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# Two Spin One-Half Particles

Consider a system consisting of two spin one-half particles. Suppose that the system does not possess any orbital angular momentum. Let and be the spin angular momentum operators of the first and second particles, respectively, and let
 (839)

be the total spin angular momentum operator. By analogy with the previous analysis, we conclude that it is possible to simultaneously measure either , , , and , or , , , , and . Let the quantum numbers associated with measurements of , , , , , and be , , , , , and , respectively. In other words, if the spinor is a simultaneous eigenstate of , , , and , then
 (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.

Table 4: Clebsch-Gordon coefficients for adding spin one-half to spin one-half. Only non-zero coefficients are shown.

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 , etc. Likewise, the singlet state is written:
 (853)

Subsections

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Richard Fitzpatrick 2010-07-20