin matrices emerge only in the limit of aligned quantization axes, and what does this imply about their generality?
• RQ4 Can non-factorizable observables in entangled systems be consistently described within matrix mechanics, or is a more general formalism required?
• RQ5 To what extent is the standard angular momentum coupling formalism incomplete compared to a probability amplitude-based generalization?

Key findings

• The standard matrix treatment of two spin-1/2 particles is derived as a limiting case of a more general probability amplitude formalism, confirming its consistency with the broader framework.
• The generalized spin operators [r^(1)]₁ and [r^(2)]₂ are expressed in terms of arbitrary quantization directions via angles θ and φ, yielding non-standard matrix elements.
• In the limit where the quantization direction aligns with the z-axis (θ_d = θ₁, φ_d = φ₁), the derived matrices reduce exactly to the standard Pauli matrices.
• The singlet and triplet states are represented using 4×4 matrices in the generalized formalism, showing that the total Hilbert space is not decomposed into product spaces of the subsystems.
• The new formalism allows for non-factorizable observables, unlike the standard treatment, which assumes separable operators acting on individual subsystems.
• The derivation confirms that the Landé interpretation provides a deeper foundation for quantum mechanics, with the standard matrix formalism being a special case of a more general probability amplitude theory.