MatrixPotential1S¶

About the `MatrixPotential1S` class¶

The WaveBlocks Project

Inheritance diagram¶

Inheritance diagram of MatrixPotential1S

Class documentation¶

class WaveBlocks.MatrixPotential1S(expression, variables)¶

This class represents a scalar potential $V\left(x\right)$ . The potential is given as an analytical $1 \times 1$ matrix expression. Some symbolic calculations with the potential are supported. For example calculation of eigenvalues and exponentials and numerical evaluation. Further, there are methods for splitting the potential into a Taylor expansion and for basis transformations between canonical and eigenbasis.

calculate_eigenvalues()¶: Calculate the eigenvalue $\lambda_0\left(x\right)$ of the potential $V\left(x\right)$ . In the scalar case this is just the matrix entry $V_{0,0}$ .

Note

This function is idempotent and the eigenvalues are memoized for later reuse.

calculate_eigenvectors()¶: Calculate the eigenvector $nu_0\left(x\right)$ of the potential $V\left(x\right)$ . In the scalar case this is just the value $1$ .

Note

This function is idempotent and the eigenvectors are memoized for later reuse.

calculate_exponential(factor=1)¶

Calculate the matrix exponential $E = \exp\left(\alpha M\right)$ . In this case the matrix is of size $1 \times 1$ thus the exponential simplifies to the scalar exponential function.

Parameters:	factor – A prefactor $\alpha$ in the exponential.

Note

This function is idempotent.

calculate_hessian()¶: Calculate the hessian matrix for component $V_{0,0}$ of the potential. For potentials which depend only one variable $x$ , this equals the second derivative.

calculate_jacobian()¶: Calculate the jacobian matrix for the component $V_{0,0}$ of the potential. For potentials which depend only one variable $x$ , this equals the first derivative.

calculate_local_quadratic(diagonal_component=None)¶

Calculate the local quadratic approximation $U$ of the potential’s eigenvalue $\lambda$ .

Parameters:	diagonal_component – Dummy parameter that has no effect here.

Note

This function is idempotent.

calculate_local_remainder(diagonal_component=None)¶

Calculate the non-quadratic remainder $W$ of the quadratic approximation $U$ of the potential’s eigenvalue $\lambda$ . This function is used for the homogeneous case and takes into account the leading component $\chi$ .

Parameters:	diagonal_component – Dummy parameter that has no effect here.

Note

This function is idempotent.

evaluate_at(nodes, component=0, as_matrix=False)¶

Evaluate the potential matrix elementwise at some given grid nodes $\gamma$ .

Parameters:	nodes – The grid nodes $\gamma$ we want to evaluate the potential at. component – The component $V_{i,j}$ that gets evaluated or ‘None’ to evaluate all. as_matrix – Dummy parameter which has no effect here.
Returns:	A list with the single entry evaluated at the nodes.

evaluate_eigenvalues_at(nodes, component=None, as_matrix=False)¶

Evaluate the eigenvalue $\lambda_0\left(x\right)$ at some grid nodes $\gamma$ .

Parameters:	nodes – The grid nodes $\gamma$ we want to evaluate the eigenvalue at. diagonal_component – Dummy parameter that has no effect here. as_matrix – Dummy parameter which has no effect here.
Returns:	A list with the single eigenvalue evaluated at the nodes.

evaluate_eigenvectors_at(nodes)¶

Evaluate the eigenvector $nu_0\left(x\right)$ at some grid nodes $\gamma$ .

Parameters:	nodes – The grid nodes $\gamma$ we want to evaluate the eigenvector at.
Returns:	A list with the eigenvector evaluated at the given nodes.

evaluate_exponential_at(nodes)¶

Evaluate the exponential of the potential matrix $V$ at some grid nodes $\gamma$ .

Parameters:	nodes – The grid nodes $\gamma$ we want to evaluate the exponential at.
Returns:	The numerical approximation of the matrix exponential at the given grid nodes.

evaluate_hessian_at(nodes, component=None)¶

Evaluate the potential’s hessian at some grid nodes $\gamma$ .

Parameters:	nodes – The grid nodes $\gamma$ the hessian gets evaluated at. component – Dummy parameter that has no effect here.
Returns:	The value of the potential’s hessian at the given nodes.

evaluate_jacobian_at(nodes, component=None)¶

Evaluate the potential’s jacobian at some grid nodes $\gamma$ .

Parameters:	nodes – The grid nodes $\gamma$ the jacobian gets evaluated at. component – Dummy parameter that has no effect here.
Returns:	The value of the potential’s jacobian at the given nodes.

evaluate_local_quadratic_at(nodes, diagonal_component=None)¶

Numerically evaluate the local quadratic approximation $U$ of the potential’s eigenvalue $\lambda$ at the given grid nodes $\gamma$ . This function is used for the homogeneous case.

Parameters:	nodes – The grid nodes $\gamma$ we want to evaluate the quadratic approximation at.
Returns:	An array containing the values of $U$ at the nodes $\gamma$ .

evaluate_local_remainder_at(position, nodes, diagonal_component=None, component=None)¶

Numerically evaluate the non-quadratic remainder $W$ of the quadratic approximation $U$ of the potential’s eigenvalue $\lambda$ at the given nodes $\gamma$ . This function is used for the homogeneous and the inhomogeneous case and just evaluates the remainder $W$ .

Parameters:	position – The point $q$ where the Taylor series is computed. nodes – The grid nodes $\gamma$ we want to evaluate the potential at. component – Dummy parameter that has no effect here.
Returns:	A list with a single entry consisting of an array containing the values of $W$ at the nodes $\gamma$ .

get_number_components()¶

Returns:	The number $N$ of components the potential supports. In the one dimensional case, it’s just 1.

potential = None¶: The matrix of the potential $V\left(x\right)$ .

project_to_canonical(nodes, values, basis=None)¶

Project a given vector from the potential’s eigenbasis to the canonical basis.

Parameters:	nodes – The grid nodes $\gamma$ for the pointwise transformation. values – The list of vectors $\varphi_i$ containing the values we want to transform. basis – A list of basis vectors $nu_i$ . Allows to use this function for external data, similar to a static function.
Returns:	This method does nothing and returns the values.

project_to_eigen(nodes, values, basis=None)¶

Project a given vector from the canonical basis to the eigenbasis of the potential.

Parameters:	nodes – The grid nodes $\gamma$ for the pointwise transformation. values – The list of vectors $\varphi_i$ containing the values we want to transform. basis – A list of basisvectors $nu_i$ . Allows to use this function for external data, similar to a static function.
Returns:	This method does nothing and returns the values.

x = None¶: The variable $x$ that represents position space.

MatrixPotential1S¶

About the `MatrixPotential1S` class¶

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Class documentation¶

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MatrixPotential1S¶

About the MatrixPotential1S class¶

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Class documentation¶

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About the `MatrixPotential1S` class¶