1. The problem statement, all variables and given/known data If electric charge did not exist, and protons and electrons were only bound together by gravitational forces to form hydrogen, derive the expressions for a_0 and E_n and compute the energy and frequency of the H_alpha line and limit of Balmer series. 2. Relevant equations E_n=mc^2 * alpha * 1 / (2n^2) E_n=-mk^2Z^2e^4/(2hbar*n^2)=-E_0Z^2/n^2 a_0=hbar^2/(mc*alpha)=hbar^2/mke^2 1/lambda=Z^2R(1/nf^2-1/ni^2) 3. The attempt at a solution If electric charge did not exist, then the balance of electron orbit and distance would change. Bohr's radius would just be zero i would imagine since it depends on electron charge. E_n would suffer the same fate... Would E_n just become mc^2? The radius would have to still be a number as the two do still have mass and gravity would affect them. Edit: F=Gm1m2/r^2 is the force on either exerted by the other and E=-Gm1m2/r is the gravitational potential energy The energy of the electron E_n must be strictly dependent on this. How do I "derive" an expression from E_n or a_0 when their original formulas must be completely nixed? Is there supposed to be a way of expressing it with n levels? Otherwise I propose the energy of the electron is -Gm1m2/r as there is no kinetic energy.