calc_radial_matrix_element_from_w_z

rydstate.radial.calc_radial_matrix_element_from_w_z(z1, w1, z2, w2, k_radial=0, integration_method='sum')[source]

Calculate the radial matrix element of two wavefunctions w1(z1) and w2(z2).

Computes the integral

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

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:

  • z1 (ndarray[tuple[Any, ...], dtype[Any]]) – z coordinates of the first wavefunction

  • w1 (ndarray[tuple[Any, ...], dtype[Any]]) – w(z) values of the first wavefunction

  • z2 (ndarray[tuple[Any, ...], dtype[Any]]) – z coordinates of the second wavefunction

  • w2 (ndarray[tuple[Any, ...], dtype[Any]]) – w(z) values of the second wavefunction

  • 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

Return type:

float