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Bohr-Sommerfeld Quantization In the Schwarzschild (Reissner-Nordström) Metric

Bohr-Sommerfeld Quantization In the Schwarzschild (Reissner-Nordström) Metric. Weldon J. Wilson Department of Physics University of Central Oklahoma Edmond, Oklahoma. Email: wwilson@ucok.edu WWW: http://www.physics.ucok.edu/~wwilson. OUTLINE. Physical Motivation

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Bohr-Sommerfeld Quantization In the Schwarzschild (Reissner-Nordström) Metric

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  1. Bohr-Sommerfeld QuantizationIn the Schwarzschild (Reissner-Nordström) 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 • Physical Motivation • Charged Schwarzschild Metric (Reissner-Nordström Metric) • Hamiltonian-Jacobi Equation • Contour Integration • Bohr-Sommerfeld Quantization • Energy Levels • Summary

  3. PHYSICALMOTIVATION M Q m q A mass m with charge q = 0 bound gravitationally to the mass M with charge Q ≠ 0. Ultimate Goal - exact H-Atom energy levels including general relativistic correction.

  4. REISSNER-NORDSTRÖMMETRIC with Leads to planar orbits with Choosing The metric becomes

  5. ConservedQuantities Time independence of ds2 means that p0 is constant along the motion. As customary we denote the constant by Independence of ds2 of the angle  implies that p is constant. As customary,

  6. MASS-ENERGYRELATION The metric yields So the mass energy relation Yields or

  7. HAMILTON-JACOBIEQUATION The mass energy relation or Leads to the (separable !!) H-J equation And the integrable (!!!) action integral

  8. CONTOURINTEGRAL The action integral can be evaluated using the contour integral method of Sommerfeld. There are two poles, both of order two - one at r = 0 and the other at r = . Evaluation of the residues one obtains ...

  9. BOUNDSTATEENERGY The contour integral evaluates to Which can be solved for the (classical) bound state energy

  10. QUANTIZATION Using the Bohr-Sommerfeld quantization condition One obtains with

  11. SUMMARY The Reissner-Nordström metric Lead to the Bohr-Sommerfeld energy levels

  12. 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

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