Next: 10. Basic Integral for Up: Classical and Quantum Shell Previous:
8. The FGG-method and
9. Hamiltonian for
,
For the sake of notational simplicity, let us set GN=1.
We start with the expression of the momentum in the case6
Ain > 0 and
Aout > 0:
Then, enlisting the equalities
 |
(112) |
 |
(113) |
so that
we find the explicit form of the Hamiltonian quoted in the text
(Eq.55):
![\begin{displaymath}H = \kappa R ^{2}
-
R \left [
A_{in} +
A_{out} -
2 \sigm...
... \left( \frac{P_R}{R} \right)
\right ] ^{\frac{1}{2}}
\quad .
\end{displaymath}](img271.gif) |
(116) |
We now repeat the same steps for the case
Ain > 0 and
Aout < 0. Letting
,
we find
Enlisting now the equalities
 |
(118) |
 |
(119) |
so that
we find
![\begin{displaymath}H = \kappa R ^{2} -
R \left [
A_{in} +
A_{out} -
2 \sigma...
... \left( \frac{P_R}{R} \right)
\right ] ^{\frac{1}{2}}
\quad .
\end{displaymath}](img284.gif) |
(122) |
Evidently, we can follow the same procedure in the cases
Ain < 0,
Aout < 0;
Ain < 0,
Aout > 0.
The corresponding expressions are
![$\displaystyle H = \kappa R ^{2} -
R \left [
A_{in} +
A_{out} -
2 \sqrt{A_{in} A_{out}}
\cosh \left( \frac{P_R}{R} \right)
\right ] ^{\frac{1}{2}}$](img285.gif) |
|
|
(123) |
 |
|
|
|
![$\displaystyle H = \kappa R ^{2} -
R \left [
A_{in} +
A_{out} -
2 \sigma _{out} \sqrt{- A_{in} A_{out}}
\sinh \left( \frac{P_R}{R} \right)
\right ] ^{\frac{1}{2}}$](img287.gif) |
|
|
(124) |
 |
|
|
|
or, in a more compact notation
H |
= |
 |
|
|
|
![$\displaystyle \cdot
\left[\,
A_{in} +
A_{out} -
2 \sigma _{in} \sigma _{out}
\l...
...\vert A_{in} A_{out} \vert}
}
}
\right) ^{\frac{1}{2}}
\right] ^{\frac{1}{2}}
.$](img290.gif) |
(125) |
The Hamiltonian for the shell of dust quoted in Section IV can be
derived along the same steps with little change, and one obtains
H |
= |
 |
|
|
|
![$\displaystyle \cdot
\left[\,
A_{in} +
A_{out} -
2 \sigma _{in} \sigma _{out}
\l...
...\vert A_{in} A_{out} \vert}
}
}
\right) ^{\frac{1}{2}}
\right] ^{\frac{1}{2}}
.$](img290.gif) |
(126) |
Next: 10. Basic Integral for Up: Classical and Quantum Shell Previous:
8. The FGG-method and
Stefano Ansoldi
Department of Theoretical Physics
University of Trieste
TRIESTE - ITALY