22
2 The Discrete Spectrum and the Continuum
Deduce that the eigenstates of Eq. (2.26) can be written in terms of r 1 , r 2 or
r rel , R CM coordinates.
B. Deduce from Eq. (2.26) that an eigenstate of the Hamiltonian written in
laboratory coordinates is a finite linear combination of those written in
relative/center-of-mass coordinates. One will use Eq. (2.25) for that matter.
From the latter result, prove the existence of the Talmi-Brody-Moshinsky
transformation and the energy condition of Eq. (2.25).
Consequently, the use of harmonic oscillator states in the Gamow shell model is
not anecdotal in the Gamow Shell Model but, on the contrary, has been essential to
its development.
2.3
Coulomb Potential and Coulomb Wave Functions
Coulomb wave functions are among the most fundamental tools of quantum physics.
They describe the behavior of a charged particle in a Coulomb field, and are
therefore present in almost all domains of quantum physics. Their dimensionless
Schrödinger equation reads:
W
(z) =
+ 1)
z 2
+
2η
z
− 1
W (z) ,
(2.27)
where W (z) is a Coulomb wave function, is the orbital angular momentum, and η
the Sommerfeld parameter:
η =
e Z m
¯
h 2 k
,
(2.28)
where e is the elementary charge, m is its mass, k is its linear momentum and Z
is the charge of the external Coulomb potential. All parameters are considered as
complex in this section (see Refs. [6,7] for details about the numerical evaluation of
Coulomb wave functions).
In the following, one will discuss how to define and calculate Coulomb wave
functions with complex parameters. For this, it is convenient to consider that (() ≥
−1/2. There is no loss of generality in this restriction because Eq. (2.27) is invariant
with the change → − − 1.
The Coulomb wave functions are initially defined with confluent hypergeometric
functions [8]. The regular Coulomb wave function takes the form:
F ,η (z) = C (η) z
e
iωz
1 F 1 (1 + + iωη; 2 + 2; −2iωz)
(2.29)
C (η) = 2
exp
−πη + [ln(Γ (1 + + iη)) + ln(Γ (1 + − iη))]
2
(2.30)
− ln(Γ (2 + 2))] ,
2 The Discrete Spectrum and the Continuum
Deduce that the eigenstates of Eq. (2.26) can be written in terms of r 1 , r 2 or
r rel , R CM coordinates.
B. Deduce from Eq. (2.26) that an eigenstate of the Hamiltonian written in
laboratory coordinates is a finite linear combination of those written in
relative/center-of-mass coordinates. One will use Eq. (2.25) for that matter.
From the latter result, prove the existence of the Talmi-Brody-Moshinsky
transformation and the energy condition of Eq. (2.25).
Consequently, the use of harmonic oscillator states in the Gamow shell model is
not anecdotal in the Gamow Shell Model but, on the contrary, has been essential to
its development.
2.3
Coulomb Potential and Coulomb Wave Functions
Coulomb wave functions are among the most fundamental tools of quantum physics.
They describe the behavior of a charged particle in a Coulomb field, and are
therefore present in almost all domains of quantum physics. Their dimensionless
Schrödinger equation reads:
W
(z) =
+ 1)
z 2
+
2η
z
− 1
W (z) ,
(2.27)
where W (z) is a Coulomb wave function, is the orbital angular momentum, and η
the Sommerfeld parameter:
η =
e Z m
¯
h 2 k
,
(2.28)
where e is the elementary charge, m is its mass, k is its linear momentum and Z
is the charge of the external Coulomb potential. All parameters are considered as
complex in this section (see Refs. [6,7] for details about the numerical evaluation of
Coulomb wave functions).
In the following, one will discuss how to define and calculate Coulomb wave
functions with complex parameters. For this, it is convenient to consider that (() ≥
−1/2. There is no loss of generality in this restriction because Eq. (2.27) is invariant
with the change → − − 1.
The Coulomb wave functions are initially defined with confluent hypergeometric
functions [8]. The regular Coulomb wave function takes the form:
F ,η (z) = C (η) z
e
iωz
1 F 1 (1 + + iωη; 2 + 2; −2iωz)
(2.29)
C (η) = 2
exp
−πη + [ln(Γ (1 + + iη)) + ln(Γ (1 + − iη))]
2
(2.30)
− ln(Γ (2 + 2))] ,