5.5.4 Isospin Representation of Particle-Hole Nuclei
We have shown that the Isospin representation for two-particle and two-hole states.
5.5.3 Isospin Representation of Two-Particle and Two-Hole Nuclei
Based on the Isospin Representation, we can express the Two-Particle and Two-Hole Nuclei: $$ |a\, b; J \, M; T \, M_T \rangle = \mathcal{N}_{ab}(JT) [c_a^\dagger c_b^\dagger]_{JM}^{T M_T} | {\rm CORE} \rangle, $$ The normalization factor is given by $$ \ma
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Similarly, we can express the particle-hole nuclei in the isospin representation.
General formula of the particle-hole wavefunction
The general wave function of the particle-hole nuclei is
\begin{eqnarray} |a \, b^{-1}; J \, M ; T \, M_T \rangle = [c_a^\dagger h_b^\dagger]_{JM}^{T M_T} | {\rm HF} \rangle = [c_a^\dagger \hat{c}_b]_{JM}^{T M_T} | {\rm HF} \rangle. \end{eqnarray}
Since the $a \neq b$, the overlapping of the wavefunction is given as
$$\begin{eqnarray} \langle a \, b^{-1} ; J \, M ; T\, M_T | c \, d^{-1}; J', \, M'\, ; T' \, M_T' \rangle &=& \sum_{ \substack{m_\alpha, m_\beta, m_\gamma, m_\delta \\ m_{t\alpha}, m_{t\beta}, m_{t\gamma}, m_{t \delta}} } (j_a \, m_\alpha \, j_b \, m_\beta | J M) (j_c \, m_\gamma \, j_d \, m_\delta | J' \, M') \\[12pt] && \times (\frac{1}{2} \, m_{t\alpha} \, \frac{1}{2} \, m_{t\beta} | T M ) (\frac{1}{2} \, m_{t\gamma} \, \frac{1}{2} \, m_{t\delta} | T' M' ) \delta_{\alpha \gamma} \delta_{\beta \delta} \\[12pt] &=& \sum_{ m_\alpha, m_\beta, m_{t\alpha}, m_{t\beta} } (j_a \, m_\alpha \, j_b \, m_\beta | J M) (j_c \, m_\alpha \, j_d \, m_\beta | J' \, M') \\[12pt] && \times (\frac{1}{2} \, m_{t\alpha} \, \frac{1}{2} \, m_{t\beta} | T M ) (\frac{1}{2} \, m_{t\alpha} \, \frac{1}{2} \, m_{t\beta} | T' M' ) \delta_{a c} \delta_{b d} \\[12pt] &=& \delta_{JJ'} \delta_{MM'} \delta_{TT"} \delta_{M_T M_T'} \delta_{ac} \delta_{bd}, \end{eqnarray}$$
here I used $\langle {\rm HF} | h_b c_a c_c^\dagger h_d^\dagger | {\rm HF} \rangle = \delta_{ac}\delta_{bc}$.
Symmetry properties
Symmetry properties can be derived as follows:
$$\begin{eqnarray} | b^{-1} \, a ; J, \, M\, ; T \, M_T \rangle &=& \sum_{ m_\alpha, m_\beta, m_{t\alpha}, m_{t\beta}, } (j_b \, m_\beta\, j_a \, m_\alpha | J M) (\frac{1}{2} \, m_{t\beta} \, \frac{1}{2} \, m_{t\alpha} | T M ) h_{\beta}^\dagger c_{\alpha}^\dagger | {\rm HF} \rangle \\[12pt] &=& \sum_{ m_\alpha, m_\beta, m_{t\alpha}, m_{t\beta}, } (-1)^{j_a + j_b + J} (j_a \, m_\alpha\, j_b \, m_\beta | J M) (-1)^{1+T}(\frac{1}{2} \, m_{t\alpha} \, \frac{1}{2} \, m_{t\beta} | T M ) h_{\beta}^\dagger c_{\alpha}^\dagger | {\rm HF} \rangle \\[12pt] &=& \sum_{ m_\alpha, m_\beta, m_{t\alpha}, m_{t\beta}, } (-1)^{j_a + j_b + J} (j_a \, m_\alpha\, j_b \, m_\beta | J M) (-1)^{2+T}(\frac{1}{2} \, m_{t\alpha} \, \frac{1}{2} \, m_{t\beta} | T M ) c_{\alpha}^\dagger h_{\beta}^\dagger | {\rm HF} \rangle \\[12pt] &=& (-1)^{j_a + j_b + J + T} | a \, b^{-1} ; J \, M\, ; T \, M_T \rangle. \end{eqnarray}$$
Relation between isospin and proton-neutron representations
As shown in the previous section, we can calculate the Clebsch-Gordan coefficients in the isospin part, which results in
\begin{eqnarray} && | a_1 \, a_2^{-1} ; J \, M ; 0 \, 0 \rangle = \frac{1}{\sqrt{2}} \left( | n_1 \, n_2^{-1} ; J \, M \rangle + | p_1 \, p_2^{-1}l J \, M \rangle \right), \\[12pt] && | a_1 \, a_2^{-1} ; J \, M ; 1 \, 0 \rangle = \frac{1}{\sqrt{2}} \left( | n_1 \, n_2^{-1} ; J \, M \rangle - | p_1 \, p_2^{-1}l J \, M \rangle \right). \end{eqnarray}
For the inverse relation,
\begin{eqnarray} &&| n_1 \, n_2^{-1} ; J \, M \rangle = \frac{1}{\sqrt{2}} \left( |a_1 \, a_2^{-1} ; J \, M; 0 \, 0 \rangle + | a_1 \, a_2^{-1} ; J \, M ; 1 \, 0 \rangle \right), \\[12pt] && | p_1 \, p_2^{-1} ; J \, M \rangle = \frac{1}{\sqrt{2}} \left( |a_1 \, a_2^{-1} ; J \, M; 0 \, 0 \rangle - | a_1 \, a_2^{-1} ; J \, M ; 1 \, 0 \rangle \right). \end{eqnarray}
Finally, we establish the formulae of two-particles, two-holes, and particle-hole in isospin representation.