Spin Greater Than One-Half Systems

We have seen how to deal with a spin-half particle in quantum mechanics. But, what happens if we have a spin one or a spin three-halves particle? It turns out that we can generalize the Pauli two-component scheme in a fairly straightforward manner. Consider a spin-$s$ particle: that is, a particle for which the eigenvalue of $S^2$ is $s\,(s+1)\,\hbar^{\,2}$ . Here, $s$ is either an integer, or a half-integer. The eigenvalues of $S_z$ are written $s_z\,\hbar$ , where $s_z$ is allowed to take the values $s, s-1, \cdots, -s+1, -s$ . In fact, there are $2\,s+1$ distinct allowed values of $s_z$ . Not surprisingly, we can represent the state of the particle by $2\,s+1$ different wavefunctions, denoted $\psi_{s_z} ({\bf x}')$ . Here, $\psi_{s_z} ({\bf x}')$ specifies the probability density for observing the particle at position ${\bf x'}$ with spin angular momentum $s_z\,\hbar$ in the $z$ -direction. More exactly,

$$\psi_{s_z}({\bf x}') = \langle {\bf x'}\vert\langle s, s_z\vert \vert A\rangle\rangle,$$ (5.111)

where $\vert\vert A\rangle\rangle$ denotes a state ket in the product of the position and spin spaces. The state of the particle can be represented more succinctly by a spinor-wavefunction, $\chi$ , which is simply the $2\,s+1$ component column vector of the $\psi_{s_z} ({\bf x}')$ . Thus, a spin one-half particle is represented by a two-component spinor-wavefunction, a spin one particle by a three-component spinor-wavefunction, a spin three-halves particle by a four-component spinor-wavefunction, and so on.

In this extended Schrödinger/Pauli scheme, position space operators take the form of diagonal $(2\,s+1) \times (2\,s+1)$ matrix differential operators. Thus, we can represent the momentum operators as

$$p_k \rightarrow -{\rm i}\,\hbar \,\frac{\partial}{\partial x_k'}\, {\bf 1},$$ (5.112)

where ${\bf 1}$ is the $(2\,s+1) \times (2\,s+1)$ unit matrix. [See Equation (5.91).] We represent the spin operators as

$$S_k \rightarrow s\,\hbar \,\sigma_k,$$ (5.113)

where the $(2\,s+1) \times (2\,s+1)$ extended Pauli matrix $\sigma_k$ (which is, of course, Hermitian) has elements

$$(\sigma_k)_{j\,l} = \frac{ \langle s, j\vert\,S_k\, \vert s, l\rangle}{s\,\hbar}.$$ (5.114)

Here, $j$ , $l$ are integers, or half-integers, lying in the range $-s$ to $+s$ . But, how can we evaluate the brackets $\langle s, j\vert\,S_k \,\vert s, l\rangle$ and, thereby, construct the extended Pauli matrices? In fact, it is trivial to construct the $\sigma_z$ matrix. By definition,

$$S_z\, \vert s, j\rangle = j\,\hbar\, \vert s, j\rangle.$$ (5.115)

Hence,

$$(\sigma_3)_{j\,l} = \frac{\langle s, j\vert\,S_z\, \vert s, l\rangle}{s\,\hbar} = \frac{j}{s}\, \delta_{j\,l},$$ (5.116)

where use has been made of the orthonormality property of the $\vert s, j\rangle$ . Thus, $\sigma_z$ is the suitably normalized diagonal matrix of the eigenvalues of $S_z$ . The elements of $\sigma_x$ and $\sigma_y$ are most easily obtained by considering the ladder operators,

$$S^\pm = S_x \pm {\rm i}\, S_y.$$ (5.117)

We know, from Equations (4.55)-(4.56), that

$$S^+\, \vert s, j\rangle = [s\,(s+1) - j \,(j+1)]^{1/2} \,\hbar\, \vert s, j+1\rangle,$$ (5.118)

$$S^- \,\vert s, j\rangle = [s\,(s+1) - j \,(j-1)]^{1/2}\, \hbar \,\vert s, j-1\rangle.$$ (5.119)

It follows from Equations (5.114), and (5.117)-(5.119), that

$$(\sigma_1)_{j\,l} = \frac{[s\,(s+1) - j\,(j-1)]^{1/2} }{2\,s}\,\delta_{j\,\, l+1}+ \frac{[s\,(s+1) - j\,(j+1)]^{1/2} }{2\,s}\,\delta_{j\,\, l-1},$$ (5.120)

$$(\sigma_2)_{j\,l} = \frac{[ s\,(s+1) - j\,(j-1)]^{1/2} }{2\,{\rm i}\,s}\,\delta_{j\,\, l+1}- \frac{[s\,(s+1) - j\,(j+1)]^{1/2} }{2\,{\rm i}\,s}\,\delta_{j\,\, l-1}.$$ (5.121)

According to Equations (5.116) and (5.120)-(5.121), the Pauli matrices for a spin one-half ($s=1/2$ ) particle (e.g., an electron, a proton, or a neutron) are

$$\sigma_1 = \left(\!\begin{array}{rr} 0, &1\\ 1,&0\end{array}\!\right),$$ (5.122)

$$\sigma_2 = \left(\!\begin{array}{rr} 0, &-{\rm i}\\ {\rm i},&0\end{array}\!\right),$$ (5.123)

$$\sigma_3 = \left(\!\begin{array}{rr} 1, &0\\ 0,&-1\end{array}\!\right),$$ (5.124)

as we have seen previously. For a spin one ($s=1$ ) particle (e.g., a $Z$ -boson or a $W$ -boson), we find that

$$\sigma_1 =\frac{1}{\sqrt{2}}\left(\! \begin{array}{rrr} 0, &1,&0\\ 1,&0,&1\\ 0,&1,&0\end{array}\!\right),$$ (5.125)

$$\sigma_2 = \frac{1}{\sqrt{2}} \left(\!\begin{array}{rrr} 0, &-{\rm i},&0\\ {\rm i},&0,&{-\rm i}\\ 0,&{\rm i},& 0\end{array}\!\right),$$ (5.126)

$$\sigma_3 = \left(\!\begin{array}{rrr} 1, &0,&0\\ 0,&0,&0\\ 0,&0,&-1\end{array}\!\right).$$ (5.127)

In fact, we can now construct the Pauli matrices for a particle of arbitrary spin. This means that we can convert the general energy eigenvalue problem for a spin-$s$ particle, where the Hamiltonian is some function of position and spin operators, into $2\,s+1$ coupled partial differential equations involving the $2\,s+1$ wavefunctions $\psi_{s_z}({\bf x'})$ . Unfortunately, such a system of equations is generally too complicated to solve exactly.