Angular-Momentum Projection

A deformed solution $\vert\Phi\rangle$ of Eq. (1) is a superposition of eigenstates of the angular-momentum operator. The angular-momentum conserving wave function can be obtained from the state $\vert\Phi\rangle$ by applying the angular-momentum projector:

$$\vert IMK\rangle = \hat{P}^I_{MK} \vert\Phi\rangle.$$ (6)

The AMP operator $\hat{P}^I_{MK}$ is a projector onto angular momentum $I$ with projection $M$ along the laboratory $z$ axis, and reads:[14,1]

$$\hat{P}^I_{MK} = \frac{2I+1}{8\pi^2}\int D^{I*}_{MK}(\Omega)\; \hat{R}(\Omega)\; d\Omega,$$ (7)

where $\Omega$ represents the set of three Euler angles $(\alpha, \beta, \gamma)$, $D^{I*}_{MK}(\Omega)$ are the Wigner functions,[15] and $\hat{R}(\Omega)=e^{-i\alpha \hat{I}_z}e^{-i\beta \hat{I}_y} e^{-i\gamma \hat{I}_z}$ is the rotation operator.

The state $\vert IMK\rangle$ is no longer a product wave function, but a complicated superposition of Slater determinants. The operator $\hat{P}^I_{MK}$ is not a projector in the mathematical sense.[1] It extracts from the intrinsic wave function the component with a projection $K$ along the intrinsic $z$ axis of the nucleus. Since $K$ is not a good quantum number, all these components must be mixed, and the mixing coefficients $g^{(i)}_K$ must be determined by the minimization of energy. This $K$-mixing is taken into account by assuming the following form for the eigenstates:

$$\vert IM\rangle^{(i)} = \sum_K g^{(i)}_K \vert IMK\rangle \equiv \sum_K g^{(i)}_K \hat{P}^I_{MK}\vert\Phi \rangle,$$ (8)

and by solving the following Hill-Wheeler (HW) equation:

$$\mathcal{H} \bar{g} = E \mathcal{N} \bar{g},$$ (9)

where $\mathcal{H}_{K'K} = \langle \Phi \vert \hat H \hat{P}^I_{K'K} \vert \Phi \rangle$ and $\mathcal{N}_{K'K} = \langle \Phi \vert \hat{P}^I_{K'K} \vert \Phi \rangle$ denote the Hamiltonian and overlap kernels, respectively.