We have reviewed the Bohr atom briefly. This atom is held together by electromagnetism. We may consider the very similar problem of a gravitationally bound ``Bohr atom''. Part of the point of this problem is to give you some practice making calculations with our simple ℏ=c=1 units, so I hope you will attempt your calculations in this spirit. The gravitational potential between two objects of masses m1 and m2 is
V=−rGm1m2,
where r=∣r1−r2∣, and
GN=6.67×10−11 m3kg−1s−2,=6.71×10−39 GeV−2,(1.137)
is Newton's constant.
(a) With the same quantization condition as for Bohr's atom, find the formulas for the energy levels, relative velocities, and relative separations, for a ``gravity atom''. Let n be the quantum level (i.e., n=1,2,3,… is the orbital angular momentum).
(b) Find n for the earth-sun system. Is quantum mechanics important here?
(c) What would r be for the ground state of the earth-sun system (give answer in meters)?
(d) Consider a gravitationally bound system of two neutrons, where we suppose that we have ``turned off'' other potentially important interactions (such as the strong interaction). What is the ground state energy (in GeV) and the ground state separation (in meters) for this system? Is gravity important compared with other forces for a real system of two neutrons in a real ground state?
(e) Physicists have invented quite a few systems of units for measurement. One notion of at least theoretical value is the notion of ``natural units''. Our use of ℏ=c=1 units is such a system. Around 1900, Planck proposed such a system based on his quantum of action (now ℏ). Thus, he proposed to measure masses in units such that the gravitational strength is unity, i.e., G=1. We set GmP2=1 to find this unit of mass, mP. In SI units, we would write this equation as GmP2=ℏc, as you can check that the units match. Determine the mass mP in GeV.