Two Spin One-Half Particles

(839) |

(840) | |||

(841) | |||

(842) | |||

(843) | |||

(844) |

Likewise, if the spinor is a simultaneous eigenstate of , , , and , then

(845) | |||

(846) | |||

(847) | |||

(848) |

Of course, since both particles have spin one-half, , and . Furthermore, by analogy with previous analysis,

(849) |

Now, we saw, in the previous section, that when spin is added
to spin one-half then the possible values of the total angular momentum
quantum number are . By analogy, when spin one-half
is added to spin one-half then the possible values of the
total spin quantum number are . In other words,
when two spin one-half particles are combined, we either obtain
a state with overall spin , or a state with overall spin . To be more exact, there are
three possible states (corresponding to , 0, 1), and
one possible state (corresponding to ). The three states
are generally known as the *triplet* states, whereas the
state is known as the *singlet* state.

The Clebsch-Gordon coefficients for adding spin one-half to
spin one-half can easily be inferred from Table 2 (with ),
and are listed in Table 4. It follows from this table that the
three triplet states are:

(850) | |||

(851) | |||

(852) |

where is shorthand for ,

(853) |