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# Angular Momentum in the Hydrogen Atom

In a hydrogen atom, the wavefunction of an electron in a simultaneous eigenstate of and has an angular dependence specified by the spherical harmonic (see Sect. 8.7). If the electron is also in an eigenstate of and then the quantum numbers and take the values and , respectively, and the internal state of the electron is specified by the spinors (see Sect. 10.5). Hence, the simultaneous eigenstates of , , , and can be written in the separable form
 (806)

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,

 (807)

where and are, as yet, unknown coefficients. Note that the number of states which can appear on the right-hand side of the above expression is limited to two by the constraint that [see Eq. (801)], and the fact that can only take the values . Assuming that the eigenstates are properly normalized, we have
 (808)

Now, it follows from Eq. (804) that

 (809)

where [see Eq. (790)]
 (810)

Moreover, according to Eqs. (806) and (807), we can write
 (811)

Recall, from Eqs. (568) and (569), that
 (812) (813)

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 (i.e., ), the above expressions reduce to
 (816)

and
 (817)

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

 (818)

and
 (819)

Hence, Eqs. (809) and (811) yield
 (820) (821)

where
 (822)

Equations (820) and (821) can be solved to give
 (823)

and
 (824)

It follows that or , which corresponds to or , respectively. Once is specified, Eqs. (808) and (824) can be solved for and . We obtain
 (825)

and
 (826)

Here, we have neglected the common subscripts for the sake of clarity: i.e., , etc. The above equations can easily be inverted to give the eigenstates in terms of the eigenstates:
 (827) (828)

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 Clebsch-Gordon coefficients.

Table 2: Clebsch-Gordon coefficients for adding spin one-half to spin .

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:

 (829) (830) (831) (832) (833)

and
 (834) (835) (836) (837) (838)

Thus, if we know that an electron in a hydrogen atom is in an state characterized by and [i.e., the state represented by ] then, according to Eq. (836), a measurement of the total angular momentum will yield , with probability , and , with probability . Suppose that we make such a measurement, and obtain the result , . As a result of the measurement, the electron is thrown into the corresponding eigenstate, . It thus follows from Eq. (830) that a subsequent measurement of and will yield , with probability , and , with probability .

Table 3: Clebsch-Gordon coefficients for adding spin one-half to spin one. Only non-zero coefficients are shown.

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.

Next: Two Spin One-Half Particles Up: Addition of Angular Momentum Previous: General Principles
Richard Fitzpatrick 2010-07-20