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Lecture# Quantum Physics I: Angular Momentum Operations

Description

This lecture covers the angular momentum operations in quantum physics, including the commutation relations, rotation operators, and eigenvalue problems. It also discusses spherical coordinates, spherical harmonics, and possible values of m in general.

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PHYS-313: Quantum physics I

The objective of this course is to familiarize the student with the concepts, methods and consequences of quantum physics.

In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum problems involving rotational symmetry. Such an operator is applied to a mathematical representation of the physical state of a system and yields an angular momentum value if the state has a definite value for it.

In pure and applied mathematics, particularly quantum mechanics and computer graphics and their applications, a spherical basis is the basis used to express spherical tensors. The spherical basis closely relates to the description of angular momentum in quantum mechanics and spherical harmonic functions. While spherical polar coordinates are one orthogonal coordinate system for expressing vectors and tensors using polar and azimuthal angles and radial distance, the spherical basis are constructed from the standard basis and use complex numbers.

In special functions, a topic in mathematics, spin-weighted spherical harmonics are generalizations of the standard spherical harmonics and—like the usual spherical harmonics—are functions on the sphere. Unlike ordinary spherical harmonics, the spin-weighted harmonics are U(1) gauge fields rather than scalar fields: mathematically, they take values in a complex line bundle. The spin-weighted harmonics are organized by degree l, just like ordinary spherical harmonics, but have an additional spin weight s that reflects the additional U(1) symmetry.

In physics, angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity – the total angular momentum of a closed system remains constant. Angular momentum has both a direction and a magnitude, and both are conserved. Bicycles and motorcycles, flying discs, rifled bullets, and gyroscopes owe their useful properties to conservation of angular momentum.

This article concerns the rotation operator, as it appears in quantum mechanics. With every physical rotation , we postulate a quantum mechanical rotation operator which rotates quantum mechanical states. In terms of the generators of rotation, where is rotation axis, is angular momentum, and is the reduced Planck constant. Translation operator (quantum mechanics) The rotation operator , with the first argument indicating the rotation axis and the second the rotation angle, can operate through the translation operator for infinitesimal rotations as explained below.