calc_angular_matrix_element

ryd_numerov.angular.calc_angular_matrix_element(s1, l1, j1, m1, s2, l2, j2, m2, operator, kappa, q)[source]

Calculate the angular matrix element $bra{state_2} hat{O}_{kq} ket{state_1}$.

For the states $bra{state_2} = bra{s2,l2,j2,m2}$ and $ket{state_1} = ket{s1,l1,j1,m1}$, the angular matrix elements of the operator $\hat{O}_{kq}$ is given by

\[\bra{state_2} \hat{O}_{kq} \ket{state_1} = \bra{s2,l2,j2,m2} \hat{O}_{kq} \ket{s1,l1,j1,m1} = \langle j1, m1, k, q | j2, m2 \rangle \langle j2 || \hat{O}_{k0} || j1 \rangle / \sqrt{2 * j2 + 1} = (-1)^{j1 - \kappa + m2} wigner_3j(j1, kappa, j2, m1, q, -m2) \langle j2 || \hat{O}_{k0} || j1 \rangle\]

where we first used the Wigner-Eckhart theorem and then the Wigner 3-j symbol to express the Clebsch-Gordan coefficient.

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.
m1 (float) – The magnetic 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.
m2 (float) – The magnetic 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.
q (int) – The quantum number $q$ of the angular momentum operator.

Return type:

float

Returns:

The angular matrix element $bra{state_2} hat{O}_{kq} ket{state_1}$.