42
2 The Discrete Spectrum and the Continuum
Ginocchio potentials (see Eqs. (2.92), (2.123), and (2.124). Compare them to
those obtained with a Woods-Saxon potential using the Woods-Saxon code.
C. Run the one-particle code of radial wave functions to calculate energies
and widths of resonances with both the second-order approximation for
V PTG–mod (r) and with a direct integration of Eq. (2.2). From numerical
calculations, determine the typical domain of validity of the second-order
approximation.
D. Locate the phase shift changes in the obtained figure and relate them to
energies and widths of calculated resonances.
2.5
Basic Properties of Bound States
From a mathematical and physical point of view, bound one-body states are
important, as they are related to the properties of localized quantum systems.
Moreover, these states must always be included in the set of states entering the
completeness relations generated by a one-body Hamiltonian (2.3). Due to the
special role played by bound states in the eigenspectrum of one-body Hamiltonians,
we will state the general properties of bound states and discuss the most important of
them. We will also show that bound states form a discrete set of states and that they
are orthogonal to each other. These properties are fundamental, as they are directly
related to the fact that the experimental spectrum of bound levels of the quantum
system is discrete.
Another feature associated to bound eigenstates is that there is a finite number
of them if their generating potential is short-range or repulsive at large distances.
Conversely, one has an infinity of bound eigenstates accumulating at zero energy
if their generating potential is of infinite range and attractive, as is the case of the
hydrogen spectrum. One will also demonstrate this property due to its mathematical
and physical importance. Finally, one will state the virial theorem associated to the
bound eigenstates of the Hamiltonian of Eq. (2.3). One will see that this theorem
allows to explain why the hydrogen atom is stable, on the one hand, and, on the
other hand, why halos and resonance states of energy close to zero can develop in
atomic nuclei.
Due to the k ↔ −k symmetry of Eq. (2.2), it is sufficient to consider the upper
half of the complex plane. Thus, one can deduce from Eq. (2.7) that u(k, r) is a
bound state only if C − = 0, as H
±
,η (kr) ∼ exp(±(ikr −η ln(2kr))) up to a function
of kr behaving as a rational function for k = 0 and r → +∞ (see Sect. 2.3). Hence,
both
|u(k, r)| 2 dr and integrals of u(k, r) and of its derivatives are finite as well.
As k 2 is real for bound states (see Exercise XI), one can always write k = iκ, with
κ ≥ 0. This notation will be commonly used in the following. For real k 2 , Eq. (2.2)
involves only real numbers. Hence, u(k, r) can be always chosen so that it is a realvalued function.
2 The Discrete Spectrum and the Continuum
Ginocchio potentials (see Eqs. (2.92), (2.123), and (2.124). Compare them to
those obtained with a Woods-Saxon potential using the Woods-Saxon code.
C. Run the one-particle code of radial wave functions to calculate energies
and widths of resonances with both the second-order approximation for
V PTG–mod (r) and with a direct integration of Eq. (2.2). From numerical
calculations, determine the typical domain of validity of the second-order
approximation.
D. Locate the phase shift changes in the obtained figure and relate them to
energies and widths of calculated resonances.
2.5
Basic Properties of Bound States
From a mathematical and physical point of view, bound one-body states are
important, as they are related to the properties of localized quantum systems.
Moreover, these states must always be included in the set of states entering the
completeness relations generated by a one-body Hamiltonian (2.3). Due to the
special role played by bound states in the eigenspectrum of one-body Hamiltonians,
we will state the general properties of bound states and discuss the most important of
them. We will also show that bound states form a discrete set of states and that they
are orthogonal to each other. These properties are fundamental, as they are directly
related to the fact that the experimental spectrum of bound levels of the quantum
system is discrete.
Another feature associated to bound eigenstates is that there is a finite number
of them if their generating potential is short-range or repulsive at large distances.
Conversely, one has an infinity of bound eigenstates accumulating at zero energy
if their generating potential is of infinite range and attractive, as is the case of the
hydrogen spectrum. One will also demonstrate this property due to its mathematical
and physical importance. Finally, one will state the virial theorem associated to the
bound eigenstates of the Hamiltonian of Eq. (2.3). One will see that this theorem
allows to explain why the hydrogen atom is stable, on the one hand, and, on the
other hand, why halos and resonance states of energy close to zero can develop in
atomic nuclei.
Due to the k ↔ −k symmetry of Eq. (2.2), it is sufficient to consider the upper
half of the complex plane. Thus, one can deduce from Eq. (2.7) that u(k, r) is a
bound state only if C − = 0, as H
±
,η (kr) ∼ exp(±(ikr −η ln(2kr))) up to a function
of kr behaving as a rational function for k = 0 and r → +∞ (see Sect. 2.3). Hence,
both
|u(k, r)| 2 dr and integrals of u(k, r) and of its derivatives are finite as well.
As k 2 is real for bound states (see Exercise XI), one can always write k = iκ, with
κ ≥ 0. This notation will be commonly used in the following. For real k 2 , Eq. (2.2)
involves only real numbers. Hence, u(k, r) can be always chosen so that it is a realvalued function.