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Bohr-Sommerfeld Quantization In the Schwarzschild (Reissner-Nordström) Metric
Weldon J. Wilson
Department of Physics
University of Central Oklahoma
Edmond, Oklahoma
Illustrated with Microsoft PowerPoint 97
120 MHz Pentium Toshiba laptop PC running Windows 95
$8000 Panasonic LCD projector
Entire presentation available on WWW
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2. OUTLINE Physical Motivation
4/15/2017
OUTLINE
Physical Motivation
Charged Schwarzschild Metric (Reissner-Nordström Metric)
Hamiltonian-Jacobi Equation
Contour Integration
Bohr-Sommerfeld Quantization
Energy Levels
Summary

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PHYSICAL MOTIVATION
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-NORDSTRÖM METRIC
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REISSNER-NORDSTRÖM METRIC
with
Leads to planar orbits with
Choosing
Equivalence (diffeomorphism) between systems with:
1 variable with many time delays
Multiple variables and no delay
Takens' theorem (1981)
A generalization of the Mandelbrot set equations
The metric becomes

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Conserved Quantities
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,
Equivalence (diffeomorphism) between systems with:
1 variable with many time delays
Multiple variables and no delay
Takens' theorem (1981)
A generalization of the Mandelbrot set equations

6. MASS-ENERGY RELATION The metric yields So the mass energy relation
4/15/2017
MASS-ENERGY RELATION
The metric
yields
So the mass energy relation
Equivalence (diffeomorphism) between systems with:
1 variable with many time delays
Multiple variables and no delay
Takens' theorem (1981)
A generalization of the Mandelbrot set equations
Yields or

7. HAMILTON-JACOBI EQUATION
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HAMILTON-JACOBI EQUATION
The mass energy relation or
Leads to the (separable !!) H-J equation
Equivalence (diffeomorphism) between systems with:
1 variable with many time delays
Multiple variables and no delay
Takens' theorem (1981)
A generalization of the Mandelbrot set equations
And the integrable (!!!) action integral

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CONTOUR INTEGRAL
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 ...
Equivalence (diffeomorphism) between systems with:
1 variable with many time delays
Multiple variables and no delay
Takens' theorem (1981)
A generalization of the Mandelbrot set equations

9. BOUND STATE ENERGY The contour integral evaluates to
4/15/2017
BOUND STATE ENERGY
The contour integral evaluates to
Which can be solved for the (classical) bound state energy
Equivalence (diffeomorphism) between systems with:
1 variable with many time delays
Multiple variables and no delay
Takens' theorem (1981)
A generalization of the Mandelbrot set equations

10. QUANTIZATION Using the Bohr-Sommerfeld quantization condition
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QUANTIZATION
Using the Bohr-Sommerfeld quantization condition
One obtains
Equivalence (diffeomorphism) between systems with:
1 variable with many time delays
Multiple variables and no delay
Takens' theorem (1981)
A generalization of the Mandelbrot set equations
with

11. SUMMARY The Reissner-Nordström metric
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SUMMARY
The Reissner-Nordström metric
Lead to the Bohr-Sommerfeld energy levels
Equivalence (diffeomorphism) between systems with:
1 variable with many time delays
Multiple variables and no delay
Takens' theorem (1981)
A generalization of the Mandelbrot set equations

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REFERENCES
Robert M. Wald, General Relativity (Univ of Chicago Press, 1984) pp
Bernard F. Schutz, A First Course in General Relativity (Cambridge Univ Press, 1985) pp
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