Communication and Network, 2010, 2, 62-64
doi:10.4236/cn.2010.21009 Published Online January 2010 (http://www.scirp.org/journal/cn)
On Solvable Potentials, Supersymmetry, and the
One-Dimensional Hydrogen Atom
R. P. Martínez-y-Romero1, H. N. Núñez-Yépez2, A. L. Salas-Brito3*
1Facultad de Ciencias, Universidad Nacional Autónoma de México, Apartado Postal, Coyoacán, México
2Departamento Física, Universidad Autónoma Metropolitana-Iztapalap a,
Apartado Postal, Iztapalapa, México
3Laboratorio de Sistemas Dinámicos, Departamento de Ciencias Básicas, Universidad Autónoma
Metr opolitana-Azcapotzalco, Apartado Postal, Coyoacán, México
Received November 20, 2009; accepted December 15, 2009
Abstract: The ways for improving on techniques for finding new solvable potentials based on supersym-
metry and shape invariance has been discussed by Morales et al. [1] In doing so they address the peculiar
system known as the one-dimensional hydrogen atom. In this paper we show that their remarks on such
problem are mistaken. We do this by explicitly constructing both the one-dimensional Coulomb potential and
the superpotential associated with the problem, objects whose existence are denied in the mentioned paper.
Keywords: one-dimensional hydrogen atom, one-dimensional Coulomb potential, supersymmetric quantum
mechanics.
A paper of Morales et. al. [1] has discussed the use of
supersymmetric and shape invariance techniques and of
Darboux and intertwining transformations, for building
new solvable potentials.
To illustrate these ideas they apply them to hydro-
gen-like potentials and to radial and one-dimensional
problems. They assert [page 23 of [1], in the paragraph
after Equation (39)] that the potential corresponding to a
one-dimensional hydrogen atom, i. e. a one-dimensional
Coulomb potential, is nonexistent. They further claim
that there is no superpotential associated with the -1/|x|
potential energy term [page 22 of [1], in the paragraph
just before their Equation (36)]. In this letter we want to
challenge these two affirmations. Throughout this work
we use atomic units qe = ~ = m = 1. In this paper we want
to discuss their results concerning such 1D problem.
We recognize from the start that the potential deserv-
ing the name one-dimensional Coulomb potential is not
the one usually alluded to in the literature—i. e. it is not
-1/|x|. The true Coulomb potential in one dimension must
be the solution of the corresponding Poisson equation
214()DC x
  (1)
where δ(x) is a Dirac delta function which is really not a
function but a distribution also termed a generalized
function [2]. As it is very easy to realize, just solving
Equation (3), the 1D Coulomb potential definitively exist
and is given by
12DC x

  (2)
so the potential energy function needed in the Schr-
ödinger equation should be
1() 2 DC
Vx x

 (3)
In this sense the one-dimensional Coulomb potential
does indeed exist. However, V1DC (x) is not the potential
energy usually referred to as the one-dimensional hydro-
gen atom potential. But even if Morales et. al. are refer-
ring to this potential, namely V1DH = -1/|x|, corresponding
to a Hamiltonian
2
12
11
2
DC d
H
x
dx

 (4)
the existence of a superpotential is beyond doubt, as we
intend to exhibit in this work, see also [3,4]. The result
[Equation (36) in [1]] they base their argument on the
nonexistence of a superpotential for the one-dimensional
hydrogen atom cannot be right since it does not have any
explicit r-dependence. Even though this problem is
surely just a misprint, the limit l → 0 has no meaning for
discrediting Hamiltonian (4) because the problem really
comes from the need to describe Coulomb systems con-
strained to one-dimensional motions with no spherical
symmetry and hence described by states with no well
defined angular momentum.

R. P. Y. ROMER ET AL. 63
Any system described by Hamiltonian (4) is one with
baffling properties [5–7]. Its properties are so peculiar
that people is prone to express erroneous concepts about
it. For example, it has been claimed that the potntial en-
ergy term in Hamiltonian (4) is its own supersymmetric
partner [8], or, as in [1], that the Hamiltonian itself can-
not really be written since its potential energy function
does not exist. On the other hand, it has been proven that
it violateswhich the nondegeneracy theorem for one-
dimensional quantum problems [5], and it has been
shown that a superselection rule, analogous to the one
preventing the so-called paradox of optical isomers of
quantum chemistry, operates in the system [6,9–13]; see
also [14,15] for other similar points of view. The Hamil-
tonian (4) is not in general self-adjoint (in conventional
physics parlance, is not Hermitian). Self-adjoint 4 pa-
rameter extensions have been derived in [16], such ex-
tension admits Hamiltonian (4) as one of its members
[7,16–19]. Let us emphasize that Hamiltonian HD to-
gether with the matching condition φ(x)|x=0=0 is self-
adjoint.
We think the misconception in the Morales et. al. pa-
per could have arisen from their ideas on how the one-
dimensional hydrogen atom problem come to be. As they
say that, according to certain authors [1], its equation
arises from the radial Schrödinger equation of the (3D)
hydrogen atom merely by substituting r by x and a van-
ishing angular momentum l = 0. Given such assertion,
we assume that they think the 1D hydrogen atom is a
purely formal problem with little or no relation to any
actual systems. This, however, is not so. There are spe-
cific problems which lead to essentially one-dimensional
quantum motions which may be described by Hamilto-
nian (4). Examples of such problems are an hydrogen
atom placed in a constant but super-strong magnetic field
B [20–22], or the problem of the motion of an electron
sitting on a surface producing an image charge as hap-
pens to electrons over a pool of liquid helium. In this last
case, given the charge and its image is hence clear that
the electron is acted by a Coulomb interaction [23]. In
the case of the hydrogen atom within a B field, any elec-
tron state may be expressed as a product of transverse
Landau states times a state depending on a coordinate
parallel to B — states with no spherical symmetry [21].
The motion tranverse to the magnetic field is classically
restrained to distances of the order of ρc = ( c/B)1/2. In the
quantum case ρc may be called the mean size of the
Landau states. So, as the intensity of the magnetic field is
increased, ρc → 0 leaving only the motion along B for a
dynamical description [20]. When the (x-pointing) mag-
netic field is super-strong the potential felt by the elec-
tron can be approximated as
22
11
() lim
Bc
Vr
x
x 

 
 (5)
This is the potential used in Equation (4). Hence the
name one-dimensional hydrogen atom is justified: it is
just an hydrogen atom constrained to move in one direc-
tion and under the assumption that any transverse mo-
tions can be disregarded for field strenghts B ~109 Gauss
typical of neutron stars [24] they are certainly very small.
It is worth noting that an hydrogen atom in a magnetic
field has two integrable cases: 1) when B=0, and, 2)
when B=∞.
As we have shown previously [3–6], the two eigen-
functions describing the ground state of the one dimen-
sional hydrogen atom are
1
0
0
2(2)exp() 0
() 00
xLxxif x
xif x





 (6)
and
01
0
00
() 2(2)exp() 0
if x
xxLxxif x








 (7)
where the L1
0 (x) are generalized Laguerre polynomials
[17]. Notice the vanishing of the eigenfunctions at x = 0
and the explicit separation between the x > 0 and the x <
0 regions. This is one of the manifestations of the su-
perselection rule which, among other things, prohibits
any superposition of the right ψ0
+ with the left ψ0 - eigen-
states. The energy eigenstates of the problem are given
by a Balmer-like formula [4,13,25,26] En = -1/2 n2,
n=1,2,3, …, so the ground state energy is E1 = -1/2.
With the ground eigenstates given above, the superpo-
tential can be easily calculated as [3,27,28]
'
0
'
0
() 1
() sgn()
()
x
Wx x
x
x



 (8)
where sgn(x) is the signum function and we have in-
cluded in a single formula the consequences of both the
right and the left eigenfunctions. Using the superpoten-
tial, the corresponding partner potentials are readily
evaluated
2
11 111
(), ()
22
Vxand Vx
xx
x

  (9)
where, clearly, V- is the one-dimensional hydrogen atom
potential, V1DH, but shifted so that its ground state energy
is zero, and V+ is the partner potential. Also, the raising
and lowering operators are
d
A
W
dx


 (10)
and
d
A
W
dx

 (11)

R. P. Y. ROMER ET AL. 64
where, as it is easy to show,
[,] 2
dW
AA dx
 (12)
and
2
2VV W

 (13)
The results (8) to (13) establish that the one dimen-
sional potential V1DH can be regarded as stemming
from the superpotential W(x) in Equation (8). In [4,29],
we have discussed a complete supersymmetric extension
of the one-dimensional hydrogen atom problem, with
Hamiltonian
2
22 2
11111
22
22
s
usy z
Hx
xx x


 
 (14)
where σz is a standard Pauli matrix which is needed to
operate on both the the fermionic and bosonic sectors of
the system. But, as the motivations of [29] were the
similarities between light-cone singularities in quantum
field theory with the singularity in (4), the results in [29]
are not all related to the present discussion. Second, that
Morales et al. have mistaken the paper they cite (refer-
ence [21] in their paper, reference [30] in this work) for
other of our papers dealing with the one-dimensional
hydrogen atom, since [30] has nothing to do with the
problem at hand. It deals with a solvable model in rela-
tivistic quantum mechanics, the Dirac oscillator, which at
the time was thought to have applications in QCD. They
should have cited [3,5,29] instead.
Acknowledgements
We acknowledge with thanks the comments and sugges-
tions of P. M. Schwartz, P. N. Zeus, P. M. Mec, and G. R.
Maya.
REFERENCES
[1] J. Morales, J. J. Peña, J. L. López-Bonilla, and J. Mol.
Struct. (Teochem), Vol. 621, pp. 19, 2003.
[2] M. J. Lighthill, “Fourier analysis and generalized func-
tions, cambridge university press,” Cambridge, 1975.
[3] R. P. Martínez-Romero, C. A. Vargas, A. L. Salas-Brito,
and H. N. Núñez-Yépez, Rev. Mex. Fis., Vol. 35, pp. 617,
1989.
[4] B. Jaramillo, R. P. Martínez-y-Romero, H. N. Núñez-
Yépez,and A. L. Salas-Brito, Phys. Lett. A, Vol. 374, pp.
150, 2009.
[5] H. N. Núñez-Yépez, C. A. Vargas, A. L. Salas-Brito, Eur.
J. Phys., Vol. 8, pp. 189, 1987.
[6] H. N. Núñez-Yépez, C. A. Vargas, A. L. Salas-Brito, J.
Phys. A, Math. Gen., Vol. 21, pp. L651, 1988.
[7] I. Tsutsui, T. Fülöp, and T. Cheon, J. Phys. A: Math. Gen.
Vol. 36, pp. 275, 2003.
[8] T. D. Imbo and U. P. Sukhatme, Phys. Rev. Lett., Vol. 54,
pp. 2184, 1985.
[9] H. N. Núñez-Yépez, C. A. Vargas, A. L. Salas-Brito, Phys.
Rev. A, Vol. 39, pp. 4307, 1989.
[10] P. Pfeifer, “Dissertation, Eidgenössiche Technisch Hoch-
schule,” Zürich, 1980.
[11] P. Pfeifer, in J. Hinze ed. Energy Storage and Redistribu-
tion in Molecules, Plenum Press, New York, pp. 315,
1983.
[12] R. P. Martínez-Romero, H. N. Núñez-Yépez, A. L.
Salas-Brito, “A simple introduction to superselection
rules in nonrelativistic quantum mechanics can be found
in C. Cisneros,” Eur. J. Phys. Vol. 19, pp. 237, 1998.
[13] L. J. Boya, M. Kmiecik, A. Bohm, Phys. Rev. A., Vol. 37,
pp. 3567, 1988.
[14] R. G. Newton, J. Phys. A: Math. Gen., Vol. 27, pp. 4717,
1994.
[15] U. Oseguera, M. de Llano, and J. Math. Phys. Vol. 43, pp.
4575, 1993.
[16] W. Fischer, H. Leschke, P. Muller, and J. Math. Phys., Vol.
36, pp. 2313, 1995.
[17] M. M. Nieto, Phys. Rev. A, Vol. 61, pp. 034901, 2000.
[18] C. R. de Oliveira and A. A. Verria, Annals of Physics Vol.
324, pp. 251, 2009.
[19] S. Nouri, Phys. Rev. A, Vol. 65, pp. 062108, 2002.
[20] R. P. Martínez-Romero, H. N. Núñez-Yépez, A. L.
Salas-Brito, and C. A. Vargas, “Actas de la 3a. Reunión
Latinoamericana de Colisiones Atómicas, Moleculares y
Electrónicas,” CNEA, Bariloche, Argentina, 1989.
[21] W. Rössner, G. Wunner, H. Herold, and H. Ruder, J. Phys.
B: At. Mol. Opt. Phys., Vol. 17, pp. 29, 1984.
[22] W. Edelstein and H. N. Spector, Phys. Rev. B, Vol. 39, pp.
7697, 1989.
[23] M. W. Cole and M. H. Cohen, Phys. Rev. Lett. Vol. 23 pp.
1238, 1969. M. W. Cole, Phys. Rev. B, Vol. 2, pp. 4239.
1970.
[24] H. Ruder, G. Wunner, H. Herold, and F. Geyer, “Atoms in
strong magnetic fields,” Springer Verlag, Berlin, 1994.
[25] S. H. Patil, Phys. Rev. A, Vol. 64, pp. 064902, 2001.
[26] G. Abramovici and Y. Avishai, J. Phys. A: Math. Theory,
Vol. 42, pp. 28532, 2009.
[27] F. Cooper and J. G. Ginocchio, A. Khare, Phys. Rev. D,
Vol. 36, pp. 2458, 1987.
[28] R. Montemayor and L. D. Salem, Phys. Rev. A, Vol. 40,
pp. 2170, 1989.
[29] R. P. Martínez-Romero and H. N. Núñez-Yépez, A. L.
Salas-Brito, Phys. Lett. A, Vol. 142, pp. 318, 1989.
[30] J. Benítez, R. P. Martínez-y-Romero, H. N. Núñez-Yépez,
and A. L. Salas-Brito, Phys. Rev. Lett. Vol. 64, pp. 1643.
1990.