Harmonic Oscillator Wave Functions in Nuclear Spectroscopy

Pairing energy calculations are generally carried out assuming the nucleon wave functions are those of an harmonic oscillator. The two-body interaction is assumed to be some function of ${\sigma }_1$, ${\sigma }_2$ and ${\mathbf{r}}_1$-${\mathbf{r}}_2$. Since the coordinates appear only in the form ${\mathbf{r}}_1$-${\mathbf{r}}_2$, it is convenient to write the two-body wave function in terms of the relative coordinate, r=${\mathbf{r}}_1$-${\mathbf{r}}_2$, and the center-of-mass coordinate, 2R=${\mathbf{r}}_1$+${\mathbf{r}}_2$. The eigenfunction in the new coordinates can be determined by noting that if the two particles are in the same oscillator level, then $\psi ({\mathrm{r}}_1, {\mathrm{r}}_2)$, which is an eigenfunction of $H_1+H_2$, is also an eigenfunction of $H_1$ alone. Transforming $H_1$ to relative and center-of-mass coordinates implies that the operator $\mathrm{p}·\mathrm{P}+mk\mathrm{r}·\mathrm{R}$ (where $m$ is the mass of the particle and $k$ is the spring constant of the oscillator) must give zero when operating on the wave function. This condition plus certain requirements arising from the radial form of the oscillator eigenfunctions is sufficient to determine the wave function in the new coordinate system.