ISSUE 2007 VOLUME 2 - The World of Mathematical Equations

April, 2007 PROGRESS IN PHYSICS Volume 2
orbiting body, respectively, and [3]:
�
H = h 2π f Mmn
m2 �
, (2)
0
V (r) = − GMm
r
By assuming that R takes the form:
, (3)
E ′ = E
. (4)
m
−α r
R = e
and substituting it into equation (1), and using simplified
terms only of equation (1), one gets:
Ψ = α 2 e −α r r 2α e−α
−
r
(5)
+ 8πGMm2 −α r e
r H2 . (6)
After factoring this equation (7) and solving it by equating
the factor with zero, yields:
RR = − 2 � 4πGMm 2 − H 2 α �
α 2 H 2 = 0 , (7)
or
RR = 4πGMm 2 − H 2 α = 0 , (8)
and solving for α, one gets:
a = 4π2 GMm 2
H 2 . (9)
Gravitational Bohr radius is defined as inverse of this
solution of α, then one finds (in accordance with Rubcic &
Rubcic [3]):
r1 =
H2 4π2 , (10)
GMm2 and by substituting back equation (2) into (10), one gets [3]:
r1 =
� 2π f
αc
Equation (11) can be rewritten as follows:
r1 = GM
ν 2 0
� 2
GM . (11)
, (11a)
where the “specific velocity” for the system in question can
be defined as: � �−1 2π f
ν0 = = αg c .
αc
(11b)
The equations (11a)-(11b) are equivalent with Nottale’s
result [1, 2], especially when we introduce the quantization
number: rn = r1n 2 [3]. For complete Maple session of these
all steps, see Appendix 1. Furthermore, equation (11a) may
be generalised further to include multiple nuclei, by rewriting
it to become: r1 =(GM)/v 2 ⇒ r1 =(G ΣM)/v 2 , where
ΣM represents the sum of central masses.
Solution of time-dependent gravitational Schrödinger
equation is more or less similar with the above steps, except
that we shall take into consideration the right hand side
of Schrödinger equation and also assuming time dependent
form of r:
R = e −α r(t) . (12)
Therefore the gravitational Schrödinger equation now
reads:
d2R 2 dR
+
dr2 r dr + 8πm2E ′
H2 R +
+ 2 4π
r
2GMm2 H2 ℓ (ℓ + 1)
R −
r2 dR
R = H
dt ,
(13)
or by using Leibniz chain rule, we can rewrite equation
(15) as:
−H dR dr (t)
dr (t) dt + d2R 2 dR
+
dr2 r dr + 8πm2E ′
H2 R +
+ 2 4π
r
2GMm2 H2 ℓ (ℓ + 1)
R −
r2 (14)
R = 0 .
The remaining steps are similar with the aforementioned
procedures for time-independent case, except that now one
gets an additional term for RR:
RR ′ = H 3 �
d
α
dt r(t)
�
r(t) − α 2 r(t)H 2 +
(15)
+ 8πGMm 2 − 2H 2 α = 0 .
At this point one shall assign a value for d
dt r(t) term,
because otherwise the equation cannot be solved. We choose
d
dt r(t) = 1 for simplicity, then equation (15) can be rewritten
as follows:
RR ′ : = rH3 α
2 + rH2 α 2
2 +4π2 GMm 2 −H 2 α = 0 . (16)
The roots of this equation (16) can be found as follows:
a1 : = −r2 H+2H+
a2 : = −r2 H+2H−
√ r 4 H 4 −4H 3 r+4H 2 −32rGMm 2 π 2
2rH
√
r4H 4−4H 3r+4H2−32rGMm2π 2
2rH
,
.
(17)
Therefore one can conclude that there is time-dependent
modification factor to conventional gravitational Bohr radius
(10). For complete Maple session of these steps, see Appendix
2.
3 Gross-Pitaevskii effect. Bogoliubov-deGennes approximation
and coupled time-independent gravitational
Schrödinger equation
At this point it seems worthwhile to take into consideration a
proposition by Moffat, regarding modification of Newtonian
acceleration law due to phion condensate medium, to include
Yukawa type potential [5, 6]:
a(r) = − G∞ M
r 2
+ K exp (−μφ r)
r 2 (1 + μφ r) . (18)
V. Christianto, D. L. Rapoport and F. Smarandache. Numerical Solution of Time-Dependent Gravitational Schrödinger Equation 57