2.1 Definition of One-Body States
17
where v c is the strength of the Coulomb potential for r → +∞, which vanishes
for neutral particles but is proportional to the charge of the potential v l (r) for
charged particles, and where v c 0 and v 0 are real. One demands that potentials have a
Coulomb asymptote for both r → 0 and r → +∞. These two conditions typically
occur in practical applications.
0 and v c 0 in (2.4) can be different from and v c , as for example in the PöschlTeller-Ginocchio potential (see Sect. 2.4), but one must have 0 ≥ 0. The resonant
and scattering u(k, r) states form the spectrum of h, in which only bound states are
eigenstates of h, as they are integrable. However, as is commonly done in quantum
physics, the function u(k, r) of Eq. (2.2) and its energy proportional to k 2 will
always be referred to as eigenstate and eigenvalue of h in Eq. (2.3), respectively.
For simplicity, one will consider in the following that Eqs. (2.4) and (2.5) are
obtained for 0 < r < r 0 and r > R, respectively, with 0 < r 0 < R. Indeed, v l (r) −
v c /r typically vanishes exponentially for r → +∞, so that it can be neglected after
a finite radius R. Moreover, v l (r) can only have a divergence of Coulomb type, that
is, of the form 1/r, when r → 0. In fact, the reduction of Eq. (2.2) to Eqs. (2.4)
and (2.5) for 0 < r < r 0 and r > R, respectively, covers all practical purposes. As
all calculations are done up to a given numerical precision, taking sufficiently small
r 0 and sufficiently large R provides with the same result as that arising by taking the
values r 0 = 0 and R = +∞. For the same reason, one can also demand v l (r) to be
twice differentiable ∀r ≥ 0 (see Eq. (2.2)). Indeed, even though v l (r) is often only
piecewise continuous on the real axis, as for a square well potential, it can always be
approximated up to an arbitrarily small precision by a twice differentiable function.
Hamiltonians bearing an effective mass when r 0 < r ≤ R are implicitly
accounted for in Eq. (2.2) as they can always be rewritten via a point-canonical transformation to verify a Schrödinger equation without effective mass [3]. Evidently, the
new Schrödinger equation induced by the point-canonical transformation must also
reduce to Eqs. (2.4) and (2.5) when 0 < r < r 0 and r > R, respectively.
φ(k, r) in Eq. (2.1) must be finite for all r. As a consequence, the boundary
conditions for u(k, r) read:
u(k, r) = C 0 F 0 ,η 0 (k 0 r) , 0 ≤ r ≤ r 0
(2.6)
u(k, r) = C
+ H
+
,η (kr) + C
− H
−
,η (kr) , r ≥ R
(2.7)
where F 0 ,η 0 (k 0 r) and H
±
,η (kr) are Coulomb wave functions (see Sect. 2.3). In
Eqs. (2.6) and (2.7), k 0 = k
1 − v 0 /k 2 , η and η 0 are the Sommerfeld parameters
equal to v c /(2k) and v c 0 /(2k 0 ), respectively. C 0 , C + , and C − are constants to
be determined from the continuity, derivability, and normalization of u(k, r).
Equations (2.2) and (2.6) univocally define u(k, r) up to a normalization constant.
17
where v c is the strength of the Coulomb potential for r → +∞, which vanishes
for neutral particles but is proportional to the charge of the potential v l (r) for
charged particles, and where v c 0 and v 0 are real. One demands that potentials have a
Coulomb asymptote for both r → 0 and r → +∞. These two conditions typically
occur in practical applications.
0 and v c 0 in (2.4) can be different from and v c , as for example in the PöschlTeller-Ginocchio potential (see Sect. 2.4), but one must have 0 ≥ 0. The resonant
and scattering u(k, r) states form the spectrum of h, in which only bound states are
eigenstates of h, as they are integrable. However, as is commonly done in quantum
physics, the function u(k, r) of Eq. (2.2) and its energy proportional to k 2 will
always be referred to as eigenstate and eigenvalue of h in Eq. (2.3), respectively.
For simplicity, one will consider in the following that Eqs. (2.4) and (2.5) are
obtained for 0 < r < r 0 and r > R, respectively, with 0 < r 0 < R. Indeed, v l (r) −
v c /r typically vanishes exponentially for r → +∞, so that it can be neglected after
a finite radius R. Moreover, v l (r) can only have a divergence of Coulomb type, that
is, of the form 1/r, when r → 0. In fact, the reduction of Eq. (2.2) to Eqs. (2.4)
and (2.5) for 0 < r < r 0 and r > R, respectively, covers all practical purposes. As
all calculations are done up to a given numerical precision, taking sufficiently small
r 0 and sufficiently large R provides with the same result as that arising by taking the
values r 0 = 0 and R = +∞. For the same reason, one can also demand v l (r) to be
twice differentiable ∀r ≥ 0 (see Eq. (2.2)). Indeed, even though v l (r) is often only
piecewise continuous on the real axis, as for a square well potential, it can always be
approximated up to an arbitrarily small precision by a twice differentiable function.
Hamiltonians bearing an effective mass when r 0 < r ≤ R are implicitly
accounted for in Eq. (2.2) as they can always be rewritten via a point-canonical transformation to verify a Schrödinger equation without effective mass [3]. Evidently, the
new Schrödinger equation induced by the point-canonical transformation must also
reduce to Eqs. (2.4) and (2.5) when 0 < r < r 0 and r > R, respectively.
φ(k, r) in Eq. (2.1) must be finite for all r. As a consequence, the boundary
conditions for u(k, r) read:
u(k, r) = C 0 F 0 ,η 0 (k 0 r) , 0 ≤ r ≤ r 0
(2.6)
u(k, r) = C
+ H
+
,η (kr) + C
− H
−
,η (kr) , r ≥ R
(2.7)
where F 0 ,η 0 (k 0 r) and H
±
,η (kr) are Coulomb wave functions (see Sect. 2.3). In
Eqs. (2.6) and (2.7), k 0 = k
1 − v 0 /k 2 , η and η 0 are the Sommerfeld parameters
equal to v c /(2k) and v c 0 /(2k 0 ), respectively. C 0 , C + , and C − are constants to
be determined from the continuity, derivability, and normalization of u(k, r).
Equations (2.2) and (2.6) univocally define u(k, r) up to a normalization constant.