Orthogonal Classification of Alpha-Particle Wave Functions

A four-nucleon wave function has four parts: spin ($S$), isospin ($T$), angular-momentum ($L$), and internal ($I$), combined into a totally antisymmetric function. The nonrelativistic $\alpha$-particle ground state is described by 32 independent internal functions. Cohen, using an overcomplete set of 34 internal functions, has previously classified the $\alpha$-particle ground state into three ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$, eight ${}_{}{}^3{}_{}{}^{}P_0^{}{}_{}{}^{}$, and six ${}_{}{}^5{}_{}{}^{}D_0^{}{}_{}{}^{}$ states, not all mutually orthogonal (or even independent). With the help of the concept of $S$ independence, the present paper describes a grouping of the 32 independent internal functions into an orthogonal and complete set of three ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$, six ${}_{}{}^3{}_{}{}^{}P_0^{}{}_{}{}^{}$, and five ${}_{}{}^5{}_{}{}^{}D_0^{}{}_{}{}^{}$ states.