The angular momentum is a physical variable that is characteristic of rotating systems. In classical mechanics, angular momentum is associated with macroscopic objects that are rotating, such as the movement of the planets of the solar system around the sun. This rotational movement leads to the formation of angular momentum that is the vector product of the distance with the momentum. In the language of mathematics, the angular momentum L is equal to rxp where p is the momentum associated with the rotating object and r is the distance which is also a vector that has a value and a direction.
Our earth movement around the sun is also associated with the production of an angular momentum. This angular momentum is a function of the velocity of the planet around the sun and also a function of its distance from the center of the sun. The angular momentum of rotating objects is a vector that has a value and a direction. The direction of the angular momentum is perpendicular to the plane that is made by the momentum vector and the distance vector.
The angular momentum in classical mechanics is associated with an operator in quantum mechanics. This operator describes the angular momentum in terms of the theory of quantum mechanics. The angular momentum of rotating objects has three components x, y and z that each is along one of the three cartesian axises. Thus Lx is the angular momentum component along the x axis and Ly is that component along the y axis and Lz is that component along the z axis.
The quantum mechanical representation of the angular momentum by the operator that is designated L has the property that only the square of the total angular momentum commutes with the Hamiltonian of the system that is under discussion. This means mathematically that the Hamiltonian and the square of the total angular momentum operators have a common basis of eigen functions. The components of the angular momentum Lx and Ly do not commute with the Hamiltonian. Therefore they have separate eigen functions than those of the Hamiltonian.
The eigen values of the total angular momentum operator can be obtained by using the commutation relations of the components of the angular momentum with each other and by using the raising the lowering operators that is made by the combination and subtraction of the components of the angular momentum. This technique of using raising and lowering operators is common in finding eigen values of operators such as the case with harmonic oscillator operator in quantum mechanics.
The classical angular momentum has an analogue in quantum mechanics. There is in quantum mechanics another type of angular momentum that is called the spin. This type of angular momentum does not have a classical analogue but can be compared with the rotation of planets around their axises. This rotation of the planets around their axises can be considered as global spin. The spin describes the movement of the particle around its axis. This phenomenon was not discovered until the formulation of the relativistic quantum mechanical equation by Dirac. Solution of this equation for an electron gives an explicit expression for the spin.
The Schroedinger equation is also a quantum mechanical equation that does not give description of the spin in its solution of the free electron. Solving the Dirac equation for the hydrogen atom gives mathematical terms that describe the phenomenon of spin-orbit coupling which is also a relativistic quantum mechanical effect. In mathematics, linear operators can be represented conveniently using matrices. This is also the case with quantum mechanical linear operators. The spin which is an intrinsic angular momentum of the electron can be represented using matrices of the dimension 2*2. These matrices are called Pauli matrices.