The Expectation Value of S(1).S(2)-Wave Functions Don't Matter

TL;DR

This paper demonstrates that the expectation value of a specific spin operator in two-nucleon systems depends only on angular momentum quantum numbers and is independent of detailed wave function structures.

Contribution

It reveals that the expectation value for two-nucleon systems has a universal form depending solely on angular momentum quantum numbers, regardless of wave function specifics.

Findings

Expectation value depends on J(J+1) for jj coupling.

Expectation value per pair is independent of wave function details.

The structure is similar for two-proton-two-neutron configurations with I(I+1).

Abstract

We consider the expectation value of the quantity [3+ \sigma (1).\sigma (2)]/4 . This has a value +1 for 2 nucleons with spin S=1and zero for S=0. We show that for the jj coupling 2 particle configuration [j(1) j(2)]^{J} the expectation value has the structure A+B J(J+1) where A and B are constants. We then show that for a 2proton-2neutron configuration with total angular momentum I the expectation value per pair is independent of the details of the wave function and has a similar structure A' +B' I(I+1) with B'=B/6.