Bohr Model of Particle Motion In the Schwarzschild Metric

1. Bohr Modelof Particle MotionIn 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 • Schwarzschild Metric • Effective Potential • Bound States - Circular Orbits • Bohr Quantization • Summary

3. SCHWARZSCHILDMETRIC where Leads to the action And corresponding Lagrangian

4. HAMILTONIANFORMULATION Using the standard procedure, the Lagrangian With Yields the Hamiltonian

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

6. CIRCULARORBITS For circular orbits And the Hamiltonian becomes 0

7. EFFECTIVEPOTENTIAL 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. EFFECTIVEPOTENTIAL

9. RADIALFORCEEQUATION 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. ALLOWEDRADII R- R+

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

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

14. ENERGYQUANTIZATION From the energy One obtains the quantized energy levels where

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