10.3: Two Spin One-Half Particles
( \newcommand{\kernel}{\mathrm{null}\,}\)
Consider a system consisting of two spin one-half particles. Suppose that the system does not possess any orbital angular momentum. Let S1 and S2 be the spin angular momentum operators of the first and second particles, respectively, and let S=S1+S2 be the total spin angular momentum operator. By analogy with the previous analysis, we conclude that it is possible to simultaneously measure either S21, S22, S2, and Sz, or S21, S22, S1z, S2z, and Sz. Let the quantum numbers associated with measurements of S21, S1z, S22, S2z, S2, and Sz be s1, ms1, s2, ms2, s, and ms, respectively. In other words, if the spinor χ(1)s1,s2;ms1,ms2 is a simultaneous eigenstate of S21, S22, S1z, and S2z, then S21χ(1)s1,s2;ms1,ms2=s1(s1+1)ℏ2χ(1)s1,s2;ms1,ms2,S22χ(1)s1,s2;ms1,ms2=s2(s2+1)ℏ2χ(1)s1,s2;ms1,ms2,S1zχ(1)s1,s2;ms1,ms2=ms1ℏχ(1)s1,s2;ms1,ms2,S2zχ(1)s1,s2;ms1,ms2=ms2ℏχ(1)s1,s2;ms1,ms2,Szχ(1)s1,s2;ms1,ms2=msℏχ(1)s1,s2;ms1,ms2. Likewise, if the spinor χ(2)s1,s2;s,ms is a simultaneous eigenstate of S21, S22, S2, and Sz, then S21χ(2)s1,s2;s,ms=s1(s1+1)ℏ2χ(2)s1,s2;s,ms,S22χ(2)s1,s2;s,ms=s2(s2+1)ℏ2χ(2)s1,s2;s,ms,S2χ(2)s1,s2;s,ms=s(s+1)ℏ2χ(2)s1,s2;s,ms,Szχ(2)s1,s2;s,ms=msℏχ(2)s1,s2;s,ms. Of course, because both particles have spin one-half, s1=s2=1/2, and s1z,s2z=±1/2. Furthermore, by analogy with previous analysis, ms=ms1+ms2.
Now, we saw, in the previous section, that when spin l is added to spin one-half then the possible values of the total angular momentum quantum number are j=l±1/2. By analogy, when spin one-half is added to spin one-half then the possible values of the total spin quantum number are s=1/2±1/2. In other words, when two spin one-half particles are combined, we either obtain a state with overall spin s=1, or a state with overall spin s=0. To be more exact, there are three possible s=1 states (corresponding to ms=−1, 0, 1), and one possible s=0 state (corresponding to ms=0). The three s=1 states are generally known as the triplet states, whereas the s=0 state is known as the singlet state.
−1/2,−1/2 | −1/2,1/2 | 1/2,−1/2 | 1/2,1/2 | ms1,ms2 | |
[0.5ex] 1,−1 | 1 | ||||
[0.5ex] 1,0 | 1/√2 | 1/√2 | |||
[0.5ex] 0,0 | 1/√2 | −1/√2 | |||
[0.5ex] 1,1 | 1 | ||||
s,ms |
The Clebsch-Gordon coefficients for adding spin one-half to spin one-half can easily be inferred from Table [t2] (with l=1/2), and are listed in Table [t4]. It follows from this table that the three triplet states are: χ(2)1,−1=χ(1)−1/2,−1.2,χ(2)1,0=1√2(χ(1)−1/2,1/2+χ(1)1/2,−1/2),χ(2)1,1=χ(1)1/2,1/2, where χ(2)s,ms is shorthand for χ(2)s1,s2;s,ms, et cetera. Likewise, the singlet state is written: χ(2)0,0=1√2(χ(1)−1/2,1/2−χ(1)1/2,−1/2).
Contributors and Attributions
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)