SWPEigenstates

Here are defined the classes allowing one to create and use the eigenstates of the 1-dimendional (1D) Square Well Potential (SWP) case.

The two classes are:

• SWPSiegert, the class of Siegert states of the 1DSWP
• SWPContinuum, the class of Continuum states

Both classes derive from the SWPEigenstate abstract class, that forces to redefine the scalar product so that it is computed analytically when possible.

class siegpy.swpeigenstates.SWPEigenstate(k, parity, potential, grid, analytic)

Bases: siegpy.analyticeigenstates.AnalyticEigenstate

This is the base class for any eigenstate of the 1D Square-Well Potential (1DSWP). It defines some generic methods, while leaving some others to be defined by its subclasses, that are:

• the SWPSiegert class, defining the Siegert states of the Hamiltonian defined by the 1D SWP,
• the SWPContinuum class, defining the continuum states of the same problem.

In addition to those of an AnalyticEigenstate, one of the main characteristics of a SWPEigenstate is its parity (the eigenstate must be an even or odd function).

Parameters:

• k (complex) – Wavenumber of the state.
• parity (str) – Parity of the state ('e' for even, 'o' for odd).
• potential (SWPotential) – 1D Square-Well Potential giving rise to this eigenstate.
• grid (list or set or numpy array) – Discretization grid.
• analytic (bool) – If True, the scalar products must be computed analytically.

Raises:

TypeError – If the potential is not a SWPotential instance.

parity

Returns:Parity of the eigenstate ('e' if it is even or 'o' if it is odd). Return type:str

is_even

Returns:True if the eigenstate is even. Return type:bool

is_odd

Returns:True if the eigenstate is odd. Return type:bool

_compute_values(grid)

Evaluate the wavefunction of the eigenstate discretized over the whole grid.

Parameters:grid (list or set or numpy array) – Discretization grid. Returns:Wavefunction discretized over the grid. Return type:numpy array

_even_wf_1(grid_1)

Note

This is an asbtract method.

Evaluate the even eigenstate wavefunction over the grid points in region I.

Parameters:grid_1 (numpy array) – Grid in region I. Returns:Even eigenstate wavefunction discretized over the grid in region I. Return type:numpy array

_even_wf_2(grid_2)

Note

This is an asbtract method.

Evaluate the even eigenstate wavefunction over the grid points in region II.

Parameters:grid_2 (numpy array) – Grid in region II. Returns:Even eigenstate wavefunction discretized over the grid in region II. Return type:numpy array

_odd_wf_1(grid_1)

Note

This is an asbtract method.

Evaluate the odd eigenstate wavefunction over the grid points in region I.

Parameters:grid_1 (numpy array) – Grid in region I. Returns:Odd eigenstate wavefunction discretized over the grid in region I. Return type:numpy array

_odd_wf_2(grid_2)

Note

This is an asbtract method.

Evaluate the odd eigenstate wavefunction over the grid points in region II.

Parameters:grid_2 (numpy array) – Grid in region II. Returns:Odd eigenstate wavefunction discretized over the grid in region II. Return type:numpy array

scal_prod(other, xlim=None)

Evaluate the scalar product of an eigenstate with another function. It can be computed analytically or not.

Note

If it has to be computed analytically, a TypeError may be raised if the analytic scalar product with the test function other is analytically unknown (at present, only scalar products with Gaussian and rectangular functions are possible).

Parameters:

• other (Function) – Another Function.
• xlim (tuple(float or int, float or int)) – Range of the x axis for the integration (optional).

Returns:

Result of the scalar product.

Return type:

complex

Raises:

TypeError – If the value of the analytical scalar product with the other function is unknown.

_scal_prod_with_Gaussian(gaussian)

Parameters:gaussian (Gaussian) – Gaussian function. Returns:Value of the analytic scalar product of an eigenstate with a Gaussian test function. Return type:complex or float

_scal_prod_with_Rectangular(rect)

Parameters:rect (Rectangular) – Rectangular function. Returns:Value of the analytic scalar product of an eigenstate with a Rectangular test function. Return type:complex or float

_sp_R_1(rect)

Note

This is an asbtract method.

Parameters:rect (Rectangular) – Rectangular function. Returns:Value of the analytic scalar product between an eigenstate and a rectangular function spreading over region I. Return type:float or complex

_sp_R_2(rect)

Parameters:rect (Rectangular) – Rectangular function. Returns:Value of the analytic scalar product between an eigenstate and a rectangular function spreading over region II. Return type:float or complex

_sp_R_2_other_cases(rect)

Note

This is an asbtract method.

Parameters:rect (Rectangular) – Rectangular function. Returns:Value of the analytic scalar product of an eigenstate with a rectangular test function in region II when the result is not obviously 0. Return type:complex or float

_sp_R_3(rect)

Parameters:rect (Rectangular) – Rectangular function. Returns:Value of the analytic scalar product of an eigenstate with a rectangular test function in region III. Return type:complex or float

class siegpy.swpeigenstates.SWPSiegert(ks, parity, potential, grid=None, analytic=True)

Bases: siegpy.swpeigenstates.SWPEigenstate, siegpy.analyticeigenstates.AnalyticSiegert

This class defines a Siegert state in the case of the 1D SWP.

Parameters:

• ks (complex) – Wavenumber of the Siegert state.
• parity (str) – Parity of the Siegert state ('e' for even, 'o' for odd).
• potential (SWPotential) – 1D Square-Well Potential giving rise to this eigenstate.
• grid (list or set or numpy array) – Discretization grid (optional).
• analytic (bool) – If True, the scalar products must be computed analytically (default to True).

Raises:

ParityError – If the parity is inconsistent with the Siegert state wavenumber.

_even_wf_1(grid_1)

Evaluate the even Siegert state wavefunction over the grid points in region I.

Parameters:grid_1 (numpy array) – Grid in region I. Returns:Even Siegert state wavefunction discretized over the grid in region I. Return type:numpy array

_even_wf_2(grid_2)

Evaluate the even Siegert state wavefunction over the grid points in region II.

Parameters:grid_2 (numpy array) – Grid in region II. Returns:Even Siegert state wavefunction discretized over the grid in region II. Return type:numpy array

_odd_wf_1(grid_1)

Evaluate the odd Siegert state wavefunction over the grid points in region I.

Parameters:grid_1 (numpy array) – Grid in region I. Returns:Odd Siegert state wavefunction discretized over the grid in region I. Return type:numpy array

_odd_wf_2(grid_2)

Evaluate the odd Siegert state wavefunction over the grid points in region II.

Parameters:grid_2 (numpy array) – Grid in region II. Returns:Odd Siegert state wavefunction discretized over the grid in region II. Return type:numpy array

_sp_even_gauss(gaussian, expkp, expkm, expqp, expqm, zkpp, zkmp, zkpm, zkmm, zqpp, zqmp, zqpm, zqmm)

Parameters:gaussian (Gaussian) – Gaussian function. Returns:Analytical value of the c-product between an even Siegert state and a Gaussian test function. Return type:float or complex

_sp_odd_gauss(gaussian, expkp, expkm, expqp, expqm, zkpp, zkmp, zkpm, zkmm, zqpp, zqmp, zqpm, zqmm)

Parameters:gaussian (Gaussian) – Gaussian function. Returns:Analytical value of the c-product between an odd Siegert state and a Gaussian test function. Return type:float or complex

_sp_R_1(rect)

Parameters:rect (Rectangular) – Rectangular function. Returns:Value of the analytic scalar product of a Siegert state with a rectangular function spreading over region I. Return type:complex or float

_sp_R_2_other_cases(rect)

Parameters:rect (Rectangular) – Rectangular function. Returns:Value of the analytic scalar product of a Siegert state with a rectangular function spreading over region II. Return type:complex or float

MLE_strength_function(test, kgrid)

Evaluate the contribution of a Siegert state to the Mittag-Leffler expansion of the strength function, for a given test function discretized on a grid of wavenumbers kgrid.

Parameters:

• test (Wavefunction) – Test function.
• kgrid (numpy array) – Discretization grid of the wavenumber.

Returns:

Contribution of the Siegert state to the strength function.

Return type:

numpy array

class siegpy.swpeigenstates.SWPContinuum(k, parity, potential, grid=None, analytic=True)

Bases: siegpy.swpeigenstates.SWPEigenstate, siegpy.analyticeigenstates.AnalyticContinuum

Class defining a continuum state of the 1D Square-Well potential.

Parameters:

• k (complex) – Wavenumber of the continuum state.
• parity (str) – Parity of the continuum state ('e' for even, 'o' for odd).
• potential (SWPotential) – 1D Square-Well Potential giving rise to this eigenstate.
• grid (list or set or numpy array) – Discretization grid (optional).
• analytic (bool) – If True, the scalar products must be computed analytically (default to True).

Raises:

ParityError – If the parity is not 'e' (even) or 'o' (odd).

_even_wf_1(grid_1)

Evaluate the even eigenstate wavefunction over the grid points in region I.

Parameters:grid_1 (numpy array) – Grid in region I. Returns:Even eigenstate wavefunction discretized over the grid in region I. Return type:numpy array

_even_wf_2(grid_2)

Evaluate the even Siegert state wavefunction over the grid points in region II.

Parameters:grid_2 (numpy array) – Grid in region II. Returns:Even Siegert state wavefunction discretized over the grid in region II. Return type:numpy array

_odd_wf_1(grid_1)

Evaluate the odd eigenstate wavefunction over the grid points in region I.

Parameters:grid_1 (numpy array) – Grid in region I. Returns:Odd eigenstate wavefunction discretized over the grid in region I. Return type:numpy array

_odd_wf_2(grid_2)

Evaluate the odd Siegert state wavefunction over the grid points in region II.

Parameters:grid_2 (numpy array) – Grid in region II. Returns:Odd Siegert state wavefunction discretized over the grid in region II. Return type:numpy array

_sp_even_gauss(gaussian, expkp, expkm, expqp, expqm, zkpp, zkmp, zkpm, zkmm, zqpp, zqmp, zqpm, zqmm)

Parameters:gaussian (Gaussian) – Gaussian function. Returns:Analytical value of the c-product between an even continuum state and a Gaussian test function. Return type:float or complex

_sp_odd_gauss(gaussian, expkp, expkm, expqp, expqm, zkpp, zkmp, zkpm, zkmm, zqpp, zqmp, zqpm, zqmm)

Parameters:gaussian (Gaussian) – Gaussian function. Returns:Analytical value of the c-product between an odd continuum state and a Gaussian test function. Return type:float or complex

_sp_R_1(rect)

Parameters:rect (Rectangular) – Rectangular function. Returns:Value of the analytic scalar product of a continuum state with a Rectangular function spreading over region I. Return type:complex or float

_sp_R_2_other_cases(rect)

Parameters:rect (Rectangular) – Rectangular function. Returns:Value of the analytic scalar product of an even continuum state with a Rectangular function spreading over region II. Return type:complex or float

exception siegpy.swpeigenstates.ParityError

Bases: Exception

Error thrown if the parity of an eigenstate is incorrect.