Bohr-Sommerfeld QuantizationIn the Schwarzschild (Reissner-Nordström) Metric
Weldon J. Wilson
Department of Physics
University of Central Oklahoma
Edmond, Oklahoma
Email: [email protected]
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