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Illinois Tech Professor of Physics Grant Bunker’s latest published paper shows how his unconventional computational approach can exponentially increase the accuracy and range for determining the thresholds at which subatomic particles become bound or unbound.  

In the single-author paper, “” based on solo, self-funded research and published in the peer-reviewed open-access journal Atoms, Bunker shows how the , a computational method that he developed, can be used to calculate the energy levels and critical binding parameters for a wide variety of screened Coulomb potentials to an accuracy of 60 decimal digits. 

“Essentially, you’re adjusting a knob that controls a parameter, the screening length, which affects the interaction between particles, such as ions in a plasma with a cloud of mobile electrons all around, or an electron-hole pair in a semiconductor in a sea of partial mobile electrons,” Bunker says. “And you’re trying to very precisely determine the values at which quantum states just barely become bound or unbound. It’s hard to do accurately for these potentials because you can’t get analytical solutions for them.” 

The results of this research show that the Phase Method can be applied to many subfields in physics and chemistry including nuclear and particle physics, plasma physics, inertial confinement fusion, astrophysics, cosmology, solid-state physics, physical chemistry, quantum computing, material science, and nanotechnology. 

Bunker used the Phase Method and his Mathematica programs written in “Wolfram Language” to organize 2,167 precise critical binding parameters into 84 tables. 

“This paper demonstrates that the Phase Method can solve problems that were previously very difficult,” Bunker says. “The binding of quantum states affects the behavior and properties, such as spectra and thermodynamics, of the systems you’re studying. It’s relevant to understanding the quantum mechanics of how stars work and look, to fusion energy, to trapped ion quantum computing, to materials properties, and many other areas.” 

Bunker says he ran most of the computations on several ordinary M4 Mac Minis, which, he argues opens new opportunities in research and pedagogy. 

“Extraordinary claims, such as 60-digit accuracy, require extraordinary evidence, and I have done the work, self-checking results, cross-checking, and reproducing others’ work,” he says. “When I submitted my 60-digit values to the journal, a referee was the first one to confirm that my values agreed with a contemporary Chinese group’s 30-digit values to all their digits, independently obtained by very different methods. It was a blind test, which is a scientific gold standard.” 

In addition to matching exactly known values to 60 digits, Bunker’s results also match previous 30-digit results from papers written by , a research group led by Li , and a paper written by , work that Bunker says he considers unjustly overlooked. 

Bunker says that he admires the Demiralp paper because it achieved an extraordinary accomplishment despite very modest resources. Its publication in 1989 also blows up the ahistorical “steady improvement” narrative that later developed in that research area.   

“Demiralp is a mathematician and wasn’t part of the scientific in-group who was normally concerned with this kind of problem, and he moved on afterward,” he says. “You have to advocate for the work if people don’t get it at first.”