calc_radial_matrix_element

ryd_numerov.radial.calc_radial_matrix_element(state1, state2, k_radial=0, integration_method='sum')[source]

Calculate the radial matrix element between two Rydberg states.

Computes the integral

\[\int_{0}^{\infty} dr r^2 r^\kappa R_1(r) R_2(r) = a_0^\kappa \int_{0}^{\infty} dx x^\kappa \tilde{u}_1(x) \tilde{u}_2(x) = a_0^\kappa \int_{0}^{\infty} dz 2 z^{2 + 2\kappa} w_1(z) w_2(z)\]

where R_1 and R_2 are the radial wavefunctions of the two states and w(z) = z^{-1/2} tilde{u}(z^2) = (r/_a_0)^{1/4} sqrt{a_0} r R(r).

Parameters:

state1 (RydbergState) – First Rydberg state
state2 (RydbergState) – Second Rydberg state
k_radial (int) – Power of r in the matrix element (default=0, this corresponds to the overlap integral int dr r^2 R_1(r) R_2(r))
integration_method (Literal['sum', 'trapezoid', 'scipy_simpson', 'scipy_trapezoid']) – Integration method to use, one of [“sum”, “trapezoid”, “scipy_simpson”, “scipy_trapezoid”] (default=”sum”)

Returns:

The radial matrix element in atomic units.

Return type:

float