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arXiv:quant-ph/9906002

Generalized Spin-1/2 Operators and Their Eigenvectors

Habatwa Vincent Mweene

Published Jun 1, 1999• 0 citations
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Abstract

Recently, we have shown how the interpretation of quantum mechanics due to Lande' can be used to derive from first principles generalized formulas for the operators and some eigenvectors for spin 1/2 Though we gave the operators for all the components of the spin, we did not give the eigenvectors of the operators for the x and y components of the spin. We now give these vectors. In addition, we present a new and simple method of deriving the operators for the x and y components of the spin as well as their vectors from those for the z component. We give a general proof that the operator for the square of the spin is the unit matrix multiplied by the value of the square of the spin.

DOI
10.48550/arxiv.quant-ph/9906002

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Recently, we have shown how the interpretation of quantum mechanics due to Lande' can be used to derive from first principles generalized formulas for the operators and some eigenvectors for spin 1/2 Though we gave the operators for all the components of the spin, we did not give the eigenvectors of the operators for the x and y components of the spin. We now give these vectors. In addition, we present a new and simple method of deriving the operators for the x and y components of the spin as well as their vectors from those for the z component. We give a general proof that the operator for the square of the spin is the unit matrix multiplied by the value of the square of the spin.
The interpretation of quantum mechanics due to Lande' is applied to the connection between wave mechanics and matrix mechanics. The connection between the differential eigenvalue equation and the matrix eigenvalue equation for an operator is elucidated. In particular, we show that the elements of a matrix vector state are probability amplitudes with a structure rather than being mere constants. We obtain the most general expressions for the probability amplitudes for the description of spin-1/2 measurements. As a result, we derive spin-1/2 operators and vectors from first principles. The procedure used is analogous to that by which orbital angular momentum wavefunctions and operators are transformed to matrix mechanics vectors and matrices. The most generalized forms of the spin operators and their eigenvectors for spin-1/2 are derived and shown to reduce to the Pauli spin matrices and vectors in an appropriate limit.
For a given eigenvalue of the square of the total spin operator, the spin eigenvectors with various eigenvalues of the z-component are obtained systematically by successive operations of the spin lowering operator on the eigenvector proposed in a previous paper for a one-dimensional many-fermion system with delta-function potential. The property of the spin eigenvector under the operation of the spin raising operator is investigated to clarify the mechanism how it has the given eigenvalue of the square of the total spin operator. It is shown that the spin eigenvector can be modified in various forms under some condition presented in the previous paper.
The approach to quantum mechanics which we have used to derive the matrix treatment of spin from first principles is now employed to treat systems of compounded angular momentum. A general treatment is first given, which is then applied to the concrete cases of a spin-0 and a spin-1 system obtained by adding the spins of two spin-1/2 systems. Thus the probability amplitudes for measurements on the systems are derived, as well as the matrix vectors and operators corresponding to the systems. The matrix operators and states obtained are different from the standard forms and are much more generalized. The new results are applied to the case of joint measurements on the subsystems of such a system; this is a problem that has been made very topical by the high level of interest in the foundations of quantum mechanics. As a consequence of the insights arising from this treatment, we show that the Clebsch-Gordan coefficients are amenable to generalization, and we give the generalized forms for these cases.
Elementary treatments of a single angular momentum having components Jk (k =1, 2, 3) usually go as far as discussions of the spectra of these operators and the derivations of their standard, i.e., J2, J3-diagonal, representatives. On the other hand, the standard representatives of their eigenvectors are not usually obtained, leaving J3 aside, for which the problem is trivial. In fact, this part of the theory is most often dealt with only after a lengthy excursion into the theory of matrix representations. The representatives in question are obtained, in the first part of this paper, by means of essentially elementary arguments. Once these representatives are known, the standard representative of the unitary operator U which generates the state vector of a system after it has undergone a rotation R from the state vector prior to the rotation are deduced; this representative comprises just the familiar matrix representations of the rotation group. Finally, the relevance of harmonic oscillator theory to the present problem is briefly considered.