20
2 The Discrete Spectrum and the Continuum
In this expression L
(α)
n (x) is the generalized Laguerre polynomial and N nn is a
normalization constant:
N nn =
2
b
Γ (n + 1)
Γ (n + + 3/2)
,
(2.18)
where Γ (z) is the Euler Gamma function. The energy e nn of the considered
harmonic oscillator state reads:
e nn =
2n + +
3
2
¯
hω .
(2.19)
Comparison of the analytical formulas of energy and wave functions in the one- and
three-dimensional cases will be effected in Exercise II.
Exercise II
One will compare the harmonic oscillator wave functions and energies in the
one- and three-dimensional cases and point out analogies between them.
For this, show that the formulas of Eqs. (2.12), (2.14), and (2.17)–(2.19)
provide with normalized solutions of Eqs. (2.11) and (2.16).
Explain from Eq. (2.15) why Eqs. (2.12), (2.14), and (2.17)–(2.19) are very
similar in the one- and three-dimensional cases.
As the harmonic oscillator potential goes to +∞ for increasing x and r (see
Eqs. (2.10) and (2.15)), it possesses only bound states. Its set of eigenstates is then
discrete. Therefore, it is straightforward to show their completeness properties from
variational arguments [1]. However, due to the rapid decrease of harmonic oscillator
wave functions on the real axis (see Eqs. (2.12) and (2.17)), the harmonic oscillator
states in practice are efficient to expand only well-bound states.
The basis of harmonic oscillator states is fundamental to the standard shell
model, which has been very effective to describe well-bound states of stable nuclei.
However, weakly bound states and resonances cannot be expanded in this basis
as the squared modulus of their wave functions is either slowly decreasing or
increasing along the real axis. For that purpose, the use of the Berggren basis,
possessing bound states, resonances, and scattering states, is much better suited.
Nevertheless, harmonic oscillator states are very important for weakly bound and
resonance nuclei as well, not for the expansion their wave functions, but to expand
two-body matrix elements of nuclear interaction. Indeed, on the one hand, the
nuclear interaction is localized, so that it can be efficiently expanded with harmonic
oscillator states. On the other hand, two-body harmonic oscillator states whose
coordinates are defined with laboratory coordinates, can always be written as a
finite sum of products of two-body harmonic oscillator states in the relative or
center-of-mass coordinates of the two-particle system. This is the objective of
the Talmi-Brody-Moshinsky transformation [4, 5]. Indeed, as nuclear interactions
depend only on relative degrees of freedom, their matrix elements which are
2 The Discrete Spectrum and the Continuum
In this expression L
(α)
n (x) is the generalized Laguerre polynomial and N nn is a
normalization constant:
N nn =
2
b
Γ (n + 1)
Γ (n + + 3/2)
,
(2.18)
where Γ (z) is the Euler Gamma function. The energy e nn of the considered
harmonic oscillator state reads:
e nn =
2n + +
3
2
¯
hω .
(2.19)
Comparison of the analytical formulas of energy and wave functions in the one- and
three-dimensional cases will be effected in Exercise II.
Exercise II
One will compare the harmonic oscillator wave functions and energies in the
one- and three-dimensional cases and point out analogies between them.
For this, show that the formulas of Eqs. (2.12), (2.14), and (2.17)–(2.19)
provide with normalized solutions of Eqs. (2.11) and (2.16).
Explain from Eq. (2.15) why Eqs. (2.12), (2.14), and (2.17)–(2.19) are very
similar in the one- and three-dimensional cases.
As the harmonic oscillator potential goes to +∞ for increasing x and r (see
Eqs. (2.10) and (2.15)), it possesses only bound states. Its set of eigenstates is then
discrete. Therefore, it is straightforward to show their completeness properties from
variational arguments [1]. However, due to the rapid decrease of harmonic oscillator
wave functions on the real axis (see Eqs. (2.12) and (2.17)), the harmonic oscillator
states in practice are efficient to expand only well-bound states.
The basis of harmonic oscillator states is fundamental to the standard shell
model, which has been very effective to describe well-bound states of stable nuclei.
However, weakly bound states and resonances cannot be expanded in this basis
as the squared modulus of their wave functions is either slowly decreasing or
increasing along the real axis. For that purpose, the use of the Berggren basis,
possessing bound states, resonances, and scattering states, is much better suited.
Nevertheless, harmonic oscillator states are very important for weakly bound and
resonance nuclei as well, not for the expansion their wave functions, but to expand
two-body matrix elements of nuclear interaction. Indeed, on the one hand, the
nuclear interaction is localized, so that it can be efficiently expanded with harmonic
oscillator states. On the other hand, two-body harmonic oscillator states whose
coordinates are defined with laboratory coordinates, can always be written as a
finite sum of products of two-body harmonic oscillator states in the relative or
center-of-mass coordinates of the two-particle system. This is the objective of
the Talmi-Brody-Moshinsky transformation [4, 5]. Indeed, as nuclear interactions
depend only on relative degrees of freedom, their matrix elements which are