Spin Greater Than One-Half Systems

(5.111) |

where 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, , which is simply the component column vector of the . 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 matrix differential operators. Thus, we can represent the momentum operators as

(5.112) |

where is the unit matrix. [See Equation (5.91).] We represent the spin operators as

(5.113) |

where the extended Pauli matrix (which is, of course, Hermitian) has elements

Here, , are integers, or half-integers, lying in the range to . But, how can we evaluate the brackets and, thereby, construct the extended Pauli matrices? In fact, it is trivial to construct the matrix. By definition,

(5.115) |

Hence,

where use has been made of the orthonormality property of the . Thus, is the suitably normalized diagonal matrix of the eigenvalues of . The elements of and are most easily obtained by considering the ladder operators,

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

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

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

(5.122) | ||

(5.123) | ||

(5.124) |

as we have seen previously. For a spin one ( ) particle (e.g., a -boson or a -boson), we find that

(5.125) | ||

(5.126) | ||

(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- particle, where the Hamiltonian is some function of position and spin operators, into coupled partial differential equations involving the wavefunctions . Unfortunately, such a system of equations is generally too complicated to solve exactly.