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# 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- particle: that is, a particle for which the eigenvalue of is . Here, is either an integer, or a half-integer. The eigenvalues of are written , where is allowed to take the values . In fact, there are distinct allowed values of . Not surprisingly, we can represent the state of the particle by different wavefunctions, denoted . Here, specifies the probability density for observing the particle at position with spin angular momentum in the -direction. More exactly, (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 (5.114)

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, (5.116)

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, (5.117)

We know, from Equations (4.55)-(4.56), that  (5.118)  (5.119)

It follows from Equations (5.114), and (5.117)-(5.119), that  (5.120)  (5.121)

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.   Next: Exercises Up: Spin Angular Momentum Previous: Factorization of Spinor-Wavefunctions
Richard Fitzpatrick 2016-01-22