calc_reduced_angular_matrix_element

ryd_numerov.angular.calc_reduced_angular_matrix_element(s1, l1, j1, s2, l2, j2, operator, kappa, _lazy_evaluation=True)[source]

Calculate the reduced matrix element \(\langle j2 || \hat{O}_{k0} || j1 \rangle\).

The reduced matrix elements \(\langle j2 || \hat{O}_{k0} || j1 \rangle\) for \(\bra{j2} = \bra{\gamma_2, s2, l2, j2}\) and \(\ket{j1} = \ket{\gamma_1, s1, l1, j1}\) simplify for the special cases \(s2 = s1\) or \(l2 = l1\) to the following expressions: (see https://www.phys.ksu.edu/reu2015/danielkeylon/Wigner.pdf, and Edmonds: “Angular Momentum in Quantum Mechanics”)

For \(s2 = s1\) (i.e. when \(\hat{O}_{k0}\) only acts on \(l\)), the reduced matrix element is given by

\[\langle \gamma_{2}, s2, l2, j2 || \hat{O}_{k0} || \gamma_{1}, s2, l1, j1 \rangle = (-1)^{s2 + l2 + j1 + \kappa} \sqrt{2j + 1} \sqrt{2j1 + 1} \text{wigner_6j}(l2, j2, s2, j1, l1, \kappa) \langle \gamma_{2}, l2 || \hat{O}_{k0} || \gamma_{1}, l1 \rangle\]

And for \(l2 = l1\) (i.e. when \(\hat{O}_{k0}\) only acts on \(s\)), the reduced matrix element is given by

\[\langle \gamma_{2}, s2, l2, j2 || \hat{O}_{k0} || \gamma_{1}, s1, l2, j1 \rangle = (-1)^{s2 + l2 + j1 + \kappa} \sqrt{2j + 1} \sqrt{2 * j1 + 1} \text{wigner_6j}(s2, j2, l2, j1, s1, \kappa) \langle \gamma_{2}, s2 || \hat{O}_{k0} || \gamma_{1}, s1 \rangle\]

Note we changed the formulas to match the pairinteraction paper convention: https://doi.org/10.1088/1361-6455/aa743a

Parameters:

  • s1 (float) – The spin quantum number of the initial state.

  • l1 (int) – The orbital quantum number of the initial state.

  • j1 (float) – The total angular momentum quantum number of the initial state.

  • s2 (float) – The spin quantum number of the final state.

  • l2 (int) – The orbital quantum number of the final state.

  • j2 (float) – The total angular momentum quantum number of the final state.

  • operator (Literal['MAGNETIC', 'ELECTRIC', 'SPHERICAL', 'MAGNETIC_S', 'MAGNETIC_L']) – The operator type \(\hat{O}_{kq}\) for which to calculate the matrix element. Can be one of “MAGNETIC”, “ELECTRIC”, “SPHERICAL”.

  • kappa (int) – The quantum number \(\kappa\) of the angular momentum operator.

  • _lazy_evaluation (bool)

Return type:

float

Returns:

The reduced matrix element \(\langle j2 || \hat{O}_{k0} || j1 \rangle\).