About 45 years ago or so I was a Physics department staff computer programmer when I was a physics major undergrad. I worked with a professor doing research on nuclear shell models -- he was attempting to develop a numerical method that would solve particular potentials using the BG equation as the mathematical foundation. My job was to write the programs, run jobs, plot graphic output on a nice 27 inch Calcomp flat bed plotter, and other such things. However, whatever I might have known, I have forgotten.

But, I do have one form of this equation written below: $$ \left[\lambda + \frac{\mathrm{d}^2}{\mathrm{d}\rho^2}-V(\rho)\right]U(\rho) = (\lambda-\lambda^0)R(\rho) \langle R {\mid} U \rangle - \sum_a C^aR^a(\rho) \langle R^a {\mid} \nu {\mid} U \rangle $$ Unfortunately, I do not have definitions of the variables other than I knew that $V(\rho)$ is of course the potential and I believe the $\lambda$ are eigenvalues.

The professor also notes in a hand-written note he prepared for me that the numerical method we were investigating is based on the Fox-Goodwin method. He wrote in the margin, just barely legible that the Fox-Goodwin method he refers to is described in a paper cited as "Proc. of Cambridge Phil. Soc. 45 (1949) 373".

So, with this skimpy information out of the past, can someone give me more information on this so-called BG equation? I believe that B and G are the initials of last names (?) of the authors of some earlier paper.

By the way, I have googled various aspects of these facts I know and I found some possible hits but they referred to papers that I could not find on-line and probably predated any form of publication on the Internet.

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    $\begingroup$ According to Google, BG refers to Bethe-Goldstone. $\endgroup$ May 28, 2016 at 2:56
  • $\begingroup$ Yes, I found a link for Bethe-Goldstone before but it was a reference to a paper I had no access to. However, just googling it again, I did find a book reference that seems to give me a lot of information that I didn't have before: books.google.com/… $\endgroup$
    – K7PEH
    May 28, 2016 at 3:00
  • $\begingroup$ The BG equation is frequently called the BBG (Brueckner-Bethe-Goldstone) equation. It is used to calculate corrections to the mean field or Hartree-Fock model of a many body system. $\endgroup$ May 28, 2016 at 20:47

2 Answers 2


The Fox-Goodwin reference is to a method for numerical integration of ordinary differential equations, which must refer to the algorithm you were to use.

Possibly the problem was taken from a paper which referenced this method; that means you can use a scientific citation index to find papers which cite Proc. of Cambridge Phil. Soc. 45 (1949) 373.

You can probably do this from any university library; ask the reference librarian for assistance.


The equation is the Bethe-Goldstone equation for nuclear matter or for finite nuclei. The sum is over the occupied nuclear states preventing two interacting nuclei from scattering into other occupied states, i,e., Pauli exclusion principle. This problem was solved around 1967 by Mackeller (PhD dissertation 1966 Texas A&M universtity and by Grillot and McManus 1969 Nuclear Physics. Grillot and McManus solve the integro-differential equation by expanding the wave function in Taylor series and iterated the equation starting from a trial wave function. BTW this is not an eigenvalue problem because the solution is buried inside an integral


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