5.8: Spin Greater Than One-Half Systems
( \newcommand{\kernel}{\mathrm{null}\,}\)
In the absence of spin, the Hamiltonian can be written as some function of the position and momentum operators. Using the Schrödinger representation, in which p→−iℏ∇, the energy eigenvalue problem,
H|E⟩=E|E⟩,
can be transformed into a partial differential equation for the wavefunction x′ . In general, we find
H is now a partial differential operator. The boundary conditions (for a bound state) are obtained from the normalization constraint ψ+(x′), specifies the probability density of observing the particle at position x′ with spin angular momentum z -direction. The second, x′ with spin angular momentum z -direction. In the Pauli scheme, these wavefunctions are combined into a spinor, ψ+ and Hχ=Eχ, ???where 2×1 matrix of wavefunctions) and H is a ψ+ and 2×2 matrix partial differential operator in the Schrödinger/Pauli scheme [see Equation ???]. In other words, the partial differential equation for ψ− . In fact, both equations have the same form, so there is only really one differential equation. In this situation, the most general solution to Equation ??? can be written
ψ(x′) is determined by the solution of the differential equation, and the 2×2 matrix of complex numbers in the Schrödinger/Pauli scheme [see Equation ???], and the spinor eigenvalue equation ??? reduces to a straightforward matrix eigenvalue problem. The most general solution can again be written c+/c− is determined by the matrix eigenvalue problem, and the wavefunction ψ+ and s particle: i.e., a particle for which the eigenvalue of s(s+1)ℏ2 . Here, Sz are written sz is allowed to take the values 2s+1 distinct allowed values of 2s+1 different wavefunctions, denoted


In this extended Schrödinger/Pauli scheme, position space operators take the form of diagonal pk→−iℏ∂∂x′k1,
???where 1 is the Sk→sℏσk,
???where the σk has elements
j,l are integers, or half-integers, lying in the range +s . But, how can we evaluate the brackets σz matrix. By definition, (σ3)jl=⟨s,j|Sz|s,l⟩sℏ=jsδjl, ???where use has been made of the orthonormality property of the σz is the suitably normalized diagonal matrix of the eigenvalues of σx and S±=Sx±iSy.
???We know, from Equations ???-???, that
=[s(s+1)−j(j+1)]1/2ℏ|s,j+1⟩, ??? =[s(s+1)−j(j−1)]1/2ℏ|s,j−1⟩. ???It follows from Equations ???, and ???-???, that
=[s(s+1)−j(j−1)]1/22sδjl+1+[s(s+1)−j(j+1)]1/22sδjl−1, ??? =[s(s+1)−j(j−1)]1/22isδjl+1−[s(s+1)−j(j+1)]1/22isδjl−1. ???According to Equations ??? and ???-???, the Pauli matrices for a spin one-half (σ1
σ2 σ3 s=1 ) particle, we find that =1√2(010101010), ??? =1√2(0−i0i0−i0i0), ??? =(10000000−1). ???In fact, we can now construct the Pauli matrices for a spin anything particle. This means that we can convert the general energy eigenvalue problem for a spin-2s+1 coupled partial differential equations involving the ψsz(x′) . Unfortunately, such a system of equations is generally too complicated to solve exactly.
Contributors
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)