Bohr-Sommerfeld%20Quantization%20In%20the%20Schwarzschild%20(Reissner-Nordstr

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Bohr-Sommerfeld QuantizationIn the Schwarzschild
(Reissner-Nordström) Metric

• Weldon J. Wilson
• Department of Physics
• University of Central Oklahoma
• Edmond, Oklahoma

Email wwilson_at_ucok.edu WWW http//www.physics.
ucok.edu/wwilson
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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.
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REISSNER-NORDSTRÖM METRIC
with
Leads to planar orbits with
Choosing
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,
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MASS-ENERGY RELATION
The metric
yields
So the mass energy relation
Yields
or
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HAMILTON-JACOBI EQUATION
The mass energy relation
or
Leads to the (separable !!) H-J equation
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 ...
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BOUND STATE ENERGY
The contour integral evaluates to
Which can be solved for the (classical) bound
state energy
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QUANTIZATION
Using the Bohr-Sommerfeld quantization condition
One obtains
with
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SUMMARY
The Reissner-Nordström metric
Lead to the Bohr-Sommerfeld energy levels
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REFERENCES

• 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 slides http//www.physics.ucok.edu/wwilson