Angular momentum in n-dimensional spaces

Sami M. Al-Jaber

Stephen P. Martin

Mary Dale Alexander Cox

The solution of the three-dimensional angular momentum problem based on the commutation rules is well-known. 1-5 The commuting operators are the square of the total angular momentum and any one of the three components, called M₂. The matrix elements of these are found to be ²(j + 1) and m, respectively.Gallup has worked out the orbital angular momentum problem in n-dimensional space using the Mij commutation.

John D. McGervey

Ana Marı́a Cetto, Luis de la Peña

Chapter 11 is devoted to the quantum theory of angular momentum. As in classical mechanics, working with angular momentum requires leaving the one-dimensional space, with all the complications that this entails. The importance of angular momentum resides in at least three facts: i) central problems are of particular interest; (ii) many particles have an intrinsic angular momentum (the spin); and iii) the physics of angular momentum involves features such as the so-called space quantization. The first part of the chapter deals with the orbital angular momentum and its eigenfunctions, the spherical harmonics. Its algebraic properties, embodied in the matrix representation, allow for an extension of the theory to half-integer angular momentum in terms of Pauli matrices, as well as the addition of angular momenta.

Kazunari Sugiyama

We derive the two-valuedness and the angular momentum of a spin-1/2 from a rotation of 3-dimensional surface of a sphere existing in extra 4-dimensional space other than normal 3-dimensional space, in this paper. We will derive the two-valuedness of the spin as follows. We introduce 3-dimensional surface of a sphere S3 existing in extra 4-dimensional space (W, X, Y, Z) other than normal 3-dimensional space (x, y, z). We interpret the angle of rotation of the 3-sphere S3 as the phase of a wave function. We interpret the 3-sphere S3 as the absolute value of a wave function. We can express 3-sphere as the manifold with a constant sum of squares of the radius of two circles. When one circle's radius becomes the maximum, the other circle's radius becomes zero. Therefore, we can turn the circle inside out naturally. If we combine the circle turned inside out with the original circle, the manifold becomes a torus with a node. If we rotate the node of the torus by 360 degrees, we can turn the torus inside out. If we rotate the node of the torus 720 degrees, we can return the torus to the original state. This property is consistent with the property of the spin. We derive the angular momentum of the spin as follows. We make 3-dimensional solid sphere by removing one point from 3-sphere S3. On the other hand, we can make boundary like a 3-sphere S3 by removing one point from normal 3-dimensional space (x, y, z). We combine the boundaries of them. By repeating this, we can construct 3-dimensional helical space. The angle of rotation of the 3-sphere S3 is the angle of rotation of 3-dimensional helical space. On the other hand, we can interpret the angle of the rotation in the helical space as the coordinates (x, y, z) of the normal 3-dimensional space. Therefore, we can interpret the angular momentum of the 3-sphere S3 as the angular momentum of normal space.