Communication and Network, 2010, 2, 62-64
doi:10.4236/cn.2010.21009 Published Online January 2010 (
Copyright © 2010 SciRes 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
E-mail: asb@correo.azc.uam. mx, nyhn@xanum
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
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
Vx x
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
DC d
 (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.
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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-
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
() lim
 
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
2(2)exp() 0
() 00
xLxxif x
xif x
() 2(2)exp() 0
if x
xxLxxif x
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]
() 1
() sgn()
Wx 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
11 111
(), ()
Vxand Vx
  (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
 (10)
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where, as it is easy to show,
[,] 2
AA dx
 (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
22 2
usy z
xx x
 
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.
We acknowledge with thanks the comments and sugges-
tions of P. M. Schwartz, P. N. Zeus, P. M. Mec, and G. R.
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