Projection of good quantum numbers for reaction fragments

In reactions, the wave packets of the emerging products typically are not eigenstates of particle number operators or any other conserved quantities, and their properties are entangled. I describe a particle projection technique in parts of space, which eschews the need to evaluate Pfaffians in the case of overlap of generalized Slater determinants or Hartree-Fock-Bogoliubov type of vacua. The extension of these formulas for calculating either angular momentum or particle projected energy distributions of the reaction fragments are presented as well. The generalization to simultaneous particle and angular momentum projection of various reaction fragment observables is straightforward.

Figure 1

The probability $P\left(N\right)$ of finding $N$ particles in a BCS wave function, derived from the function $Z\left(\eta \right)$ in Fig. 2.

Figure 2

The function $Z\left(\eta \right)$ with the factor $exp(-i\eta N)$ included or not, evaluated for $|v_n{|}^2=\frac{1}{2}\left[1-\frac{{\varepsilon }_n-\mu }{\sqrt{{({\varepsilon }_n-\mu )}^2+{\mathrm{\Delta }}^2}}\right]$ for a uniform single-particle spectrum ${\varepsilon }_n\propto n,n=1,2,...,\infty ,\mu =34$, and $\mathrm{\Delta }=2.5$.