Source code for ryd_numerov.radial.wavefunction

import logging
from abc import ABC, abstractmethod
from typing import TYPE_CHECKING, Literal, Optional

import numpy as np
from mpmath import whitw
from scipy.special import gamma

from ryd_numerov.radial.numerov import _run_numerov_integration_python, run_numerov_integration

if TYPE_CHECKING:
    from ryd_numerov.model import Model
    from ryd_numerov.radial.grid import Grid
    from ryd_numerov.rydberg import _CommonRydbergState
    from ryd_numerov.units import NDArray

logger = logging.getLogger(__name__)

WavefunctionSignConvention = Literal[None, "positive_at_outer_bound", "n_l_1"]


[docs] class Wavefunction(ABC): r"""An object containing all the relevant information about the radial wavefunction.""" def __init__( self, state: "_CommonRydbergState", grid: "Grid", ) -> None: """Create a Wavefunction object. Args: state: The RydbergState object. grid: The grid object. """ self.state = state self.grid = grid self._w_list: Optional[NDArray] = None @property def w_list(self) -> "NDArray": r"""The dimensionless scaled wavefunction w(z) = z^{-1/2} \tilde{u}(x=z^2) = (r/a_0)^{-1/4} sqrt(a_0) r R(r).""" if self._w_list is None: self.integrate() assert self._w_list is not None return self._w_list @property def u_list(self) -> "NDArray": r"""The dimensionless wavefunction \tilde{u}(x) = sqrt(a_0) r R(r).""" return np.sqrt(self.grid.z_list) * self.w_list @property def r_list(self) -> "NDArray": r"""The radial wavefunction \tilde{R}(r) in atomic units \tilde{R}(r) = a_0^{-3/2} R(r).""" return self.u_list / self.grid.x_list
[docs] @abstractmethod def integrate(self) -> None: """Integrate the radial Schrödinger equation and store the wavefunction in the w_list attribute."""
[docs] def apply_sign_convention(self, sign_convention: WavefunctionSignConvention) -> None: """Set the sign of the wavefunction according to the sign convention. Args: sign_convention: The sign convention for the wavefunction. - None: Leave the wavefunction as it is. - "n_l_1": The wavefunction is defined to have the sign of (-1)^{(n - l - 1)} at the outer boundary. - "positive_at_outer_bound": The wavefunction is defined to be positive at the outer boundary. """ if self._w_list is None: raise ValueError("The wavefunction has not been integrated yet.") if sign_convention is None: return current_outer_sign = 1 for w in self._w_list[::-1]: if w != 0 and not np.isnan(w): current_outer_sign = np.sign(w) break if sign_convention == "n_l_1": assert self.state.n is not None, "n must be given to apply the n_l_1 sign convention." if current_outer_sign != (-1) ** (self.state.n - self.state.l - 1): self._w_list = -self._w_list elif sign_convention == "positive_at_outer_bound": if current_outer_sign != 1: self._w_list = -self._w_list else: raise ValueError(f"Unknown sign convention: {sign_convention}")
[docs] class WavefunctionNumerov(Wavefunction): def __init__( self, state: "_CommonRydbergState", grid: "Grid", model: "Model", ) -> None: """Create a Wavefunction object. Args: state: The RydbergState object. grid: The grid object. model: The model object. """ super().__init__(state, grid) self.model = model
[docs] def integrate(self, run_backward: bool = True, w0: float = 1e-10, *, _use_njit: bool = True) -> None: r"""Run the Numerov integration of the radial Schrödinger equation. The resulting radial wavefunctions are then stored as attributes, where - w_list is the dimensionless and scaled wavefunction w(z) - u_list is the dimensionless wavefunction \tilde{u}(x) - r_list is the radial wavefunction R(r) in atomic units The radial wavefunction are related as follows: .. math:: \tilde{u}(x) = \sqrt(a_0) r R(r) .. math:: w(z) = z^{-1/2} \tilde{u}(x=z^2) = (r/a_0)^{-1/4} \sqrt(a_0) r R(r) where z = sqrt(r/a_0) is the dimensionless scaled coordinate. The resulting radial wavefunction is normalized such that .. math:: \int_{0}^{\infty} r^2 |R(x)|^2 dr = \int_{0}^{\infty} |\tilde{u}(x)|^2 dx = \int_{0}^{\infty} 2 z^2 |w(z)|^2 dz = 1 Args: run_backward (default: True): Wheter to integrate the radial Schrödinger equation "backward" of "forward". w0 (default: 1e-10): The initial magnitude of the radial wavefunction at the outer boundary. For forward integration we set w[0] = 0 and w[1] = w0, for backward integration we set w[-1] = 0 and w[-2] = (-1)^{(n - l - 1) % 2} * w0. _use_njit (default: True): Whether to use the fast njit version of the Numerov integration. """ if self._w_list is not None: raise ValueError("The wavefunction was already integrated, you should not integrate it again.") # Note: Inside this method we use y and x like it is used in the numerov function # and not like in the rest of this class, i.e. y = w(z) and x = z grid = self.grid glist = ( 8 * self.state.element.reduced_mass_factor * grid.z_list * grid.z_list * (self.state.get_energy(unit="a.u.") - self.model.calc_total_effective_potential(grid.x_list)) ) if run_backward: # During the Numerov integration we define the wavefunction such that it should always stop # at the inner boundary with positive weight # Note: n - l - 1 is the number of nodes of the radial wavefunction # Thus, the sign of the wavefunction at the outer boundary is (-1)^{(n - l - 1) % 2} # You can choose a different sign convention by calling the method apply_sign_convention() afterwards. y0, y1 = 0, w0 x_start, x_stop, dx = grid.z_max, grid.z_min, -grid.dz g_list_directed = glist[::-1] # We set x_min to the classical turning point # after x_min is reached in the integration, the integration stops, as soon as it crosses the x-axis again # or it reaches a local minimum (thus going away from the x-axis) # the reason for this is that the second derivative of the wavefunction d^2/dz^2 w(z) (= concavity) # can only vanish at either # i) where w(z) = 0 or ii) where the potential is equal to the energy (-> classical turning point) # If we further assume, that the wavefunction converges to zero at the inner boundary, # we know that after the inner classical turning point # the wavefunction should never increase the distance from the x-axis again. x_min = self.model.calc_turning_point_z(self.state.get_energy("a.u.")) else: # forward y0, y1 = 0, w0 x_start, x_stop, dx = grid.z_min, grid.z_max, grid.dz g_list_directed = glist n = self.state.n if self.state.n is not None else self.state.get_n_star() x_min = np.sqrt(n * (n + 15)) if _use_njit: w_list_list = run_numerov_integration(x_start, x_stop, dx, y0, y1, g_list_directed, x_min) else: logger.warning("Using python implementation of Numerov integration, this is much slower!") w_list_list = _run_numerov_integration_python(x_start, x_stop, dx, y0, y1, g_list_directed, x_min) w_list = np.array(w_list_list) if run_backward: w_list = w_list[::-1] grid.set_grid_range(step_start=grid.steps - len(w_list)) else: grid.set_grid_range(step_stop=len(w_list)) # normalize the wavefunction, see docstring norm = np.sqrt(2 * np.sum(w_list * w_list * grid.z_list * grid.z_list) * grid.dz) w_list /= norm self._w_list = w_list self.sanity_check(x_stop, run_backward)
[docs] def sanity_check(self, z_stop: float, run_backward: bool) -> bool: # noqa: C901, PLR0915, PLR0912 """Do some sanity checks on the wavefunction. Check if the wavefuntion fulfills the following conditions: - The wavefunction is positive (or zero) at the inner boundary. - The wavefunction is close to zero at the inner boundary. - The wavefunction is close to zero at the outer boundary. - The wavefunction has exactly (n - l - 1) nodes. - The integration stopped before z_stop (for l>0) """ warning_msgs: list[str] = [] grid = self.grid state = self.state # Check and Correct if divergence of the wavefunction w_list_abs = np.abs(self.w_list) idmax = np.argmax(w_list_abs) w_abs_max = w_list_abs[idmax] outer_max = np.max(w_list_abs[int(0.1 * grid.steps) :]) if idmax <= 5 and w_abs_max / outer_max > 10: warning_msgs.append( f"Wavefunction diverges at the inner boundary, w_abs_max / outer_max={w_abs_max / outer_max:.2e}", ) warning_msgs.append("Trying to correct the wavefunction.") first_ind = np.argwhere(w_list_abs < outer_max)[0][0] self._w_list = self._w_list[first_ind:] # type: ignore [index] grid.set_grid_range(step_start=first_ind) norm = np.sqrt(2 * np.sum(self.w_list * self.w_list * grid.z_list * grid.z_list) * grid.dz) self._w_list /= norm # Check the maximum of the wavefunction idmax = np.argmax(np.abs(self.w_list)) if idmax < 0.05 * grid.steps: warning_msgs.append( f"The maximum of the wavefunction is close to the inner boundary (idmax={idmax}) " "probably due to inner divergence of the wavefunction. " ) # Check the weight of the wavefunction at the inner boundary inner_ind = 10 inner_weight = ( 2 * np.sum( self.w_list[:inner_ind] * self.w_list[:inner_ind] * grid.z_list[:inner_ind] * grid.z_list[:inner_ind] ) * grid.dz ) inner_weight_scaled_to_whole_grid = inner_weight * grid.steps / inner_ind tol = 1e-4 # for low n the wavefunction converges not as good and still has more weight at the inner boundary n = state.n if state.n is not None else state.get_n_star() + 5 if n <= 10: tol = 8e-3 elif n <= 16: tol = 2e-3 if inner_weight_scaled_to_whole_grid > tol: warning_msgs.append( f"The wavefunction is not close to zero at the inner boundary" f" (inner_weight_scaled_to_whole_grid={inner_weight_scaled_to_whole_grid:.2e})" ) # Check the wavefunction at the outer boundary outer_ind = int(0.95 * grid.steps) outer_wf = self.w_list[outer_ind:] if np.mean(outer_wf) > 1e-7: warning_msgs.append( f"The wavefunction is not close to zero at the outer boundary, mean={np.mean(outer_wf):.2e}" ) outer_weight = 2 * np.sum(outer_wf * outer_wf * grid.z_list[outer_ind:] * grid.z_list[outer_ind:]) * grid.dz outer_weight_scaled_to_whole_grid = outer_weight * grid.steps / len(outer_wf) if outer_weight_scaled_to_whole_grid > 1e-10: warning_msgs.append( f"The wavefunction is not close to zero at the outer boundary," f" (outer_weight_scaled_to_whole_grid={outer_weight_scaled_to_whole_grid:.2e})" ) # Check the number of nodes nodes = np.sum(np.abs(np.diff(np.sign(self.w_list)))) // 2 if state.n is not None and nodes != state.n - state.l - 1: warning_msgs.append(f"The wavefunction has {nodes} nodes, but should have {state.n - state.l - 1} nodes.") # Check that numerov stopped and did not run until x_stop if state.l > 0: if run_backward and z_stop > grid.z_list[0] - grid.dz / 2 and inner_weight_scaled_to_whole_grid > 1e-6: warning_msgs.append(f"The integration did not stop before z_stop, z={grid.z_list[0]}, z_stop={z_stop}") if not run_backward and z_stop < grid.z_list[-1] + grid.dz / 2: warning_msgs.append(f"The integration did not stop before z_stop, z={grid.z_list[-1]}") elif state.l == 0 and run_backward: if grid.z_list[0] > 0.035: # z_list[0] should run almost to zero for l=0 warning_msgs.append(f"The integration for l=0 did stop at {grid.z_list[0]} (should be close to zero).") if warning_msgs: msg = f"The wavefunction for the state {state} has some issues:" msg += "\n ".join(["", *warning_msgs]) logger.warning(msg) return False return True
[docs] class WavefunctionWhittaker(Wavefunction):
[docs] def integrate(self) -> None: logger.warning("Using Whittaker to get the wavefunction is not recommended! Use this only for comparison.") l = self.state.l nu = self.state.get_n_star() whitw_vectorized = np.vectorize(whitw, otypes=[float]) whitw_list = whitw_vectorized(nu, l + 0.5, 2 * self.grid.x_list / nu) u_list: NDArray = whitw_list / np.sqrt(nu**2 * gamma(nu + l + 1) * gamma(nu - l)) w_list: NDArray = u_list / np.sqrt(self.grid.z_list) self._w_list = w_list