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High-Fidelity Solver

SchroedingerEquation is a high-fidelity (HF), Schrödinger-equation solver for local, complex interactions.

By default, rose will provide HF solution using scipy.integrate.solve_ivp. For details about providing your own solutions, see Basis documentation.

SchroedingerEquation(interaction, rk_tols=[1e-09, 1e-09], s_0=None, domain=None)

Solver for the single-channel, reduced, radial Schrödinger equation using scipy.integrate https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.solve_ivp.html

All other solvers for the single-channel, reduced, radial Schrödinger equation inherit from this class

Solves the Shrödinger equation for local, complex potentials.

Parameters:

Name Type Description Default
interaction Interaction

See Interaction documentation.

 required
rk_tols list

2-element list of numbers specifying tolerances for the Runge-Kutta solver: the relative tolerance and the absolute tolerance

 [1e-09, 1e-09]
s_0 float)
None
domain ndarray)
None

Returns:

Name Type Description
solver SchroedingerEquation

instance of SchroedingerEquation
Source code in src/rose/schroedinger.py 
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def __init__(
    self,
    interaction: Interaction,
    rk_tols: list = [1e-9, 1e-9],
    s_0=None,
    domain=None,
):
    r"""Solves the Shrödinger equation for local, complex potentials.

    Parameters:
        interaction (Interaction): See [Interaction documentation](interaction.md).
        rk_tols (list): 2-element list of numbers specifying tolerances for the
            Runge-Kutta solver: the relative tolerance and the  absolute tolerance
        s_0 (float) :
        domain (ndarray) :

    Returns:
        solver (SchroedingerEquation): instance of `SchroedingerEquation`

    """
    if domain is None:
        domain = np.array([self.DEFAULT_S_MIN, self.DEFAULT_S_MAX])

    if s_0 is None:
        s_0 = domain[1] - np.pi
        assert s_0 > 0

    assert domain[1] > s_0

    self.s_0 = s_0
    self.domain = domain.copy()
    self.interaction = interaction
    self.rk_tols = rk_tols

    if self.interaction is not None:
        if self.interaction.k_c == 0:
            self.eta = 0
        else:
            # There is Coulomb, but we haven't (yet) worked out how to emulate
            # across energies, so we can precompute H+ and H- stuff.
            self.eta = self.interaction.eta(None)

        self.Hm = H_minus(self.s_0, self.interaction.ell, self.eta)
        self.Hp = H_plus(self.s_0, self.interaction.ell, self.eta)
        self.Hmp = H_minus_prime(self.s_0, self.interaction.ell, self.eta)
        self.Hpp = H_plus_prime(self.s_0, self.interaction.ell, self.eta)

        self.domain[0], self.init_cond = self.initial_conditions(
            self.eta,
            self.PHI_THRESHOLD,
            self.interaction.ell,
            self.domain[0],
        )

        assert self.domain[0] < self.s_0

delta(alpha, **kwargs)

Calculates the $\ell$-th partial wave phase shift

Parameters:

Name Type Description Default
alpha ndarray

parameter vector

 required

Returns:

Name Type Description
delta float

phase shift extracted from the reduced, radial wave function
Source code in src/rose/schroedinger.py 
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def delta(
    self,
    alpha: np.array,
    **kwargs,
):
    r"""Calculates the $\ell$-th partial wave phase shift

    Parameters:
        alpha (ndarray): parameter vector

    Returns:
        delta (float): phase shift extracted from the reduced, radial
            wave function

    """
    sl = self.smatrix(alpha, **kwargs)

    return np.log(sl) / 2j

initial_conditions(eta, phi_threshold, l, rho_0=None)

Returns:

Type Description

initial_conditions (tuple) : initial conditions [phi, phi'] at rho_0

Parameters:

Name Type Description Default
eta float

sommerfield param

 required
phi_threshold float

minimum $\phi$ value; The wave function is considered zero below this value.

 required
rho_0 float

starting point for the solver

 None
Source code in src/rose/schroedinger.py 
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def initial_conditions(self, eta: float, phi_threshold: float, l: int, rho_0=None):
    r"""
    Returns:
        initial_conditions (tuple) : initial conditions [phi, phi'] at rho_0

    Parameters:
        eta (float): sommerfield param
        phi_threshold (float): minimum $\phi$ value; The wave function is
            considered zero below this value.
        rho_0 (float): starting point for the solver
    """

    C_l = Gamow_factor(l, eta)
    min_rho_0 = (phi_threshold / C_l) ** (1 / (l + 1))

    if min_rho_0 > rho_0:
        rho_0 = min_rho_0

    phi_0 = C_l * rho_0 ** (l + 1)
    phi_prime_0 = C_l * (l + 1) * rho_0**l

    if self.interaction.is_complex:
        initial_conditions = np.array([phi_0 * (1 + 0j), phi_prime_0 * (1 + 0j)])
    else:
        initial_conditions = np.array([phi_0, phi_prime_0])

    return rho_0, initial_conditions

phi(alpha, s_mesh, **kwargs)

Computes the reduced, radial wave function $\phi$ (or $u$) on s_mesh using the Runge-Kutta method

Parameters:

Name Type Description Default
alpha ndarray

parameter vector

 required
s_mesh ndarray

values of $s$ at which $\phi$ is calculated

 required
kwargs dict)

passed to scipy.integrate.solve_ivp

 {}

Returns:

Name Type Description
phi ndarray

reduced, radial wave function
Source code in src/rose/schroedinger.py 
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def phi(
    self,
    alpha: np.array,
    s_mesh: np.array,
    **kwargs,
):
    r"""Computes the reduced, radial wave function $\phi$ (or $u$) on `s_mesh` using the
    Runge-Kutta method

    Parameters:
        alpha (ndarray): parameter vector
        s_mesh (ndarray): values of $s$ at which $\phi$ is calculated
        kwargs (dict) : passed to scipy.integrate.solve_ivp

    Returns:
        phi (ndarray): reduced, radial wave function

    """
    solution = np.zeros_like(s_mesh, dtype=np.complex128)
    mask = np.logical_and(s_mesh >= self.domain[0], s_mesh <= self.domain[1])
    phi = self.solve_se(alpha, **kwargs)
    solution[mask] = phi(s_mesh)[0][mask]
    return solution

rmatrix(alpha, **kwargs)

Calculates the $\ell$-th partial wave R-matrix element at the specified energy. using the Runge-Kutta method for integrating the Radial SE. kwargs are passed to solve_se.

Parameters:

Name Type Description Default
alpha ndarray

parameter vector

 required
kwargs dict)

passed to scipy.integrate.solve_ivp

 {}

Returns:

Type Description

rl (float) : r-matrix element, or logarithmic derivative of wavefunction at the channel radius; s_0
Source code in src/rose/schroedinger.py 
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def rmatrix(
    self,
    alpha: np.array,
    **kwargs,
):
    r"""Calculates the $\ell$-th partial wave R-matrix element at the specified energy.
        using the Runge-Kutta method for integrating the Radial SE. kwargs are passed to
        solve_se.

    Parameters:
        alpha (ndarray): parameter vector
        kwargs (dict) : passed to scipy.integrate.solve_ivp

    Returns:
        rl (float)  : r-matrix element, or logarithmic derivative of wavefunction at the channel
            radius; s_0

    """
    solution = self.solve_se(alpha, **kwargs)
    u = solution(self.s_0)
    rl = 1 / self.s_0 * (u[0] / u[1])
    return rl

solve_se(alpha, **kwargs)

Solves the reduced, radial Schrödinger equation using the builtin in Runge-Kutta solver in scipy.integrate.solve_ivp

Parameters:

Name Type Description Default
alpha ndarray

parameter vector

 required
kwargs dict)

passed to scipy.integrate.solve_ivp

 {}

Returns:

Type Description

sol (scipy.integrate.OdeSolution) : the radial wavefunction

and its first derivative; see https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.OdeSolution.html#scipy.integrate.OdeSolution
Source code in src/rose/schroedinger.py 
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def solve_se(
    self,
    alpha: np.array,
    **kwargs,
):
    r"""Solves the reduced, radial Schrödinger equation using the builtin in Runge-Kutta
        solver in scipy.integrate.solve_ivp

    Parameters:
        alpha (ndarray): parameter vector
        kwargs (dict) : passed to scipy.integrate.solve_ivp

    Returns:
        sol  (scipy.integrate.OdeSolution) : the radial wavefunction
        and its first derivative; see https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.OdeSolution.html#scipy.integrate.OdeSolution
    """

    args = self.interaction.bundle_gcoeff_args(alpha)
    sol = solve_ivp(
        lambda s, phi: np.array(
            [
                phi[1],
                -1 * g_coeff(s, *args) * phi[0],
            ]
        ),
        self.domain,
        self.init_cond,
        rtol=self.rk_tols[0],
        atol=self.rk_tols[1],
        dense_output=True,
        **kwargs,
    )

    return sol.sol