The algebraic relations between the components of an augular momentum operator are of key importance in solving Schrodinger.In this paper we have exploited the dyadic product between vector products to deduce the coordinate-independent formula of scalar product and vector product between angular momentum operators.Then the formula for the scalar product of angular momentum operators is used to deduce the Schrodinger equation for one particle in spherical coordinate.In the final,we make a interest contrast among vector product,commutation relation and classical Poisson brackets,and find out their common properties that they all have the anti-commutative relationship and the Jacohi identity.The algebraic relations between the components of an augular momentum operator are of key importance in solving Schrodinger.In this paper we have exploited the dyadic product between vector products to deduce the coordinate-independent formula of scalar product and vector product between angular momentum operators.Then the formula for the scalar product of angular momentum operators is used to deduce the Schrodinger equation for one particle in spherical coordinate.In the final,we make a interest contrast among vector product,commutation relation and classical Poisson brackets,and find out their common properties that they all have the anti-commutative relationship and the Jacohi identity.

Journal of Liaoning Normal UniversityJan 1, 20060 citations

Angular Momentum Operators and Their Matrix Elements

Springer eBooksJan 1, 20020 citations

The Angular Momentum Operators

Elsevier eBooksJan 1, 20150 citations

The exact meaning of the angular-momentum and spin operators in quantum mechanics

G. Coddens

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 ${\hat{L}}_{z} = {\hbar\over{\imath}}\,(x{\partial\over{\partial y}} - y{\partial\over{\partial x}} )$ is not that it is the operatorfor the $z$ -component $L_{z}$ of the angular momentum ${\mathbf{L}}$ , but rather the expression of the operator for the angular momentum ${\mathbf{L}}$ when it is aligned with the $z$ -axis. Hence what we are used to note (erroneously) as ${\hat{L}}_{z}$ 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.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 ${\hat{L}}_{z} = {\hbar\over{\imath}}\,(x{\partial\over{\partial y}} - y{\partial\over{\partial x}} )$ is not that it is the operatorfor the $z$-component $L_{z}$ of the angular momentum ${\mathbf{L}}$, but rather the expression of the operator for the angular momentum ${\mathbf{L}}$ when it is aligned with the $z$-axis. Hence what we are used to note (erroneously) as ${\hat{L}}_{z}$ 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.

Sep 11, 20210 citations

Rotations et Moments Angulaires en Mecanique Quantique - Rotations and angular moments in quantum mechanics

Jacques Van de Wiele

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.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.

Jan 1, 20010 citations

{\hat{L}}_{z} = {\hbar\over{\imath}}\,(x{\partial\over{\partial y}} - y{\partial\over{\partial x}} )

is not that it is the operatorfor the

z

-component

L_{z}

of the angular momentum

{\mathbf{L}}

, but rather the expression of the operator for the angular momentum

{\mathbf{L}}

when it is aligned with the

z

-axis. Hence what we are used to note (erroneously) as

{\hat{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.