Solutions to Exercises
77
from Eq. (2.182) by differentiating its denominator only and evaluating it at k = k n :
Res [S (k)] k=k n = (−2ik n C
+ )
dJ +
dk
(k)
k=k n
−1
= (−2ik n C
+ )
2k n
+∞
0
u(k n , r)u
+ (k n , r) dr
−1
=
i
+∞
0
u
+ (k n , r)
2 dr
−1
.
(2.223)
Exercise XV.
A. Bound wave functions are real and decrease exponentially at infinity.
Inside of a nucleus, wave function of a narrow resonance is similar to bound
state wave function, as the real part of the resonance wave function resembles
the bound state having the same number of nodes in the nuclear interior and its
imaginary part is very small therein. Real and imaginary parts become of the
same order of magnitude in the asymptotic region, as the outgoing Hankel and
Coulomb wave functions are therein equal to exp(ikr) multiplied by a smoothly
varying function. They increase exponentially with oscillations on the real axis,
but this increase can be mild if the width is very small.
Wave functions of broad resonances always have large real and imaginary parts,
and increase quickly in modulus on the real axis. There is no resemblance with
bound states.
Wave functions of low energy antibound states are similar to bound state wave
functions for very small radii, but quickly reach their asymptote, which is
exponential without oscillations. The antibound wave functions are also purely
imaginary.
B. One can see from the obtained plots that widths depend strongly on orbital
angular momentum and total charge Z. Indeed, the increase of width with
energy is less and less visible when and Z are augmented, with the effect more
pronounced for the charge than for the angular momentum . The centrifugal
barrier, in + 1)/r 2 , is less confining than the Coulomb barrier, in Z/r. In
the absence of barrier in the asymptotic region, that is, for neutron s states, one
cannot generate resonance states when the potential becomes more and more
shallow. In fact, when decreasing the potential depth one goes from the bound
to antibound region.
Exercise XVI.
A. It is necessary to slowly change potential depth because width varies quickly
with energy.
B. Calculations show that energy and width must be of the same order of magnitude
for direct integration to be able to provide both of them accurately.
77
from Eq. (2.182) by differentiating its denominator only and evaluating it at k = k n :
Res [S (k)] k=k n = (−2ik n C
+ )
dJ +
dk
(k)
k=k n
−1
= (−2ik n C
+ )
2k n
+∞
0
u(k n , r)u
+ (k n , r) dr
−1
=
i
+∞
0
u
+ (k n , r)
2 dr
−1
.
(2.223)
Exercise XV.
A. Bound wave functions are real and decrease exponentially at infinity.
Inside of a nucleus, wave function of a narrow resonance is similar to bound
state wave function, as the real part of the resonance wave function resembles
the bound state having the same number of nodes in the nuclear interior and its
imaginary part is very small therein. Real and imaginary parts become of the
same order of magnitude in the asymptotic region, as the outgoing Hankel and
Coulomb wave functions are therein equal to exp(ikr) multiplied by a smoothly
varying function. They increase exponentially with oscillations on the real axis,
but this increase can be mild if the width is very small.
Wave functions of broad resonances always have large real and imaginary parts,
and increase quickly in modulus on the real axis. There is no resemblance with
bound states.
Wave functions of low energy antibound states are similar to bound state wave
functions for very small radii, but quickly reach their asymptote, which is
exponential without oscillations. The antibound wave functions are also purely
imaginary.
B. One can see from the obtained plots that widths depend strongly on orbital
angular momentum and total charge Z. Indeed, the increase of width with
energy is less and less visible when and Z are augmented, with the effect more
pronounced for the charge than for the angular momentum . The centrifugal
barrier, in + 1)/r 2 , is less confining than the Coulomb barrier, in Z/r. In
the absence of barrier in the asymptotic region, that is, for neutron s states, one
cannot generate resonance states when the potential becomes more and more
shallow. In fact, when decreasing the potential depth one goes from the bound
to antibound region.
Exercise XVI.
A. It is necessary to slowly change potential depth because width varies quickly
with energy.
B. Calculations show that energy and width must be of the same order of magnitude
for direct integration to be able to provide both of them accurately.