MatrixPotential1S¶

About the `MatrixPotential1S` class¶

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Class documentation¶

class WaveBlocksND.MatrixPotential1S(expression, variables, **kwargs)[source]¶

This class represents a scalar potential $V(x)$ . The potential is given as an analytic $1 \times 1$ matrix expression. Some symbolic calculations with the potential are supported.

calculate_eigenvalues()[source]¶: Calculate the eigenvalue $\lambda_0(x)$ of the potential $V(x)$ . In the scalar case this is just equal to the matrix entry $V_{0,0}(x)$ . Note: This function is idempotent and the eigenvalues are memoized for later reuse.

calculate_eigenvectors()[source]¶: Calculate the eigenvector $\nu_0(x)$ of the potential $V(x)$ . 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)[source]¶

Calculate the matrix exponential $\exp(\alpha V)$ . In the case of this class the matrix is of size $1 \times 1$ thus the exponential simplifies to the scalar exponential function. Note: This function is idempotent.

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

calculate_hessian()[source]¶: Calculate the Hessian matrix $\nabla^2 V$ of the potential $V(x)$ with $x \in \mathbb{R}^D$ . For potentials which depend only one variable, this equals the second derivative and $D=1$ . Note that this function is idempotent.

calculate_jacobian()[source]¶: Calculate the Jacobian matrix $\nabla V$ of the potential $V(x)$ with $x \in \mathbb{R}^D$ . For potentials which depend only one variable, this equals the first derivative and $D=1$ . Note that this function is idempotent.

calculate_jacobian_canonical()[source]¶: Calculate the Jacobian matrix $\nabla V(x)$ of the potential $V(x)$ with $x \in \mathbb{R}^D$ . For potentials which depend only one variable, this equals the first derivative and $D=1$ . Note that this function is idempotent.

calculate_local_quadratic(diagonal_component=None)[source]¶

Calculate the local quadratic approximation $U(x)$ of the potential’s eigenvalue $\lambda(x)$ . Note that this function is idempotent.

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

calculate_local_remainder(diagonal_component=None)[source]¶

Calculate the non-quadratic remainder $W(x) = V(x) - U(x)$ of the quadratic Taylor approximation $U(x)$ of the potential’s eigenvalue $\lambda(x)$ . Note that this function is idempotent.

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

evaluate_at(grid, entry=None, as_matrix=False)[source]¶

Evaluate the potential $V(x)$ elementwise on a grid $\Gamma$ .

Parameters:

grid (A Grid instance. (Numpy arrays are not directly supported yet.)) – The grid containing the nodes $\gamma_i$ we want to evaluate the potential at.
entry (A python tuple of two integers.) – The indices $(i,j)$ of the component $V_{i,j}(x)$ we want to evaluate or None to evaluate all entries. This has no effect here as we only have a single entry $V_{0,0}$ .
as_matrix – Dummy parameter which has no effect here.

Returns:

A list containing a single numpy ndarray of shape $(1,|\Gamma|)$ .

evaluate_eigenvalues_at(grid, entry=None, as_matrix=False)[source]¶

Evaluate the eigenvalue $\lambda_0(x)$ elementwise on a grid $\Gamma$ .

Parameters:

grid (A Grid instance. (Numpy arrays are not directly supported yet.)) – The grid containing the nodes $\gamma_i$ we want to evaluate the eigenvalue at.
entry (A python tuple of two integers.) – The indices $(i,j)$ of the component $\Lambda_{i,j}(x)$ we want to evaluate or None to evaluate all entries. If $j = i$ then we evaluate the eigenvalue $\lambda_i(x)$ . This has no effect here as we only have a single entry $\lambda_0$ .
as_matrix – Dummy parameter which has no effect here.

Returns:

A list containing a single numpy ndarray of shape $(N_1, ... ,N_D)$ .

evaluate_eigenvectors_at(grid, entry=None)[source]¶

Evaluate the eigenvector $\nu_0(x)$ elementwise on a grid $\Gamma$ .

Parameters:	grid (A `Grid` instance. (Numpy arrays are not directly supported yet.)) – The grid containing the nodes $\gamma_i$ we want to evaluate the eigenvector at. entry (A singly python integer.) – The index $i$ of the eigenvector $\nu_i(x)$ we want to evaluate or `None` to evaluate all eigenvectors. This has no effect here as we only have a single entry $\nu_0$ .
Returns:	A list containing the numpy ndarrays, all of shape $(1, \|\Gamma\|)$ .

evaluate_exponential_at(grid, entry=None)[source]¶

Evaluate the exponential of the potential matrix $V(x)$ on a grid $\Gamma$ .

Parameters:	grid (A `Grid` instance. (Numpy arrays are not directly supported yet.)) – The grid containing the nodes $\gamma_i$ we want to evaluate the exponential at.
Returns:	The numerical approximation of the matrix exponential at the given grid nodes.

evaluate_hessian_at(grid, component=None)[source]¶

Evaluate the potential’s Hessian $\nabla^2 V(x)$ at some grid nodes $\Gamma$ .

Parameters:	grid – 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. The result is an `ndarray` of shape $(D,D)$ is we evaluate at a single grid node or of shape $(\|\Gamma\|,D,D)$ if we evaluate at multiple nodes simultaneously.

evaluate_jacobian_at(grid, component=None)[source]¶

Evaluate the potential’s Jacobian $\nabla \Lambda(x)$ at some grid nodes $\Gamma$ .

Parameters:	grid – 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. The result is an `ndarray` of shape $(D,1)$ is we evaluate at a single grid node or of shape $(D,\|\Gamma\|)$ if we evaluate at multiple nodes simultaneously.

evaluate_jacobian_canonical_at(grid, component=None)[source]¶

Evaluate the potential’s Jacobian $\nabla V(x)$ at some grid nodes $\Gamma$ .

Parameters:	grid – 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. The result is an `ndarray` of shape $(D,1)$ is we evaluate at a single grid node or of shape $(D,\|\Gamma\|)$ if we evaluate at multiple nodes simultaneously.

evaluate_local_quadratic_at(grid, diagonal_component=None)[source]¶

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

Parameters:	grid – The grid nodes $\Gamma$ the quadratic approximation gets evaluated at. diagonal_component – Dummy parameter that has no effect here.
Returns:	A list containing the values $V(\Gamma)$ , $\nabla V(\Gamma)$ and $\nabla^2 V(\Gamma)$ .

evaluate_local_remainder_at(grid, position, diagonal_component=None, entry=None)[source]¶

Numerically evaluate the non-quadratic remainder $W(x)$ of the quadratic approximation $U(x)$ of the potential’s eigenvalue $\lambda(x)$ at the given nodes $\Gamma$ .

Parameters:	grid – The grid nodes $\Gamma$ the remainder $W$ gets evaluated at. position – The point $q \in \mathbb{R}^D$ where the Taylor series is computed. diagonal_component – Dummy parameter that has no effect here. entry – Dummy parameter that has no effect here.
Returns:	A list with a single entry consisting of an `ndarray` containing the values of $W(\Gamma)$ . The array is of shape $(1,\|\Gamma\|)$ .

get_dimension()¶: Return the dimension $D$ of the potential $V(x)$ . The dimension is equal to the number of free variables $x_i$ where $x := (x_1, x_2, ..., x_D)$ .

get_number_components()¶: Return the number $N$ of components the potential $V(x)$ supports. This is equivalent to the number of energy levels $\lambda_i(x)$ .

MatrixPotential1S¶

About the `MatrixPotential1S` class¶

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Class documentation¶

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MatrixPotential1S¶

About the MatrixPotential1S class¶

Inheritance diagram¶

Class documentation¶

About the `MatrixPotential1S` class¶