Linear and Angular Momentum of a Particle

H. E. Moses, S. C. Wang

S. Rajasekar, R. Velusamy

A particle moving along a straight-line has a linear momentum mv. A revolving particle has angular momentum due to its mass and angular velocity. In classical mechanics, orbital angular momentum is the vector product of position (r) and linear momentum (p): L = r× p . (11.1) Let us imagine the electron as a small ball, revolving about some axis. It possesses angular momentum. Precisely, the angular momentum vector is a vector pointing along the axis of rotation with magnitude proportional to the speed of rotation. A vector which changes sign under space inversion (parity operation) is called a polar vector . r and p are polar vectors as they change sign. Since L = (−r)× (−p), L does not change sign under parity operation. Such vectors which do not change sign under parity operation are called axial vectors or psuedo-vectors. Classically, the angular momentum of the electron could have any magnitude. In quantum mechanics the orbital angular momentum operator is obtained by replacing p by −i~∇. The angular position of a particle is the angle at which it is located in two-dimensional polar coordinates.

Walter Kauzmann

Gordon Baym

The orbital angular momentum of a particle in quantum mechanics is defined exactly as in classical mechanics. Before developing any properties of the angular momentum operators, students must know the commutation relations they satisfy with themselves, with the momentum operator, and with the position operator. The angular momentum operator is very closely related to rotations of the system. Students should compare the ability of the angular momentum operator to generate infinitesimal rotations, with the corresponding property of the linear momentum p. The chapter provides the theory of the orbital angular momentum sufficiently to study the problem of a particle in a central potential.

S. Rajasekar, R. Velusamy

A particle moving along a straight-line has a linear momentum mv. A revolving particle has angular momentum due to its mass and angular velocity. Let us imagine the electron as a small ball, revolving about some axis. It possesses angular momentum. Precisely, the angular momentum vector is a vector pointing along the axis of rotation with magnitude proportional to the speed of rotation. In classical mechanics the symmetry of a physical system leads to conservation laws. If the space is isotropic, it possesses rotational symmetry. The number of elements in a group is called its order. A group with finite number of elements is called a finite group whereas an infinite group contains infinite number of elements. An infinite group may further be either discrete or continuous. In fact, it is impossible to tell whether an electron is spinning. The word spin is a convenient way of representing the intrinsic angular momentum of a particle.

Galina A. Nemova