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Eigenvalues of S⋅Π for spins 1/2, 1, and 3/2

David Leo Weaver

The eigenvalues of the matrix operator S⋅Π for a constant magnetic field are derived in a parallel way for spins 1/2, 1, and 3/2 using only the algebra of the spin matrices and the commutation relations of the components of Π.

Journal of Mathematical PhysicsJan 1, 19787 citations

Matrix Method of Eigenvalue for the Addition of Angular Momenta

Shao Jian-jun

Matrix expression of quantum mechanics is discussed. Matrix method of eigenvalue for the addition of angular momenta is studied, and the most useful matrix eigenvalue method is discussed.

Training and ResearchJan 1, 20040 citations

Angular Momentum: General Properties

Rocco Schiavilla

General derivation of the eigenvalues and eigenstates of the square and z-component of the angular momentum; relationship between the angular momentum and the harmonic oscillator in two dimensions; transformation of states and operators under rotations; algebraic derivation of the hydrogen atom spectrum.

Cambridge University Press eBooksDec 5, 20240 citations

Eigenvalues and Eigenvectors of Angular Momentum Operator Jx without the Theory of Rotations

Lorenzo M. Narducci, Miguel Orszag

We propose an algebraic diagonalization procedure for the angular momentum matrix Jx which is based on the introduction of a suitable generating function. The consideration of the analytic properties of the generating function leads to the eigenvalues of the problem. The coefficients of the power series expansion of this function generate the components of the eigenvectors of Jx. This procedure provides the beginning student of quantum mechanics with an alternative derivation of the eigenvalue spectrum of the angular momentum.We propose an algebraic diagonalization procedure for the angular momentum matrix Jx which is based on the introduction of a suitable generating function. The consideration of the analytic properties of the generating function leads to the eigenvalues of the problem. The coefficients of the power series expansion of this function generate the components of the eigenvectors of Jx. This procedure provides the beginning student of quantum mechanics with an alternative derivation of the eigenvalue spectrum of the angular momentum.

American Journal of PhysicsDec 1, 197219 citations

Angular Momentum

A. C. Sharma

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.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.

CRC Press eBooksJun 18, 20210 citations

Some atomic physics

P. C. W. Davies, David S. Betts

We know that even a single electron in a hydrogen atom generally has orbital angular momentum L (with (2 l + 1) possible orientations where l is a integer) and spin angular momentum S (with two possible orientations commonly described as spin up and spin down or s = ± 1/2). In more complicated atoms there may be many components of angular momentum; for example, the helium atom has two electrons both of which have orbital and spin angular momenta. Also, in many atomic systems of physical interest we need to know how to combine angular momenta. This chapter explains how to do it.We know that even a single electron in a hydrogen atom generally has orbital angular momentum L (with (2 l + 1) possible orientations where l is a integer) and spin angular momentum S (with two possible orientations commonly described as spin up and spin down or s = ± 1/2). In more complicated atoms there may be many components of angular momentum; for example, the helium atom has two electrons both of which have orbital and spin angular momenta. Also, in many atomic systems of physical interest we need to know how to combine angular momenta. This chapter explains how to do it.

Routledge eBooksApr 27, 20180 citations

Eigenvalues of J2

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Eigenvalues of J2

Abstract

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