significantly updated workhorse Mathematica notebook for computing eigenvalues, wave functions, and validity tests (see change log in notebook)
added notebook for an enhanced implementation of spectral method, borrowing some ideas from Phase Method, for comparison with it
added notebook for Mathematica’s NDEigensystem for 1D potentials, for comparison to Phase Method and spectral methods
a new example notebook implementing a variant Shooting Method “Shoot First Method” that requires no adjustment of initial conditions to get eigenvalues. It works on some (but not all) of the same principles as the Phase Method.
On Amazon.com, via Kindle Direct Publishing: ebook, 6”x9” paperback, and 6”x9” hardcover “print replica” versions are now available.
Just search for “phase method bunker” on Amazon.com.
Other vendors (e.g. reflowable epub) will be forthcoming, which may have recently gained some urgency. If you desire such alternative versions/vendors, email me at bunker@iit.edu.
My newest (2024) book introduces a simple, robust, somewhat unconventional, and new computational method, the Phase Method (PM), for numerically solving the time independent Schrödinger Equation for the energy levels and wave functions in 1D, 2D/3D-central, and higher dimensional separable potentials. It liberates the user from having to approximate potentials and wave functions by discretization or collocation methods (which have limited accuracy for discontinuous potentials), or having to guess locations of eigenvalues or fiddling with initial conditions as in the Shooting Method. The PM is something different. Detailed explanations are presented of why the PM works, how to use it effectively, and how it compares to conventional methods. Numerous examples are given, and the PM is used to test and validate our new and useful approximate semiclassical formulae that are first presented here.
“The Phase Method” is expected to be useful to a broad spectrum: college and university professors; theorists wishing to numerically check analytical expressions and approximations; students exploring quantum mechanics; high school physics and chemistry teachers; and working scientists, engineers, and nanotechnologists, with applications to quantum physics and chemistry, materials science, nanotechnology, nanophotonics, and related fields. The book contains something new for everyone; a basic knowledge of Quantum Wave Mechanics is a prerequisite.
Our initial release is a “Print Replica” edition, which reads well on a computer or tablet (but not Kindle E-readers) via the free Kindle Reader App. If you have previously purchased a copy of the ebook, please download a fresh copy at no additional cost because although some errata have been corrected, at least one remains (see below for known errata).
Paperback and hardcover 6”x9” (Print Replica) versions were released mid-May 2024 via Kindle Direct Publishing. Other formats and outlets will be forthcoming.
I have intended for the ebook to be inexpensive enough (about the price of a sandwich), for students and curious others to afford, but it is not open-sourced, so that AI/LLMs will not (I hope) simply ingest the contents, blend it with everything else devoured from the internet, and disgorge the result for resale, without context, vetting, or attribution of the original source.
No generative AI was used in creating this book. I (GB) wrote every word, did every computation, wrote all code (Mathematica 13/14, and Python3), created all graphics, and did all typesetting with TeXShop. Citations are made to all known pertinent work.
If you are interested in other formats and publishing platforms, e.g. epub, as I expect may well be the case given recent events, or if you have comments or questions, please email me at bunker@iit.edu with subject line “Phase Method Book”.
The Phase Method has been advancing since publication of the book. Although the principles remain the same, we now can obtain significantly greater accuracy than is described in the book, and some peculiar aspects (e.g. why is the number of good digits reliably half of the working precision? Why isn’t overflow a more serious problem?) are now better understood. Stay tuned for more exposition in the future.
fun fact: a recently discovered “killer app” turns out to be calculation of critical screening parameters in exponentially screened Coulomb Potentials (e.g. Yukawa/Debye/ECSC).
We easily get twice the number of digits of precision of the most recent state of the art, and doing so on a $700 M4 Mac Mini computer.
Results are being written up for publication.
It has come to my attention that, in 1967, Russian authors, in Russian Language, published what they called “the phase method”:
Their method is not directly related to, and is not the same as, our “Phase Method”. If there is a need to distinguish them, they can simply be referred to as “Uvarov’s Phase Method” as opposed to “Bunker’s Phase Method”.
The following error has been fixed in currently extant versions of the book, but was in early prints. In appendix E.7 (Lennard-Jones potential) the upper turning point scale factor was taken as 5, which is inadequate for the highest level. To attain full convergence of the n=4 barely bound state a larger range is needed owing to the long power-law tail of the Lennard-Jones potential. The Morse potential, by contrast, decreases exponentially with distance, and so converges more quickly.
Convergence was tested for the sequence of ranges tpsf=3, 5, 7, 10, 15, 20. tpsf=20 corresponds to xmax ~ 117. The eigenvalue search for tpsf=20 at 31 digits working precision took about 2 minutes on a MacBook Pro M2 Max computer.
A more accurate value for n=4 is then -0.0282723397921026
The other levels n=1…3 are thought to be accurate as given in the book.
--------------Working example code in Python and Mathematica is given in appendices of the book, or can be downloaded below. The Python code can be run as-is in a Jupyter notebook if desired.
If you use this code, implement the method, or make use of our new semiclassical equations given in Appendix A, please cite the book in your publications (“The Phase Method” (2024), Grant Byrd Bunker, author and publisher).
Example Mathematica code for Eigenvalue Search from Appendix D (plain text file)
Example as above, plus energy scanning, plot levels as they are found (for demo only)
Example Mathematica Code for spectralsolve (good for smooth or periodic potentials - .nb format)
Example Mathematica Code for spectralsolve (not good for discontinuous potentials - .nb format)
Example Mathematica Code for enhanced spectralsolve - .nb format
Example Mathematica Code for Shoot First Method - .nb format
Example Mathematica Code for Mathematica NDEigensystem (for comparison) - .nb format
Improved workhorse notebook latest version 12/25/2024. The following Mathematica notebook finds eigenvalues, wavefunctions, and does a lot of error checking, with (perhaps too many) bells and whistles and checks and fallbacks. It’s easy to override automatically chosen values if desired.
The code and documentation are interleaved within this notebook, but a mathematica front end palette is created that allows you to hide and show them as you wish.
This is my current main working notebook that was used to produce the output in the book. It is a work in progress - it can certainly be improved and cleaned up. If the programming style is not to your liking, just delete the chunks you don’t like, or just write your own using the principles described in the book.
Operational note: on Mac OS I use “Get Info” in the Finder to set the “Stationery pad” and “Locked” flags. This creates a freash copy automatically when double-clicked, so you don’t accidentally overwrite the original template file.
Videos on the use of this versions of this notebook for specific cases are posted here - more will be added
Here is Python3 code in plain text (as in the book), and as Jupyter notebooks (.ipynb format).
On download you may need to delete a “.txt” extension if appended after the “.ipynb” extension.
Below is a link to a short video of the Python code above running in real-time on an Apple iPad (8th gen, nothing special). It shows the code being edited to change the potential from a previous harmonic oscillator 1/2 x^2 run to the linear v-shaped potential U=1/2 |x|, and then executed (running entirely on the iPad). It takes about 16 seconds to do the eigenvalue search for lowest 38 levels. The code is serial and runs at machine precision, so 7-8 accurate digits are expected. The Mathematica version can do 15 or even 20 decimal digits.
Clicking on the following links will download zip files containing Mathematica notebooks and corresponding output files. Go nuts.
This example calculates a sequence of five identical equally spaced finite square wells that might represent a superlattice constructed for nanophotonics,
composed of a set of layers of different semiconductors. The PM can handle pretty nearly any shape or sequence of levels, of various potential heights,
tapered or sharp, that you might want to devise. Many 2D and 3D versions of the same often can be constructed from 1D calculations.
Note here that the level splittings within a “band” follow to good accuracy the pattern described in the book(appendix E.8.6)
specifically {-Sqrt[3],-1,0,1,Sqrt[3]} for 5 identical wells, independent of the shape of the well.
Also note the spikey behavior of the barely-bound highest level wave function near the continuum, which is difficult for usual discretization methods to handle.
As described in Phase Method book(Appendix E.1)we compare to results of Vanden Berghe et al (https://doi.org/10.1016/0377-0427(89)90350-6)). Note that Vanden Berghe et al write the Schrödinger equation with a prefactor of -1 instead of our -1/2 for the kinetic energy term. That is accounted for in the notebook and data files linked to below by setting the parameter gamma=2 instead of the usual gamma=1. This is an example of a “one-sided” potential that exists for r>=0. Only the l=0 example is shown here.
The following files give calculations of the eigenvalue spectra and wavefunctions for the different angular momentum manifolds l=0…3 for a 1/r (Coulomb, Hydrogen atom) central potential. As this is a homogenous potential the eigenvalues can be scaled using the exact scaling relation E ~ epsilon eta^mu for any values of the potential parameter (here taken as 1). These are “one-sided potentials”. The lower turning point scale factor is chosen to be quite small so that the lower limit xmin is well into the classically forbidden region while still exceeding zero. As is well-known, the eigenvalues of the Coulomb (-1/r) potential with different l are degenerate (because of a “hidden” 4D symmetry peculiar to the 1/r potential), which in classical physics manifests as conservation of the Laplace-Runge-Lenz vector in the Kepler problem.
The following zip file contains two calculations of homogeneous potentials of the same degree k, but different potential parameters. It demonstrates that the PM-computed values agree nicely with the exact scaling relation proved in the Phase Method book (Appendix A.1).