ETNA
Electronic Transactions on Numerical Analysis.
Volume 24, pp. 7-23, 2006.
Copyright  2006, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
ON RECURRENCE RELATIONS FOR RADIAL WAVE FUNCTIONS FOR THE
-TH DIMENSIONAL OSCILLATORS AND HYDROGENLIKE ATOMS:
ANALYTICAL AND NUMERICAL STUDY R. ÁLVAREZ-NODARSE , J. L. CARDOSO , AND N. R. QUINTERO Abstract. Using a general procedure for finding recurrence relations for hypergeometric functions and polynomials introduced by Cardoso et al. [J. Phys. A, 36 (2003), pp. 2055-2068] we obtain some new recurrence relations
for the radial wave functions of the -th dimensional isotropic harmonic oscillators as well as the hydrogenlike
atoms. A numerical analysis of such recurrences is also presented.
Key words. wave functions, linear recurrence relations, Laguerre polynomials
AMS subject classifications. 33C45, 11B37, 81V45, 81Q99
1. Introduction. There are many applications in modern physics that require the knowledge of the wave functions of hydrogenlike atoms and isotropic harmonic oscillators (see e.g.
[13, 16] and references therein). Of particular interest is the numerical implementation of
such functions. The most natural, efficient and fast way for generating such functions is to
use recurrence relations. In fact, although the explicit expressions are known, their numerical
implementations are usually very unstable.
In this paper we will continue the research started in [8] for obtaining recurrence relations and ladder-type operators for the -th dimensional isotropic harmonic oscillators and
the hydrogenlike atoms. In fact, using the technique of [8] we will obtain some new recurrences on one hand, and on the other we will present a comparative numerical analysis of the
obtained recurrence relations for generating numerically the corresponding eigenfunctions.
The numerical analysis of the linear recurrences, i.e., the rounding errors bounds, stability
of the scheme, etc. is, in general, very complicate (see the nice paper [5] and references
therein). So we will restrict ourselves to the discussion of the numerical examples. Even in
this framework of no rigorous numerical analysis it is possible to conclude what kind of relations are more useful for numerical computations and what are not. This is the main original
contribution of the present paper. Finally, let us mention that the present method for finding
recurrence relations can be extended to any quantum system whose (radial) wave function are
proportional to hypergeometric-type functions (see e.g. [4]).
The structure of the paper is as follows: In sections 2 and 3 the isotropic oscillator and
hydrogenlike atoms radial wave functions are introduced. In both cases we include some new
recurrence relations as well as the detailed discussion of their computational applications.
Finally, an appendix with the required formulas of the Laguerre polynomials is included.
2. The isotropic harmonic oscillator. The -dimensional isotropic harmonic oscillator (I.H.O.) is described by the Schrödinger equation
1
!
. -, ! *
+
"0/
- "$&
"$#&%(' "
# %
)
')
Received March 11, 2004. Accepted for publication June 3, 2005. Recommended by F. Marcellán.
Apdo. Postal 1160,
Sevilla, E-41080, Sevilla, Spain (ran@us.es).
Departamento de Matemática, Universidade de Trás-os-Montes e Alto Douro. Apartado 202, 5001 - 911 Vila
Real, Portugal (jluis@utad.pt).
Dpto. Fı́sica Aplicada I, EUP, Universidad de Sevilla, c/Virgen de África 7, 41011, Sevilla, Spain
(niurka@us.es).
Departamento de Análisis Matemático, Facultad de Matemática, Universidad de Sevilla.
7
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8
R. ÁLVAREZ-NODARSE, J. L. CARDOSO, AND N. R. QUINTERO
Its solutions are of the form 24793 8;: =<?> 8A@ :CB < , where 2793 8D:
65 by (see e.g. [3, 6]) 65
called the radial wave functions, defined
=< is the radial part, usually
9XZY 8[R S L
8HGJI6L9K MON QL P 8A7 R
S L I %
( _T
: T9<QUF7=3 8 WV \^] X
(2.1)
2793 8;: =<E;F793 8 05
_ 05
65
E
`
fehg
X
being baT /c/c/ and bad /c// , the quantum numbers, and
the dimension of
the space. The angular part > 8A@ :iB < are the so-called th-spherical or_ hyperspherical harX
monics [3, 14]. In the following, we will assume that the parameters , , are nonnegative
integers.
2.1. Recurrence relations for the I.H.O. radial functions. For the functions (2.1) the
following theorem hold [8]
_ radial
_
T HEOREM 2.1. Let 2 793 8;: =< , 2 7=3 R7 Kkj 8AR8 K : =< and 2 7=3 R7 L j 8AR8 L : =< be three different
X % X
6
5
6
5
6
5
%
functions of the -th dimensional isotropic harmonic oscillator, where
and are
integers such that
l4mon
_T;_ _T(_
e
< a
Then, there exist three all non vanishing polynomials in , prq , p %
%
(2.2)
p qs: =<t27 3 j 8
: 9< p : =<t27=3 R7 K j 8AR8 K : =< p : =<t27=3 R7
65
65
65
:
X
X % X
X
% /
, and p
, such that
L j 8[R8 L : =<Eba /
The proof of this theorem can be found in [8]. A key point on the proof was the recurrence
relation,
P 8[R S L I %
P 8AR8 K R S L I %
P 8AR8 L R S L I %
%
i: w < u i: w < 7=3 R7 K
:Cw < u :iw < 7=3 R7 L
:iw <xyad
uvq C: w < 7
(2.3)
5
5
where w T , and u0z :iw < , { |aT , are not all three vanishing polynomials. Moreover, in
[8] it was shown that
I %
p q}: =<E~€F7 3 j 8
u
65
p % : =<E~€F7=3 Rƒ7 Kkj 8[R8 K
65
p : =<E ~ F 7=3 Rƒ7 L j 8[R8 L
65
(2.4)
8 R8 L
q}: T<‚ K I %
8L

u % : d9<‚ I %
8

u : d <‚ K /
From the above theorem and the relation (2.3) it is very simple to obtain à la carte
relations between three different radial functions of the I.H.O. Let us point out that, in general,
it is not easy to obtain the coefficients u6z in (2.3), nevertheless, combining in a certain way
the known relations of the Laguerre polynomials (see the appendix A) they can be found.
This has been shown in [7, 8]. We will show here how this works in one of the new examples,
for the others the computations are similar. Here we present some examples. The first four
can be found in [8] and the other five seem to be new.
„ X % X
V
(2.5)
_
%
,
, 4_€
X X
_
ya [8, page 2059]
y‡
4
;
_€
9X
%
27†3 I j 8‚: =<
T9
‰ˆ‰2Š7 3 j 8‰: 9<
65
05
4
_€
X X
2 7=3 R % j 8t: =<0a /
<
V :
65
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_
ON RECURRENCE RELATIONS FOR REDIAL WAVE FUNCTIONS
_
„ X % X ba , % , [8, page 2059]
(_T
Œd_T
X
V 2 7 3 j 8‹I % : =<
65
(2.6)
V _
_
„ X % ya , X , % , ba [8, page 2059]
(_T
]X
X
V 2 7 3 j 8‹I % : 9<
65
(2.7)
X
X
<
V :
_
_
X
„ X % ,
, % ya , [8, page 2059]
b
a
V
X
;_T
X
(2.8)
_
%
,
, , p qs: =<t27 3 j 8
: =< p
65
z,{
For finding the polynomials_Tp‘
mials in (2.3). Putting ’ P0“ u qs:iw < 7 :Cw <
„ X % X
_
273 I % j t8 : =<
65
]X
T 2 7 3 j 8…: =<
65
(_T
X
(_d
9
ŠŽ2Š7 3 j [8 R % : 9<0a /
05
d 2 7 3 j 8…: =<
` 65
273 R % j 8€: =<Eya /
65
d9 2 7 3 8‰: =<
` 6j 5
;_d
X
V Š
27 3 j A8 R % : =<Eya /
65
, i.e., we are looking for a relation of type (2.2)
% : =<t27=3 R7 8AR8 : =< p 27=3 R7 L 8AR8 L : =<vba /
65 j
65 K$j K
ad we use (2.4) where u0z , { aT are the polynoy
we have
P “ R %
P “ I %
%
%
u :Cw < 7I :iw < u :iw < 7=R % :Cw <Ea /
P “ R %
P “ I %
Now we substitute the functions 7}I % :iw < and 7R % :iw < using the relations (A.2)
X P0“
X
P “ R %
’ Pv“ %
%

7
I
7 :Cw <
7}I :Cw <x
:Cw <
w
w
and (A.4)
P “
P “ I %
P “
7 :Cw <$
7=R % :iw <E =7 R % :iw <
respectively. This
leads to
u % :iw <
X
”
y
X
P “
P “
’ P “ %
7}I :iw <
uvq :iw <
u % :iw < u :Cw <‚• 7 :Cw < u :iw < =7 R % :Cw <Ea /
w
w
Comparing the last formula with the three-term recurrence relation TTRR (A.5) for the Laguerre polynomials we find
u q}:iw <x w
’ u % :Cw <E w u :Cw <x
X
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10
R. ÁLVAREZ-NODARSE, J. L. CARDOSO, AND N. R. QUINTERO
and therefore for the wave functions we find
– X
d927}3 I % j 8AR % : =<
65
(2.9)
—d_T
X
˜ :
d9^27 3 j 8 : =<
65
<?™273 R % j 8šI % : =<Eba /
65
Analogously, we can find the following three recurrence relations
„ X % X
,
_
% ,
8 : 9 <
p
q : =<t2 7 3 j 65
(2.10)
_
,
p % : 9 <?2 7 3 I % j 8‹I % : = <
65
p : =<?2 =7 3 R % j 8AR % : = <EbaT
5
0
where,
›
p qs: =<v
=X
:
: d9
<
p % : =<E : d9
X
:
X
<€< : T9
(_T
X
:
X
<
: T
<€< V X
X
(_T
X
(_T
X
X
X
p : =<E : d9
<V :
<
_
_
„ X % ya , X , % , a
ž
(2.12)
4_€
–
(2.11)
„ X % ya , X X
V :
X
9X
X
4_†
4_€
(_T
=X
<
Š
d9 œ ;_T
X
(_T
% : =<
j 8t: =<va /
_
%
, , 4_€
–
9 X
T9$27 3 j 8
65
4_€
4
_€
X
X
Ÿ
T 9o^2Š7 3 j 8[R %
V
05
4_€
X X X
2 7}3 I j 8[R %
<
V :
05 8 : =<
Tc^27 3 j 6
65
d9 2 7 3 j 8AR
5
6
4_€
X X
<:
<
273 R
65 X
d9
_T
X
_
: 9<
: 9<
: 9<vba /
/
+
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„ X % ON RECURRENCE RELATIONS FOR REDIAL WAVE FUNCTIONS
X
,
_
_
% ya , X
V
‡
X X :
X X
<
V :
11
,
<
4
_€
X
;
g X
_€
4
]
T X
`
2 7}3 I % j 8t: =<
65
d9$ŽT9ˆ027 3 j 8T: =<
65
4_€
4_€
X
2Š7=3 R % j 8AR % : =<Eya /
05
2.2. Ladder-type relations for the I.H.O. radial functions. Another important relations for the functions (2.1) are the so-called ladder operators. The following result was
proved in [8].
l¡m[n
radial
_dh_ functions of the -th
T HEOREM 2.2. Let 2 7 3 j 8‰: =< and 2 7=3 R7 K j 8AR8 K : =< be two
e
X
X %
X %
5 let
%
_dimensional isotropic harmonic
_ 65 oscillator 6and
:
<
a and : < X
: % < ¡b
¢ a , where % and % are integers. Then, there exist three not all vanishing polynomials
in , p…q , p % , and p , such that
(2.13)
p
q 2 7 3 j 8…: =<
65
p %£ 2 7 3 j 8‰: =<
5
£ 6
p 2 7=3 R7 K j A8 R8 K : =<Eya /
5
6
The proof of this theorem can be found in [8]. A key point on the proof was the recurrence
relation,
P 8ARŽS L I %
P 8ARŽS L
P 8AR8 R…S L I %
C: w < ¤ % :Cw < 7}I % :Cw < ¤ :Cw < 73 R7 K K
:iw <xyad
¤Žq :Cw < 7
5
where ¤Žq , ¤ % , and ¤
are not all three vanishing polynomials. Moreover, (2.13) can be
rewritten as
_
›
¤ % :iw < £
¤
q :iw <?< ¤ % :Cw < œ 2 793 8;: =<
d : ¤ % :Cw <
£
05
(2.15)
% I™8
F7 3 j 8
65 K 27=3 R7 Kkj 8AR8 K : =< /
¤ :Cw <
65
F 7=3 R7 K j 8AR8 K
65
To obtain the unknown polynomials ¤‘q , ¤ % , and ¤ in (2.14) we can proceed as in the
(2.14)
previous section, i.e., use the Eqs. (A.1)–(A.5) to transform (2.14) into one of the formulas
(A.1)–(A.5) or in a sum of linearly independent Laguerre polynomials and solve the resulting
equations for the unknown coefficients. Let us consider some examples. The first two are
taken from [8] and the other two are new.
_
„ X % ya , % [8, page 2061]
_
‡
(
_T
X
£
(2.16)
T
ˆ‰2Š7 3 j 8‰: 9<0
27 3 j 8AR % : =< /
V 05
65
£
_
„ X % , % ya [8, page 2061]
‡ 4_€
4_†
9X
X X
£
:
7
8
:
:
(2.17)
ˆ
3
2 73 R % j t8 : = < /
T
< 2 j = <E V
<
65
5
6
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12
R. ÁLVAREZ-NODARSE, J. L. CARDOSO, AND N. R. QUINTERO
_
„ X % , % ba . Introducing these values into (2.14) and putting ’ X
Pv“
P “ R %
P0“
%
%
(2.18)
¤ qJ:iw < 7 :Cw < ¤ :Cw < 7I :iw < ¤ :iw < 7=R :iw <E
a /
we get
Now, using relations (A.2) and the TTRR (A.5), (2.18) becomes into
¤ % :Cw <
X
X
X
P0“ R %
v
P
“
’
’
%
%
7I :Cw <
¤Žq :iw <
¤ i: w <
X
¤ :iw < 7 i: w <
w
w
g
9 X
w Pv“ %
’ 7R C: w <va /
¤ :Cw <
X
Comparing with the TTRR (A.5) we find
X
¤ qJ:Cw <x :
¤ % :Cw <x w
g X
<
=X
—
w g X
¤ :Cw <x :
X
w
<:
X
=X
w D
<$
and therefore, for the wave functions we find (see (2.15))
‡=
g X
¥ +_$
¥ X
X X
£
d9 T 9 :
<:
< ˆ 27 3 j 8‰: =<
65
£ 4
_€
4_€
X X X
X
<:
<
Š
2Š7=3 R j 8 : 9< /
Ž
V :
05 _
„ X % |a , % Following the same technique and using now twice, in the resulting (2.14),
formulas (A.3) and (A.2) we get
‡=
X
+_ g X
d T T …ˆ627 3 j 8‰: =<
65
_€
4
4€_ X
X
T V
2 7 3 j 8[R : =< /
05 £
£ 2.3. Numerical analysis of the recurrences. In this section we will present the numerical analysis of some of the recurrence relations and ladder-type operators for the I.H.O. We
_
have used the commercial program M ATLAB [12].
X
First of all, let us point out that if we choose the values of and large enough and
compute the radial wave functions 2 7 3 j 8…: =< using the definition (2.1) we observe a picture of
65
an unbounded function that do not corresponds
to the real ones. This picture is probably due
to numerical instabilities and consequently, we can not directly evaluate these functions for a
_ by means of (2.1). In figure 2.1 we represent the maximum value of the parameters
given
X
, for which the direct computation works, i.e., the region below the curve defined by these
points is the region of computational_ validity of (2.1). Therefore, if we want to have the
X
values of the radial functions for large enough, it is necessary to apply the recurrence
relations (RR) in order to calculate them.
From the previous sections it follows that the recurrence relations of the I.H.O. can be
_
_
_
classify in three different types:
X
X¨§ _
X
„ RRs_ which involve functions
_
_
_
=
7
¦
[
8
¦
:
:
:
with
indices
,
(see
(2.5));
2
<
<
X § X
XŽj © § X
XŽ§ © < ,
: < (see (2.6)); : < and :
< (see (2.9)); and : < and :
<
(see (2.10)). We will call them the regular recurrences.
ETNA
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ON RECURRENCE RELATIONS FOR REDIAL WAVE FUNCTIONS
13
60
lmax
40
20
0
20
25
30
35
40
n
F IG . 2.1. We plot the maximum value of ª versus « , for which the radial wave functions can be calculated by
using (2.1).
_
„ The_ other _}RRs,
e.g. the
ones
_ which
_ involve
_} the indices
_ : X _< , : X
X
X X
X
X
X
X
X
: <,: <,:
<;: <,: <,:
<;: <,:
etc. (see e.g. (2.7), (2.8), (2.11), (2.12)).
_¬
_
X _
s
_
<,: X <;
<,:
<;
„ Relations which involve the derivatives, i.e., ladder-type relations.
In this section we evaluate the radial wave functions by using different recurrence relations
with the aim to discuss their effectiveness (based on the time of the numerical simulations
and convergence of the formulas). In all the numerical computations of 2 7 j 8 : 9< we use 
g
and .
4
time (s)
3
2
1
0
0
50
100
150
n
F IG . 2.2. We represent with pluses, stars and circles the computational time (in seconds) to obtain the radial
function versus « from the RRs (2.5), (2.6), and (2.10), respectively. In this case ª­« and each function have been
evaluated at ®?¯$¯$¯ points.
¥ the¥ function 2 ¥ 7 j 8 : =< in aaJa points, corresponding
Ӵ
¥
First of all, we evaluate
to the ele
/ a +± a / a +±™ • ),
ments of the vector ¡ : / a / a c°°c° < (in the matlab notation _
by using the formulas (2.5), (2.6) and (2.10). In figure
2.2 we compare the computational
X
time to do these operations in each case when (notice that in this case we need to apply
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14
R. ÁLVAREZ-NODARSE, J. L. CARDOSO, AND N. R. QUINTERO
the corresponding RR
X
times).
30
time (s)
20
10
0
0
50
100
150
n
F IG . 2.3. Computational time versus
the circles are results from (2.11)+(2.5).
«
for
ª­²Q¯
. The pluses refer to the recurrence relation (2.5), whereas
We observe that formula (2.5) allows us to compute these functions faster. The difference
X
_ We would like to remark
in time among these methods is more appreciable when is bigger.
7 8
that in order to apply
X the formula (2.5) to obtain 2 j : =< , we fix and change the first argument from to
. In addition, we need to know the initial conditions of the RR, which in
%
_
this
P 8[R caseI % are 2 q j 8 : =< and
P 8A2R j 8 : 9I < % . Both functions are related with the Laguerre polynomials
%
:
:
and
,
respectively,
regardless
of
the
value
of
. Further0
<
‰
<
q
Z³ for using the relationZ(2.6)
³
7
s
q
:
more,
one
should
use
the
initial
conditions
and
2
=
<
2 7 j % : =< ,
j
)
)
)
X
but they can not be calculated by formula (2.1) for large values of (see figure 2.1) as already
pointed out. So, the initial conditions should be calculated using the recurrence formula (2.5).
Moreover, when we use the relation (2.6) we see that there exists an interval close to ´Wa
where 2 7 j 8 : 9< diverges (oscillates without any pattern) and this interval becomes larger when
X
increases. This divergence, at the vicinity of zero, also appears when we use (2.10). In the
last case, one should also use (2.5) to obtain the initial condition.
The recurrences (2.7), (2.8), (2.11) and (2.12) are even more complicated to use than (2.6)
and (2.10). They should be used together with (2.5), not only because the initial conditions,
but also because their own nature (they mix both indices). Let us check if 2 7 j 8 : 9< can be
_ (2.5)
¥ alone.
obtained faster when we combine for example (2.11) with (2.5), instead of using
X
In figure 2.3 we show the computer time to obtain 2 7 j 8 : =< versus (and fixing a ) for the
” Both functions are computed in
relation
(2.5)
alone,
and
combining
(2.11)
and
(2.5)
together.
4± / ¡±ƒ
aJaa points corresponding to the elements of the vector a / a
a a
a9• . From this figure
we conclude that the formula (2.5) is the best one for computing 2 7 j 8 : =< .
Concerning the ladder-type relations, let us_ mention that the relations (2.16) and (2.17)
X
behave numerically unstable for large values of and , respectively. Therefore, they are not
useful to compute numerically the radial wave functions but instead of this we can use them,
together with (2.5) for finding the derivative of the radial wave functions (see figure 2.4).
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ON RECURRENCE RELATIONS FOR REDIAL WAVE FUNCTIONS
15
0.15
R40,50(r)
0.05
−0.05
−0.15
0
6
12
18
r
1.0
dR40,50/dr (r)
0.5
0.0
−0.5
−1.0
0
6
12
18
r
F IG . 2.4. In the top panel we show the µ&¶‚·†¸ ¹º·»o¼Q½ computed by using the recurrence relation (2.5). In the
bottom panel we represent the derivative of this function computed from the ladder-type relation (2.16).
3. Radial functions for the Hydrogen atom. In this section we will provide a similar
study for the Hydrogen atom described by the Schrödinger equation
¾Œ
!
. ,- ! "/
¿
*
+
- "$#&% "$#&%(' "
)
'™)
The solution is given by |y24793 8;: =<t> 8A@ :CB < , where the radial part 24793 8D: =< is defined by
65
65
[2, 9]
8 ÄQŐÆ
P 8AR I
À ]X
7} I™8‹I % À X
(3.1)
2 7=3 8;: =<E(F 7 3 j 8ÁÀ X
IÂ
IÂ
IÂ /
6
5
6
5
_Ÿ
_&
_ Ã
à g
|`TÃ
hÉ
Èe
X
/c//
Here and Çad //c/ are the quantum numbers,
is the
dimension of the space, and the normalizing constant F 7 3 j 8 is
65
Ê_¬
Y
X _T
: (
<
g Y ]X F 73 j8 V X
I™Â /
:
<
05
Ë`
As we already pointed out in [8], the Laguerre
P “ polynomials that appear in the expression
of the radial functions are not the classical ones 7 : < in the sense that the parameter ’ as
)
|b
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16
R. ÁLVAREZ-NODARSE, J. L. CARDOSO, AND N. R. QUINTERO
X
well as the variable depend on the degree of the polynomials, . When the parameters of the
X
classical polynomials
) depend on , the polynomials are orthogonal with respect to a variant
weights (for more details see e.g. [10, 11, 18]). Nevertheless, as we showed in [8], using
the algebraic properties of the classical Laguerre polynomials (A.1)–(A.5) one can derive the
algebraic relations of the radial wave functions.
3.1. Recurrence relations and ladder-type operators for the radial functions. In [8]
we proved the following
IOÂ
IOÂ
]X
]X X %
T HEOREM 3.1. Let the functions 2 793 8Ì
ÎÍ , 2 7‚3 R97 K j 8 R8 K Ì
ÎÍ and
05
05
`
_ ™I  OÍ be three different
radial functions of
É
mon ` such that
integers
l4
_T;_ _T(_
e /
X
X % X
X
% :
< a
Then, there exist not all three vanishing polynomials in , p‘q , p % , and p
‡= ‡= g
X
X
X %
p…q : =<t2 793 8
Ï ˆ p % : 9<?2 73 R7 K j 8AR8 K
65
65
‡= X
X
p : 9<?273 R7 L j 8AR8 L
65
]X
X _
2 7=3 R7 L j 8AR8 X L Ì X
65 and % and %
atom
the
`
-th Hydrogen
, such
that
Ð
Š
Ï ˆ
ˆÑa /
Ï
Moreover, the expressions for the polynomial coefficients in the above formula are given by
[8, Eq. (4.7) page 2063]
I % 8 R8 L
I % 8L
p…q : =<E p q : =<E~†F 7 3 j 8
K p % : =<0 p % : 9<v~†F 7=3 Rƒ7 K j 8[R8 K 
65
65
I % 8
p : =<E p : =< ~ F 7=3 R7 L j 8AR8 L 
K 65
where prq , pr% , and pr , are the
three vanishing
of the linear relation
non all
polynomials
Pv “
P “ R 8
P “ R 8L
p _¬% : 9< @6R 7 KK I8 K : =< p : 9< @6R 7 L I8 L : =<Eyad
_d p q : 9< @ : =< Ó
X
and ’ ,Ò .
As examples of recurrences obtained by using the Theorem 3.1 we have the following
(the first two are taken from [8], the other five are seem to be new). Other cases can be
obtained in the same
_ way. _
%
„ X % ,X , ÊÐ
ÊÐ
ya [8, page 2063]
Ó_¬
_T
;
=X
‡ X
g
X
X
g 27}3 I % j 8
˜ :
<:
< =X
ˆ
65
‡= g
g
9X
X
:
=<?2 7 3 j 8
(3.2)
ˆ
05 Ê_
h_d
9X
‡= X
X
X
g 2 7=3 R % j 8
˜ :
<:
< 9X
ˆ ba /
05
_
_
„ X % X ya , % , 2064]
+_ [8,4page
_€
_†
_d
g
X
X
%
:
<c˜ :
<:
<s927 3 j 8‹I : =< :
=< Ô
6
5
‡
_
_T
g
g
g 9X
X
:
(3.3)
<t :
<:
<
‰‰
ˆ 2Š7 3 j 8‰: 9<
05
_T
+_$
4_€
X
g
X
:
<c˜ :
<:
<O92Š7 3 j 8[R % : =<Eba /
05
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17
ON RECURRENCE RELATIONS FOR REDIAL WAVE FUNCTIONS
_
_
„ X % ,X , % , a
‡= – X Ê_ X
%
:
=<tx2 7=3 R j 8
ˆ
5
0
– ;_T
_T
9X
‡9 g
g X
X
Ÿ2 7 3 j 8
:
< 9X
ˆ
65 _T
_T
– X ;_T
ÌH:
Ô
< :
9< :
=<ºÍ
‡= X
ÔÕ27=3 R % j 8AR %
ˆ baT
6
5
_
_
X
„ X % ,
, % , Ö=×HØقڑÛ^Ürݏ܉Þßdàá×HØقڑÛ^܅â߆×HØcقڑÛ^Ü
ØOßkÚ
ÞJ׋ÞJÚrØßiãtÚrØQÞJ׋äå܅Ù?ÜrÝJÜ6Þßtævç
ç¡è
éAêŸë äðÚ ÛñØ Üòâ ó ÞQô
ì ¸ íîï
ڑÛ^Ü
ö
ÜÊõ ׋äå܅ـÜ
Ø߆׋äå܅ـÜrÝ$ß ØcØcäƒ
×HØcقڑÛø܉âd܉ÞOßCÞè éAê&ë
äƒÚ Û^Ø Ü
ö Þ ô
ƒ
ä
‘
Ú
^
Û
Ü
â
v
ó
÷
ó
=
ì
™
ù
ú
ú
¸ íáû îï
ï
Ü õ ׋äå܅ً߆׋äå܅ٺڊÝQß ØQäƒÚ‘ÛøÜrÝ
×HØقڑÛ^Ürݏ܉ÞßCÞè éAêŸë
ärÚ Û^ÜrÝ Þ ôü4ý=þ
Ø ó
ì û ú ¸ í ù™ú îï
ï ØQäƒÚ‘Ûø܅â ó ÷
_
_
X
%
„ X % ,
, , DÐ
‡9 ‡= g
X
X
p…q : =<t27 3 j 8
ˆ p % : =<27}3 I % j 8‹I %
ˆ
65 6
‡ 5
(3.4)
X
p : =<t27=3 R % j 8AR %
ˆ¨baT
65
where,
p qs: =<E :
_†
4_†
X
p % : =<E ˜ :
X
(_d
”
=<
ð¥
<:
X
:
_€
< X =+<=_$O • :
= <‚9
ŠÐ
T_ 4_€
9 X
g
g
:
< 9 X
=<‚9
_†
;_T
9 X
g
X
:
g :
< 9 X
=<‚ /
g
<:
_€
<
+_$
X
: :
p : =<Eñ˜ :
<
_
_
„ X % ya , X , % , a
=‡ =‡ g
X
X
%
p qJ: =<?2Š7 3 j 8
Oˆ p : =<27 3 j 8AR %
05
65
‡9 (3.5)
X
p : =<t2 7=3 R j 8
65 where,
_†
” _€
– X 4
:
p q}: =<E
Ê
¬
_
– X
9X
%
:
p : =<E
_d
Ó_
X
p : =<E :
Q< ˜ :
<
:
X
< :
X
4_€
=<t9
Ê_T
<:
X
(_T
9<
:
=X
g
Oˆ
ˆ aT
9 X
< 9X
=<‚O•
g /
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18
R. ÁLVAREZ-NODARSE, J. L. CARDOSO, AND N. R. QUINTERO
_
_
„ X % ,X , % b
, a
=‡ ‡= g
X
X
%
p…q : =<t2 7 3 j 8
ˆ p : =<t2 7=3 R j 8
65
65 9‡ (3.6)
X
p : =<t27=3 R j 8AR %
65 where,
p qJ: 9<0
:
_€
X
<Q˜ :
Ó_T ” _T
–
X
: p % : <0
;_T
–
X
=X
:
p : ) 9<0
+_
<:
X
4_€
X
< :
Ê_¬
=<‚ /
<:
=<
X
4_†
:
9X
ˆ
g
ˆÑaT
9 X
< 9X
g
=<‚O•º
3.2. Ladder-type relations for the radial functions of the Hydrogen atom. Our start
ing point is the following
I™Â
I¬Â
]X
]X
X %
T HEOREM 3.2. Let 247=3 8 Ì
Í , and 27=3 Rƒ7 K j 8[R8 K Ì
Í two dif-
65
05
É`
l4mon `
I¬Â
]X
_ and ÿ N 2793 8 Ì
derivative
_T;_
ferent radial functions of the Hydrogen atom
Í , the X first
X %
X %
ÿ
6
5

`
%<
%
_
:
with
respect to , where
and
are integers such that
e
X %
%
:
:
a,
<
< f
¢ a . Then, there exist not all three vanishing polynomials in , p‘q ,
p % and p , such that
(3.7)
p
q : =<?2 97 3 8
05
‡=
X
g
ˆ p % : 9< £ 2
‡=£ X
p : =<t2 7=3 R7 K j 8AR8 K
65
7=3 8
65 X %
‡=
X
g
g
ˆ
ˆ ya /
The proof is similar to the one of Theorem 2.2 and was given in [8]. In fact in this case the
relation (3.7) becomes into (see [8, (4.13)
page 2064])
_
‡= g
X
¤ qJ: =<?^2793 8

Oˆ
65
‡9 (3.8)
g
F 73 j8
X
X
%
05 27=3 R7 8AR8
¤ : =<
ˆ F7=3 Rƒ7 K†j 8AR8 K 65 K j K
65
where the polynomials ¤ q , ¤ % , and ¤ are such that
P “ R %
P “
P “ R 8
%
%
¤Žq : < @ : < ¤ : < @ŽI : < ¤ : < @6R 7 KK I8 K : <xyad
(3.9)
_
Ê_¬
X
I™Â
X
and b
,’ ,Ò .
above theorem it is easy to obtain several relations for the
_ radial wave
Again, from the
X
8K
¤ % : 9< £
£ ¤ % : =<s~
functions of the Hydrogen atom and, in particular, the ladder operators in and , respectively.
We will present here some of them. The first three are taken from [8] and the other two are
seem to be new.
_
„ X % ya , % [8, page 2065]
_
‡= _T
£
ˆ 2793 8;: =<E
65
£ _T (3.10)
: <
g 2 7 3 j 8AR % : =< /
V
X
:
<
65
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_
19
ON RECURRENCE RELATIONS FOR REDIAL WAVE FUNCTIONS
„ X % , % [8, page 2065]
_
‡ _€
4_†
‡9 g
X
X
:
9
7
8
=< £
¬ : ‚
<
å
ˆ
2
3
ˆÑ
£ 4_€ 65
4_€
X
(3.11) X
: <
I %
]X
X
g ¬ 27=3 R % j 8AR % Ì
<:
< X
˜ :
:
<
65
Ë`
_
„ X % , % ya [8, page 2066]
‡ X
£
g
X
:
<‚ˆZ2 793 8;: 9<v £ +_ : 4_€ : < < 65 ‡=
(3.12)
X X :
: <
X
X
<
g ¬ 2 7=3 R % j 8
g ¬ ˆ
˜ :
<:
< X
X
:
:
<
< _å
65
_
„ X % Êa _¬, % . Introducing these values into (3.9) and putting ’ X
Ò and 7=R NSL we get
Pv“
P “ R %
P “ R
%
%
(3.13)
¤ qs: < @ : < ¤ : < @ŽI : < ¤ : < @ŽI : <xba /
Now, using three times relations (A.4) and (A.2),
(3.13) becomes
into
P “
‡
R
:Ò
’
’
< :Ò
<
¤Žq : < ¤ % : < ¤ : <
ˆ @ŽI : <
‡
P “
R
:Ò
’
< :Ò
<
¤ q: < ¤ %
¤ : <
ˆ @ŽI % : <
‡
:Ò
<Ò P “ R
¤Žq : < ¤ : <
ˆ @ : <xya /
Comparing with the TTRR (A.5) we find
<
Í /
 ¤ : x
g < :’
< : ’ <
:’
< :’
<
g :’
’
< :’
< : Ò
<
%
¤ : <x
g
’
and therefore (see (3.8)), for the wave functions we find _
‡ _T
_
_T
( £
:
< : :
<:
<€<
+_$ £ +_Q
_€
_€
X
‡= g
X
X
:
< Àå:
<:
< X
ˆ
I œ 2 7 3 j 8
+_Q
+_$
4_€
_€ à 65
X
X
‡= X
X
g
˜ :
<:
< :
<:
<
X
7
[
8
R
Š
2
3
Oˆ /
Â
IÂ
]X
0j 5 _
`
„ X % , % ya
Ø äƒÚ Ûø܉â
äƒÚ Û^ÜrÝ ÜòÞ ô Þ Ü…Ù€Ü Þ Ú äƒÚ Û^܅â ׋äƒÚ‘ÙºÚ‘ÛøÜ
Øß ô
Ø*ó ï
Ø ó
Ø ï
Ø*ó
î ï
î Þ
Ü äƒÚ Û¾Ø Üâ
׋ä4Üòً߆׋ä+ڍÙsڍ۾܍Øß4è éAê&ë à ärÚ Û¾Ø Üòâ ÞQã ü
ó0÷
ó
ì ¸í ï
ï
äƒÚ ÛÚŠÝ õ ׋äå܅ً߆׋äå܅ٺڊÝQ߆׋äƒÚ‘ÙºÚ‘ÛrÜ
ØO߆׋äƒÚ‘ÙºÚ‘ÛøÜâßtè éAê&ë
äƒÚ ÛÕÚ+Ý Þ ô
f
Ø
v
ó
÷
Ø ó
ì
ï
û ¸ íŸî‹ï
÷
¤Žq : <x Ò
Ò :Ò
g
/
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20
R. ÁLVAREZ-NODARSE, J. L. CARDOSO, AND N. R. QUINTERO
3.3. Numerical_ analysis of the recurrences. As in the previous case, the radial wave
X
functions for and large enough, calculated by using the functions (3.1) (by means of the
Laguerre polynomials), behave as an unbounded function, and therefore we can not directly
X
evaluate_ these functions for a given . In figure 3.1 we show the critical values of @ for
a given for which the radial functions can be computed using the explicit expression (3.1).
_T formula works is the one defined by the two curves in figure
The region where the explicit
X X
@ ).
3.1, (it is defined the values
100
80
nmax
60
40
20
0
0
10
20
30
40
50
60
70
l
F IG . 3.1. We represent with solid line the function «ð­ªÑ® (this line represent the minimum value of « ) and
with the stars joint by the dashed line the maximum value of « for a given ª for which the radial wave functions can
be calculated using (3.1).
Again we have studied numerically how to obtain the radial wave functions applying the
recurrence relations (RR) of two types: the first ones are represented by the formulas (3.2),
(3.3) and (3.4) and the second ones correspond to the ladder-type RRs (3.10), (3.11) and
(3.12). For the former RRs we observe that
„ the relation (3.2) is the most useful one because the initial conditions 2 8AR % 8 and
j
by_Tusing
the Laguerre polynomials (notice that in this case
2 8AR j 8 _ can be calculated
X
_
we fix and start with ).
X
„ in the relation (3.3) we fix and increase the value of in each step up to its maxX
imum value. From figure 3.1 we conclude that for large value of , we can not use
the initial condition calculated by using Laguerre polynomials (formula (3.1)) and
instead of this we should use the RR (3.2). For this reason, this relation can not be
used alone (the same is true for the RR (3.4)). In addition, notice that the initial
conditions are two radial functions, 2 7 j 8‹I % and 2 7 j 8 evaluated at , whereas (3.2)
X
gives us the same functions but at (this implies a rescaling of the argument).
„ From figure 3.2 we observe that the computational time is less when we use the
_
formula (3.2). Moreover, the formulas (3.3) and (3.4) are numerically
unstable at
X
the vicinity of zero and this region becomes larger when and increases.
„ As in the case of the I.H.O. the higher order relation (3.5) and (3.6) are not useful
for numerical evaluation of the Radial functions.
For the ladder-type relations we would like to point out the following remarks
„ Notice that the expression (3.12) involves two
” wave
evaluated at different
functions
X
%
X • we need to know 2 7 j 8 : =< ,
points, so in each step to calculate 2 7=R j 8 :
‚
<
X X
but in the previous step we calculate 2 7 j 8 at :
< , not at . For this reason
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ON RECURRENCE RELATIONS FOR REDIAL WAVE FUNCTIONS
21
1.5
time (s)
1.0
0.5
0.0
−0.5
0
20
40
60
l
F IG . 3.2. We represent with pluses, stars and circles the computational time versus ª («‘­ªŠ® ) to compute
the radial wave functions by using (3.2), (3.3) and (3.4), respectively.
this RRs it is not useful for obtaining numerically the radial wave functions.
„ As in the previous case, we can use the ladder-type relations (3.10), (3.12) and (3.11)
together to (3.2), to compute the derivative of the radial wave function. As an example see the figure 3.3, where_ we Ð plot the radial wave function and its derivative by
X
using (3.10) for a and a.
In general we verify that the ladder-type RRs are not useful in order to compute numerically the radial wave functions. Nevertheless they can be used, together with (3.2) for finding
the derivative of the radial wave functions.
Programs. For the numerical simulations presented here we have used the commercial program M ATLAB. The used source code can be obtained by request via e-mail to
niurka@euler.us.es or ran@us.es.
Acknowledgements. The authors thank J. Petronilho for interesting discussions as well
as the unknown referee for his comments that help us to improve the paper. This research
has been supported by the Programa de Acciones Integradas Hispano-Lusas HP2002-065 &
E-6/03. The authors were also partially supported by the the DGES grant BFM 2003-06335C03-01 and PAI grant FQM-0262 (RAN); CMUC (JLC); and the DGES grant FIS2005-973
and PAI grant FQM-0207 (NRQ).
Appendix. The Laguerre polynomials. The Laguerre polynomials
hypergeometric series P0“
X
7 % % :’
:’
<
<7
7 : E
F
< X Y
Z
XZY
’
)± ) ! ±
!
"
: " < q :" < "¬:"
¬< °°c° :"
$< P “
7 defined by the
"
X
!7
: < "
! Ys
<" )
"$# q : ’
g c/ /c/¬/
These polynomials satisfy the following useful recurrence and differential-recurrence relations (see e.g. [1, 15, 17])
(A.1)
(A.2)
£
£
)
Pv“ R %
X
7 : E
< :
)
)
P “ %
P “
R
7 : 0
}7 I % : <$
< )
)
P “
Pv“
v
X
’
< 7 : < :
< =7 R % : <$
)
)
ETNA
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etna@mcs.kent.edu
22
R. ÁLVAREZ-NODARSE, J. L. CARDOSO, AND N. R. QUINTERO
1.5
−5
R60,50 (r) (x 10 )
1.0
0.5
0.0
−0.5
−1.0
0
4
8
3
12
16
r (x 10 )
5
−8
dR60,50/dr (r) (x 10 )
3
1
−1
−3
−5
0
4
8
3
12
16
r (x 10 )
F IG . 3.3. In the top panel we show the µ$#‚·†¸ ¹º·9»[¼Q½ computed by using the recurrence relation (3.2), whereas in
the bottom panel we represent the derivative of this function computed from the ladder-type relation (3.10).
(A.3)
(A.4)
(A.5)
Pv“
X
%
:
’ < 7}I : < :
)
Pv“
P0“
%
7}I %
: <E 7 : <
)
)
P0“
Pv“ %
X
= X
:
’
< 7R : < :
< 7 : <
)
)
)
Pv“ R %
7 : v
< )
)
P0“ I
7
X
Pv“
< 7 : Q< )
)
: Q< )
:
X
Pv“
’ < }7 I % : E
< ya /
)
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ETNA
Kent State University
etna@mcs.kent.edu
ON RECURRENCE RELATIONS FOR REDIAL WAVE FUNCTIONS
23
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