Eigenstates of Angular Momentum

Here, the are the eigenstates in question, whereas the dimensionless quantities and parameterize the eigenvalues of and , which are and , respectively. Of course, we expect the to be both mutually orthogonal and properly normalized (see Sect. 4.9), so that

where is an element of solid angle, and the integral is over all solid angle.

Now,

(559) |

where use has been made of Eq. (543). We, thus, conclude that when the operator operates on an eigenstate of corresponding to the eigenvalue it converts it to an eigenstate corresponding to the eigenvalue . Hence, is known as the

In other words, when operates on an eigenstate of corresponding to the eigenvalue it converts it to an eigenstate corresponding to the eigenvalue . Hence, is known as the

Writing

(561) | |||

(562) |

we obtain

(563) |

(564) |

(565) | |||

(566) |

These equations are satisfied when

(567) |