56
2 The Discrete Spectrum and the Continuum
By replacing u(k a , r) by u + (k a , r) if r 1 = 0 and r 2 = r, and u(k b , r) by u + (k b , r)
if r 1 = r and r 2 > r 1 , and putting k a = k b = k n in Eq. (2.171), one obtains:
u
+ (k n , r) ˙
u(k n , r) − ˙
u
(k n , r)u
+ (k n , r) = 2k n
r
0
u(k n , r
)u
+ (k n , r
) dr
(2.172)
˙
u
+ (k n , r)u(k n , r) − u
(k n , r) ˙
u
+ (k n , r) = 2k n
+∞
r
u(k n , r
)u
+ (k n , r
) dr
,
(2.173)
where the limit r 2 → +∞ has been effected, and the dot above u indicates
differentiation with respect to k b . One has also used the fact that u(k n , 0) = 0 and
u(k n , r ) → 0 for r → +∞, arising from the bound character of u(k n , r). One then
obtains from Eqs. (2.172) and (2.173) the following equality:
dJ +
dk
(k)
k=k n
= 2k n
+∞
0
u(k n , r)u
+ (k n , r) dr ,
(2.174)
which also proves that the zeros of J + (k) are simple if k n = 0 as u(k n , r) ∝
u + (k n , r) therein.
One will show in Sect. 3.3 that Eq. (2.174) can be generalized to resonance states
as well. Consequently, the determination of bound and resonance states amounts to
finding the zeros of an analytic complex function, which is done numerically with
the quickly converging Newton method.
2.6.5 Wave Functions in a Screened Potential
The infinite range of the Coulomb potential induces an essential singularity in the
k-plane for k = 0. Consequently, it is customary to consider a screened potential,
vanishing after a finite large radius R (s) , to suppress the singularity induced by the
Coulomb potential at large distances. Unfortunately, the introduction of a screened
potential is not appropriate in all circumstances. For example, it is not sufficient to
screen the potential in Eq. (2.3) to demonstrate the completeness relation for proton
states (see Sect. 3.2). However, a screened potential is a useful intermediary function
between finite-range potentials and infinite-range potentials containing a Coulomb
potential. The centrifugal potential poses no theoretical problem as it is integrable
on the real r-axis.
To demonstrate the completeness relation involving proton states, one will start
from that generated by a screened potential, whose radius R (s) will go to infinity
afterwards to recover the infinite range of the Coulomb potential. For simplicity,
even though it is not necessary, one will also suppress the centrifugal potential when
r > R (s) , so that Eq. (2.5) becomes that of a neutron s state ( = 0) for r > R (s) .
2 The Discrete Spectrum and the Continuum
By replacing u(k a , r) by u + (k a , r) if r 1 = 0 and r 2 = r, and u(k b , r) by u + (k b , r)
if r 1 = r and r 2 > r 1 , and putting k a = k b = k n in Eq. (2.171), one obtains:
u
+ (k n , r) ˙
u(k n , r) − ˙
u
(k n , r)u
+ (k n , r) = 2k n
r
0
u(k n , r
)u
+ (k n , r
) dr
(2.172)
˙
u
+ (k n , r)u(k n , r) − u
(k n , r) ˙
u
+ (k n , r) = 2k n
+∞
r
u(k n , r
)u
+ (k n , r
) dr
,
(2.173)
where the limit r 2 → +∞ has been effected, and the dot above u indicates
differentiation with respect to k b . One has also used the fact that u(k n , 0) = 0 and
u(k n , r ) → 0 for r → +∞, arising from the bound character of u(k n , r). One then
obtains from Eqs. (2.172) and (2.173) the following equality:
dJ +
dk
(k)
k=k n
= 2k n
+∞
0
u(k n , r)u
+ (k n , r) dr ,
(2.174)
which also proves that the zeros of J + (k) are simple if k n = 0 as u(k n , r) ∝
u + (k n , r) therein.
One will show in Sect. 3.3 that Eq. (2.174) can be generalized to resonance states
as well. Consequently, the determination of bound and resonance states amounts to
finding the zeros of an analytic complex function, which is done numerically with
the quickly converging Newton method.
2.6.5 Wave Functions in a Screened Potential
The infinite range of the Coulomb potential induces an essential singularity in the
k-plane for k = 0. Consequently, it is customary to consider a screened potential,
vanishing after a finite large radius R (s) , to suppress the singularity induced by the
Coulomb potential at large distances. Unfortunately, the introduction of a screened
potential is not appropriate in all circumstances. For example, it is not sufficient to
screen the potential in Eq. (2.3) to demonstrate the completeness relation for proton
states (see Sect. 3.2). However, a screened potential is a useful intermediary function
between finite-range potentials and infinite-range potentials containing a Coulomb
potential. The centrifugal potential poses no theoretical problem as it is integrable
on the real r-axis.
To demonstrate the completeness relation involving proton states, one will start
from that generated by a screened potential, whose radius R (s) will go to infinity
afterwards to recover the infinite range of the Coulomb potential. For simplicity,
even though it is not necessary, one will also suppress the centrifugal potential when
r > R (s) , so that Eq. (2.5) becomes that of a neutron s state ( = 0) for r > R (s) .