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

OUTLINE Physical Motivation Charged Schwarzschild Metric

(Reissner-Nordström Metric) Hamiltonian-Jacobi Equation Contour Integration Bohr-Sommerfeld Quantization Energy Levels Summary

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.

REISSNER-NORDSTRÖM METRIC2222221222 sin drdrdrdtcds

4

22

2S2

2

S ,2

,1cGQ

rcGM

rr

r

r

rQ

Q

with

Leads to planar orbits withconstantL

Choosing 02

d

The metric becomes2221222 drdrdtcds

Conserved QuantitiesTime 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,

Ep 0

Lp

MASS-ENERGY RELATIONThe metric

yields

So the mass energy relation

2

210

000 0

rL

pgpddr

mp

pEcpgp

r

ppgppgppgppg

ppgcmrr

rr

00

00

22

2221222 drdrdtcxxgds

Yields 2

2221221222 )()(

rL

rpEcccm r

or2

221

2

2122 )(

rL

pcE

cm r

HAMILTON-JACOBI EQUATIONThe mass energy relation

Leads to the (separable !!) H-J equation

And the integrable (!!!) action integral

or2

221

2

2122 )(

rL

pcE

cm r

2

222

2

22)(

rL

cmcE

p r

( rS)2 E 2

c2 m2c2 L

2

r2

drrL

cmcE

J r 2

222

2

2

CONTOUR INTEGRALThe 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 ...

Residues2

2

222

2

2

i

drrL

cmcE

J r

BOUND STATE ENERGYThe contour integral evaluates to

Which can be solved for the (classical) bound state energy

2222

22

/21

||2

2cEcm

rcm

iL

r

riiJ S

Q

Sr

2/1

2

2222

||21

2

41

1

LrrJ

rcmmcE

Q

Sr

S

QUANTIZATIONUsing the Bohr-Sommerfeld quantization condition

One obtains

,3,2,1

,3,2,1,0

nnhJ

L

r

mccGQ

rcGM

r CQS

,,2

4

2

2

2/1

2

2

22

21

141

1

C

S

Q

S

n

r

r

rn

mcE

with

SUMMARYThe Reissner-Nordström metric

Lead to the Bohr-Sommerfeld energy levels

2/1

2

2

22

21

141

1

C

S

Q

S

n

r

r

rn

mcE

22222

2

1

2

222

2

22

sin

11

drdr

drr

r

r

rdtc

r

r

r

rds QSQS

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 slideshttp://www.physics.ucok.edu/~wwilson