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HFB sum rules
Since the HFB state (11) is a superposition of projected
states (13),
![\begin{displaymath}
\vert\Phi\rangle=\sum_{N=0}^{\infty}\vert\Psi_N\rangle ,
\end{displaymath}](img68.png) |
(18) |
the HFB energy
,
![\begin{displaymath}
E_{\mbox{\rm\scriptsize {HFB}}}= \langle\Phi\vert\hat{H}\vert\Phi\rangle ,
\end{displaymath}](img70.png) |
(19) |
can be
expressed as the sum of projected
energies (15),
![\begin{displaymath}
E_{\mbox{\rm\scriptsize {HFB}}}=\sum_{N=0}^{\infty}\langle\Psi_N\vert\Psi_N\rangle E^N_{\mbox{\rm\scriptsize {HFB}}} ,
\end{displaymath}](img71.png) |
(20) |
weighted by probabilities
=
=
of finding a
given PN component in the HFB state. Expression (20) constitutes
a useful sum-rule condition, which has to be obeyed by any Hamiltonian-based HFB+PNP
approach, and can be used to test the
numerical precision of PNP techniques.
A similar sum rule holds for any shifted state
![\begin{displaymath}
\vert\Phi(z_0)\rangle=\sum_{N=0}^{\infty}\vert\Psi_N(z_0)\rangle ,
\end{displaymath}](img75.png) |
(21) |
i.e.,
![\begin{displaymath}
\langle\Phi(z_0)\vert\hat{H}\vert\Phi(z_0)\rangle
=\sum_{N=...
...le\Psi_N\vert\Psi_N\rangle E^N_{\mbox{\rm\scriptsize {HFB}}} ,
\end{displaymath}](img76.png) |
(22) |
where the average energy of the shifted and unnormalized HFB state
is related to its HFB energy
as
![\begin{displaymath}
E_{\mbox{\rm\scriptsize {HFB}}}(z_0)= \frac{\langle\Phi(z_0)...
...t\Phi(z_0)\rangle}
{\langle\Phi(z_0)\vert \Phi(z_0)\rangle} .
\end{displaymath}](img78.png) |
(23) |
Finally, the sum rule
for the non-diagonal matrix elements can be written as:
![\begin{displaymath}
\langle\Phi\vert\hat{H}\vert\Phi(z_0)\rangle
=\sum_{N=0}^{\...
...le\Psi_N\vert\Psi_N\rangle E^N_{\mbox{\rm\scriptsize {HFB}}} .
\end{displaymath}](img79.png) |
(24) |
Next: Transition matrix elements and
Up: Particle-Number-Projected HFB
Previous: Projected HFB states
Jacek Dobaczewski
2007-08-08