1. Bohr Model of Particle Motion In the Schwarzschild Metric Weldon J. Wilson Department of Physics University of Central Oklahoma Edmond, Oklahoma Email: wwilson@ucok.edu WWW: http://www.physics.ucok.edu/~wwilson

2. OUTLINE n Schwarzschild Metric n Effective Potential n Bound States - Circular Orbits n Bohr Quantization n Summary

3. SCHWARZSCHILD METRIC where Leads to the action And corresponding Lagrangian

4. HAMILTONIAN FORMULATION Using the standard procedure, the Lagrangian With Yields the Hamiltonian

5. ORBITAL MOTION The Hamiltonian Leads to planar orbits with conserved angular momentum Using

6. CIRCULAR ORBITS For circular orbits And the Hamiltonian becomes 0

7. EFFECTIVE POTENTIAL The Hamiltonian for circular orbits is the total energy (rest energy + effective potential energy) of the mass m in a circular orbit of radius R in the “field” of the mass M.

8. EFFECTIVE POTENTIAL

9. RADIAL FORCE EQUATION The radial force equation can be obtained from Differentiation gives Which must vanish for the circular orbit ( )

10. ALLOWED RADII OF ORBITS Setting For the circular orbits produces the quadratic Which can be solved for the allowed radii

11. ALLOWED RADII R+R+ R-R-

12. BOHR QUANTIZATION Using the Bohr quantization condition One obtains from The quantized allowed radii

13. ENERGY – CIRCULAR ORBITS From the quadratic resulting from the radial force equation One obtains Putting this into Results in

14. ENERGY QUANTIZATION From the energy One obtains the quantized energy levels where

15. References l Robert M. Wald, General Relativity (Univ of Chicago Press, 1984) pp 136-148. l Bernard F. Schutz, A First Course in General Relativity (Cambridge Univ Press, 1985) pp 274-288. l These slides http://www.physics.ucok.edu/~wwilson