2.6 Analytical Properties of the Wave Functions
49
is an integrable function of r. Let us now assume that there is a bound state at
k = 0. The zero-energy bound state is proportional to G ,η (kr) when r > R (see
Sect. 2.3.3). As u(k, r) is orthogonal to the zero-energy bound state ∀k > 0, the
relation u(k, r) ∝ G ,η (kr) when k → 0 for r > R cannot hold. Consequently,
A k G ,η (kR) = O(F ,η (kR)) when k → 0, so that one has for r ≥ R:
u(k, r) = N k F ,η (kr) f (k, r) ,
(2.142)
where f (k, r) is bounded for k → 0, but can diverge for r → 0.
In order to devise the behavior of f (k, r) for 0 < r < R, let us rewrite u(k, r)
as:
u(k, r) = u(k, R)
F ,η (kr)
F ,η (kR)
f (k, r)
f (k, R)
.
(2.143)
f (k, R) is bounded and the ratio of Coulomb wave functions has a finite limit in
Eq. (2.143) for k → 0 (see Eq. (2.37)). u(k, r) is also continuous and finite for
k → 0 (see Exercise XII). Consequently, f (k, r) for k → 0 and 0 < r < R is
bounded as well. This implies that Eq. (2.142) is also valid for 0 < r < R.
For a fixed r > 0 and k → +∞, one will consider the uniform WKB
approximation [38], that is, based on the use of a special function removing turning
point singularity and hence valid ∀r > 0. k is supposed to be real for the moment.
For this, let us write Eq. (2.2) similarly to Eq. (2.4), with the 0 and v c 0 values
appearing explicitly:
u
(k, r) =
0 (( 0 + 1)
r 2
+
v c 0
r
+ v 0 (r) − k
2
u(k, r) ,
(2.144)
where v 0 (r) is finite ∀r ≥ 0 and v 0 (r) = v 0 for 0 ≤ r ≤ r 0 (see Sect. 2.1). One can
then choose k so that |v 0 (r)| is arbitrarily small against k 2 ∀r ≥ 0 so that one can
introduce the following functions:
u
±
app (k, r) = k
1/2
0 λ
−1/2 (r) H
±
0 ,η 0
(Λ(r)) , r > 0
(2.145)
where:
λ(r) = k
1 −
v 0 (r)
k 2
(2.146)
Λ(r) =
r
0
λ(r
) dr
.
(2.147)
Eq. (2.145) provides with an approximate solution of Eq. (2.2), what can be
checked by inserting Eq. (2.145) in Eq. (2.144). This arises due to the standard
49
is an integrable function of r. Let us now assume that there is a bound state at
k = 0. The zero-energy bound state is proportional to G ,η (kr) when r > R (see
Sect. 2.3.3). As u(k, r) is orthogonal to the zero-energy bound state ∀k > 0, the
relation u(k, r) ∝ G ,η (kr) when k → 0 for r > R cannot hold. Consequently,
A k G ,η (kR) = O(F ,η (kR)) when k → 0, so that one has for r ≥ R:
u(k, r) = N k F ,η (kr) f (k, r) ,
(2.142)
where f (k, r) is bounded for k → 0, but can diverge for r → 0.
In order to devise the behavior of f (k, r) for 0 < r < R, let us rewrite u(k, r)
as:
u(k, r) = u(k, R)
F ,η (kr)
F ,η (kR)
f (k, r)
f (k, R)
.
(2.143)
f (k, R) is bounded and the ratio of Coulomb wave functions has a finite limit in
Eq. (2.143) for k → 0 (see Eq. (2.37)). u(k, r) is also continuous and finite for
k → 0 (see Exercise XII). Consequently, f (k, r) for k → 0 and 0 < r < R is
bounded as well. This implies that Eq. (2.142) is also valid for 0 < r < R.
For a fixed r > 0 and k → +∞, one will consider the uniform WKB
approximation [38], that is, based on the use of a special function removing turning
point singularity and hence valid ∀r > 0. k is supposed to be real for the moment.
For this, let us write Eq. (2.2) similarly to Eq. (2.4), with the 0 and v c 0 values
appearing explicitly:
u
(k, r) =
0 (( 0 + 1)
r 2
+
v c 0
r
+ v 0 (r) − k
2
u(k, r) ,
(2.144)
where v 0 (r) is finite ∀r ≥ 0 and v 0 (r) = v 0 for 0 ≤ r ≤ r 0 (see Sect. 2.1). One can
then choose k so that |v 0 (r)| is arbitrarily small against k 2 ∀r ≥ 0 so that one can
introduce the following functions:
u
±
app (k, r) = k
1/2
0 λ
−1/2 (r) H
±
0 ,η 0
(Λ(r)) , r > 0
(2.145)
where:
λ(r) = k
1 −
v 0 (r)
k 2
(2.146)
Λ(r) =
r
0
λ(r
) dr
.
(2.147)
Eq. (2.145) provides with an approximate solution of Eq. (2.2), what can be
checked by inserting Eq. (2.145) in Eq. (2.144). This arises due to the standard