10.3: Two Spin One-Half Particles

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Consider a system consisting of two spin one-half particles. Suppose that the system does not possess any orbital angular momentum. Let ${\bf S}_1$ and ${\bf S}_2$ be the spin angular momentum operators of the first and second particles, respectively, and let \[{\bf S} ={\bf S}_1 + {\bf S}_2\] be the total spin angular momentum operator. By analogy with the previous analysis, we conclude that it is possible to simultaneously measure either $S_1^{\,2}$, $S_2^{\,2}$, $S^{\,2}$, and $S_z$, or $S_1^{\,2}$, $S_2^{\,2}$, $S_{1z}$, $S_{2z}$, and $S_z$. Let the quantum numbers associated with measurements of $S_1^{\,2}$, $S_{1z}$, $S_2^{\,2}$, $S_{2z}$, $S^{\,2}$, and $S_z$ be $s_1$, $m_{s_1}$, $s_2$, $m_{s_2}$, $s$, and $m_s$, respectively. In other words, if the spinor $\chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)}$ is a simultaneous eigenstate of $S_1^{\,2}$, $S_2^{\,2}$, $S_{1z}$, and $S_{2z}$, then \[\begin{aligned} S_1^{\,2}\, \chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)}&= s_1\,(s_1+1)\,\hbar^{\,2} \,\chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)},\\[0.5ex] S_2^{\,2}\, \chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)}&=s_2\,(s_2+1)\,\hbar^{\,2} \,\chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)},\\[0.5ex] S_{1z}\, \chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)}&= m_{s_1}\,\hbar \,\chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)},\\[0.5ex] S_{2z}\, \chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)}&= m_{s_2}\,\hbar \,\chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)},\\[0.5ex] S_z\, \chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)}&= m_s\,\hbar \,\chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)}.\end{aligned}\] Likewise, if the spinor $\chi_{s_1,s_2;s,m_s}^{(2)}$ is a simultaneous eigenstate of $S_1^{\,2}$, $S_2^{\,2}$, $S^{\,2}$, and $S_z$, then \[\begin{aligned} S_1^{\,2}\, \chi_{s_1,s_2;s,m_s}^{(2)}&= s_1\,(s_1+1)\,\hbar^{\,2} \,\chi_{s_1,s_2;s,m_s}^{(2)},\\[0.5ex] S_2^{\,2}\, \chi_{s_1,s_2;s,m_s}^{(2)}&= s_2\,(s_2+1)\,\hbar^{\,2} \,\chi_{s_1,s_2;s,m_s}^{(2)},\\[0.5ex] S^{2}\, \chi_{s_1,s_2;s,m_s}^{(2)}&= s\,(s+1)\,\hbar^{\,2} \,\chi_{s_1,s_2;s,m_s}^{(2)},\\[0.5ex] S_z\, \chi_{s_1,s_2;s,m_s}^{(2)}&=m_s\,\hbar \,\chi_{s_1,s_2;s,m_s}^{(2)}.\end{aligned}\] Of course, because both particles have spin one-half, $s_1=s_2=1/2$, and $s_{1z}, s_{2z}=\pm 1/2$. Furthermore, by analogy with previous analysis, \[m_s = m_{s_1}+ m_{s_2}.\]

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 $j=l\pm 1/2$. By analogy, when spin one-half is added to spin one-half then the possible values of the total spin quantum number are $s=1/2\pm 1/2$. 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 $m_s=-1$, 0, 1), and one possible $s=0$ state (corresponding to $m_s=0$). 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.
	$-1/2, -1/2$	$-1/2, 1/2$	$1/2,-1/2$	$1/2,1/2$	$m_{s_1},m_{s_2}$
[0.5ex] $1, -1$	${\scriptstyle 1}$
[0.5ex] $1, 0$		${\scriptstyle 1/\sqrt{2}}$	${\scriptstyle 1/\sqrt{2}}$
[0.5ex] $0, 0$		${\scriptstyle 1/\sqrt{2}}$	${\scriptstyle -1/\sqrt{2}}$
[0.5ex] $1, 1$				${\scriptstyle 1}$
$s, m_s$

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: \[\begin{aligned} \chi^{(2)}_{1,-1} &= \chi^{(1)}_{-1/2,-1.2},\\[0.5ex] \chi^{(2)}_{1,0} &= \frac{1}{\sqrt{2}}\left(\chi^{(1)}_{-1/2,1/2}+ \chi^{(1)}_{1/2,-1/2}\right),\\[0.5ex] \chi^{(2)}_{1,1} &= \chi^{(1)}_{1/2,1/2},\end{aligned}\] where $\chi^{(2)}_{s,m_s}$ is shorthand for $\chi^{(2)}_{s_1,s_2;s,m_s}$, et cetera. Likewise, the singlet state is written: \[\chi^{(2)}_{0,0} = \frac{1}{\sqrt{2}}\left(\chi^{(1)}_{-1/2,1/2}-\chi^{(1)}_{1/2,-1/2}\right).\]

Contributors and Attributions

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

$ \newcommand {\ltapp} {\stackrel {_{\normalsize<}}{_{\normalsize \sim}}}$ $\newcommand {\gtapp} {\stackrel {_{\normalsize>}}{_{\normalsize \sim}}}$ $\newcommand {\btau}{\mbox{\boldmath$\tau$}}$ $\newcommand {\bmu}{\mbox{\boldmath$\mu$}}$ $\newcommand {\bsigma}{\mbox{\boldmath$\sigma$}}$ $\newcommand {\bOmega}{\mbox{\boldmath$\Omega$}}$ $\newcommand {\bomega}{\mbox{\boldmath$\omega$}}$ $\newcommand {\bepsilon}{\mbox{\boldmath$\epsilon$}}$

	\(-1/2, -1/2\)	\(-1/2, 1/2\)	\(1/2,-1/2\)	\(1/2,1/2\)	\(m_{s_1},m_{s_2}\)
[0.5ex] \(1, -1\)	\({\scriptstyle 1}\)
[0.5ex] \(1, 0\)		\({\scriptstyle 1/\sqrt{2}}\)	\({\scriptstyle 1/\sqrt{2}}\)
[0.5ex] \(0, 0\)		\({\scriptstyle 1/\sqrt{2}}\)	\({\scriptstyle -1/\sqrt{2}}\)
[0.5ex] \(1, 1\)				\({\scriptstyle 1}\)
\(s, m_s\)