WavefunctionNumerov
Class Methods
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Create a Wavefunction object. |
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Set the sign of the wavefunction according to the sign convention. |
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Run the Numerov integration of the radial Schrödinger equation. |
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Do some sanity checks on the wavefunction. |
Class Attributes and Properties
The radial wavefunction tilde{R}(r) in atomic units tilde{R}(r) = a_0^{-3/2} R(r). |
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The dimensionless wavefunction tilde{u}(x) = sqrt(a_0) r R(r). |
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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). |
- class ryd_numerov.radial.WavefunctionNumerov(state, grid, model)[source]
Create a Wavefunction object.
- Parameters:
- integrate(run_backward=True, w0=1e-10, *, _use_njit=True)[source]
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:
\[\tilde{u}(x) = \sqrt(a_0) r R(r)\]\[w(z) = z^{-1/2} \tilde{u}(x=z^2) = (r/a_0)^{-1/4} \sqrt(a_0) r R(r)\]where z = sqrt(r/a_0) is the dimensionless scaled coordinate.
The resulting radial wavefunction is normalized such that
\[\int_{0}^{\infty} r^2 |R(x)|^2 dr = \int_{0}^{\infty} |\tilde{u}(x)|^2 dx = \int_{0}^{\infty} 2 z^2 |w(z)|^2 dz = 1\]- Parameters:
(default (_use_njit) – True): Wheter to integrate the radial Schrödinger equation “backward” of “forward”.
(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.
(default – True): Whether to use the fast njit version of the Numerov integration.
run_backward (bool)
w0 (float)
_use_njit (bool)
- Return type:
None
- sanity_check(z_stop, run_backward)[source]
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)
- Return type:
bool- Parameters:
z_stop (float)
run_backward (bool)
- apply_sign_convention(sign_convention)
Set the sign of the wavefunction according to the sign convention.
- Parameters:
sign_convention (
Literal[None,'positive_at_outer_bound','n_l_1']) – 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.- Return type:
None
- property r_list: ndarray[tuple[int, ...], dtype[Any]]
The radial wavefunction tilde{R}(r) in atomic units tilde{R}(r) = a_0^{-3/2} R(r).
- property u_list: ndarray[tuple[int, ...], dtype[Any]]
The dimensionless wavefunction tilde{u}(x) = sqrt(a_0) r R(r).
- property w_list: ndarray[tuple[int, ...], dtype[Any]]
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).