angular momentum in quantum mechanics

Good day, Steemit community.

Today, I am going to discuss the angular momentum in quantum mechanics. Classically, angular momentum is the vector cross product between the position vector with the linear momentum vector. It is typically associated with the circular motion of the body. In quantum mechanics, angular momentum's is not a vector anymore but an operator. And it was found to possess some important implication in the property of a quantum system.

Angular Momentum

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The vector-angular momentum has the following Cartesian components:

and, therefore,

The eigenvalue spectrum for this operator is given by

On the other hand, the eigenvalue of one of its component is

We can also show that,

And this implies that the operator and have a set of common eigenfunctions labelled by the quantum numbers l and m. This shared eigenfunction is called the spherical harmonics :

Since

only, one of the components of can be well defined.

Spherical Harmonics

The spherical harmonics are given by the expression

where the are the associated Legendre functions, defined by

The normalization constants are given by

The are polynomials of order l (power series in w containing a finite number of powers of w, the highest being $w^l$), called Legendre polynomials, and are polynomial solutions of Legendre's equation

The Legendre polynomial are defined on the interval and normalized by the requirement that

Legendre polynomials have the parity of l, that is, they are even functions of w if the integer l is even and odd function of w if l is odd.