Angular Momentum in the Hydrogen Atom

Here, it is understood that orbital angular momentum operators act on the spherical harmonic functions, , whereas spin angular momentum operators act on the spinors, .

Since the eigenstates
are (presumably)
orthonormal, and form a complete set, we can express the eigenstates
as linear combinations of them. For instance,

Now, it follows from Eq. (804) that

Moreover, according to Eqs. (806) and (807), we can write

Recall, from Eqs. (568) and (569), that

By analogy, when the spin raising and lowering operators, , act on a general spinor, , we obtain

(814) | |||

(815) |

For the special case of spin one-half spinors (

(816) |

It follows from Eqs. (810) and (812)-(817) that

(818) |

and

(819) |

Hence, Eqs. (809) and (811) yield

where

(822) |

(823) |

It follows that or , which corresponds to or , respectively. Once is specified, Eqs. (808) and (824) can be solved for and . We obtain

and

Here, we have neglected the common subscripts for the sake of clarity:

The information contained in Eqs. (825)-(828) is neatly summarized in Table 2. For instance, Eq. (825) is obtained by reading the first row of this table, whereas Eq. (828) is obtained by reading the second column. The coefficients in this type of table are generally known as

As an example, let us consider the states of a hydrogen atom.
The eigenstates of , , , and ,
are denoted
. Since can take the values ,
whereas can take the values , there are
clearly six such states: *i.e.*,
,
,
and
. The eigenstates of , , , and ,
are denoted
. Since and can be combined
together to form either or (see earlier), there are
also six such states: *i.e.*,
,
, and
. According to
Table 2, the various different eigenstates are interrelated as follows:

and

Thus, if we know that an electron in a hydrogen atom is in an state characterized by and [

The information contained in Eqs. (829)-(837) is neatly summed up in Table 3. Note that each row and column of this table has unit norm, and also that the different rows and different columns are mutually orthogonal. Of course, this is because the and eigenstates are orthonormal.