DVR Basis

In this notebook we describe how to use the BEC code to describe axially symmetric systems. The need for a new basis can be seen by considering that we now need to tabulate points at \((x, r)\) where \(r=\sqrt{y^2+z^2}\). The radial dimension \(r\) is not periodic, so the simple Fourier techniques we have been using do not work.

mmfutils.math.bases

To implement more general bases we use the mmfutils.math.bases package. This provides various classes satisfying the following interface:

from IPython.display import display
from mmfutils.interface import describe_interface
import mmfutils.math.bases.interface

---------------------------------------------------------------------------
ModuleNotFoundError                       Traceback (most recent call last)
Cell In[1], line 3
      1 from IPython.display import display
      2 from mmfutils.interface import describe_interface
----> 3 import mmfutils.math.bases.interface

ModuleNotFoundError: No module named 'mmfutils.math.bases.interface'

display(describe_interface(mmfutils.math.bases.interface.IBasisMinimal))
display(describe_interface(mmfutils.math.bases.interface.IBasis))

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
Cell In[2], line 1
----> 1 display(describe_interface(mmfutils.math.bases.interface.IBasisMinimal))
      2 display(describe_interface(mmfutils.math.bases.interface.IBasis))

NameError: name 'mmfutils' is not defined

To use this, we thus redesign our classes to call only these methods. A good exercise is to take the code in bec_basic.py and refactor it to use mmfutils.math.bases.PeriodicBasis. (This was done with revision 2abbfcb0d613 for the bec.py file if you want to see the changes)

1. Make sure that you have comprehensive testing that works for your file and make sure that all relevant lines of our source code are covered by the tests. (See revision 2abbfcb0d613 for passing tests with bec.py.)

2. Start replacing the relevant code with the basis code. We needed to do the following:

   • Create self.basis.

   • Create appropriate delegations for attributes that are now provided by the basis:

     • Nxyz→basis.Nxyz, Lxyz→basis.Lxyz, xyz→basis.xyz, metric→basis.metric, symmetric_grid→basis.symmetric_lattice.

   • Remove dependence on K. This was a little more complicated. We replaced this with K_factor which contained the factor \(-\hbar^2/2m\) and then use basis.laplacian.

   • Removed dependence on kxyz and various fft stuff which assumes a periodic basis.

   • I also removed get_H(). This could be added later, but will need the basis to provide the matrix representation of the Laplacian.

   • These changes can be seen in revision e6d500d279a6.

3. Change interface so that different bases can be used by allowing the State() object to be constructed from a basis.

   • This requires a bit more careful thought because some functionality is still assuming that Nxyz and Lxyz can be used. These had to be removed since they are not part of the IBasis interface.

4. These changes should be complete in revision 0765cab259a6 but testing is needed with the notebooks which might depend on some other features.