Loading [MathJax]/jax/output/HTML-CSS/jax.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Physics LibreTexts

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)2χ(1)s1,s2;ms1,ms2,S22χ(1)s1,s2;ms1,ms2=s2(s2+1)2χ(1)s1,s2;ms1,ms2,S1zχ(1)s1,s2;ms1,ms2=ms1χ(1)s1,s2;ms1,ms2,S2zχ(1)s1,s2;ms1,ms2=ms2χ(1)s1,s2;ms1,ms2,Szχ(1)s1,s2;ms1,ms2=msχ(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)2χ(2)s1,s2;s,ms,S22χ(2)s1,s2;s,ms=s2(s2+1)2χ(2)s1,s2;s,ms,S2χ(2)s1,s2;s,ms=s(s+1)2χ(2)s1,s2;s,ms,Szχ(2)s1,s2;s,ms=msχ(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.

Clebsch-Gordon coefficients for adding spin one-half to spin one-half. Only non-zero coefficients are shown.
  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=12(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=12(χ(1)1/2,1/2χ(1)1/2,1/2).

Contributors and Attributions

  • Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)


This page titled 10.3: Two Spin One-Half Particles is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

Support Center

How can we help?