Factorization method and new potentials with the oscillator spectrum
Bogdan Mielnik
Citation: J. Math. Phys. 25, 3387 (1984); doi: 10.1063/1.526108
View online: http://dx.doi.org/10.1063/1.526108
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Factorization method and new potentials with the oscillator spectrum
Bogdan Mielnika )
Departamento de Flsica, Centro de Investigaci6n y de Estudios Avanzados dell P. N., 07000 Mexico D. F.,
Mexico
(Received 2 March 1984; accepted for publication 15 June 1984)
A one-parameter family of potentials in one dimension is constructed with the energy spectrum
coinciding with that of the harmonic oscillator. This is a new derivation of a class of potentials
previously obtained by Abraham and Moses with the help of the Gelfand-Levitan formalism.
PACS numbers: 02.30.Em, 03.65. - w
I. INTRODUCTION
Contrasted with general relativity, where new solutions
are found every now and then, the class of exactly soluble
problems of quantum mechanics (QM) has not greatly expanded. The two most commonly used exact methods to
determine the spectra in QM are the method of the orthogonal polynomials and the algebraic method of "factorization." The potentials for which exact solutions exist form a
rather narrow family, including the elastic and Coulomb potentials (modified by the 1/r 2 terms), the Morse potential,
the square potential wells, and a few others, and the common
opinion is that this is everything exactly soluble in Schrodinger's quantum mechanics. Hence, it might be of interest
to notice that in some occasions the "factorization" method
seems not yet completely explored. In particular, it allows
the construction of a class of potentials in one dimension,
which have the oscillator spectrum, but which are different
from the potential of the harmonic oscillator. This class has
been previously derived using the Gelfand-Levitan formalism.
These relations allow the construction of the eigenvectors and eigenvalues of H. If 1/1 is an eigenvector of H
(H1/I = ,1,1/1) the functions a*1/I and a1/l [provided that they are
nonzero and belong to L 2(R )] are new eigenvectors corresponding to the eigenvalues A + 1 and A - 1, respectively:
H(a*1/I) = a*(H + 1)1/1 = (A
= a(H -
H(a1/l)
+ l)a*1/I,
(2.7)
1)1/1 = (A - l)a1/l.
(2.8)
Since the operator H is positively definite, one immediately finds the lowest energy eigenstate 1/10 as the one for
which
.1.
a'f/O
=
"----' d
x'/2.1.
"----'.1. ( )
V-r'-e
'f/o=V-r''f/OX
dx
= Coe -
x'/2
,
(2.9)
and one checks that the corresponding eigenvalue is ,1,0 = ~.
Using now the operator a*, one subsequently constructs the
ladder of other eigenvectors 1/In corresponding to the next
eigenvalues An = n + !:
1/In = C n(a*r1/lo = C n( - l)n[ ~'/2
:x:
e - X'/2]e - x'/2
(2.10)
II. CLASSICAL FACTORIZATION METHOD
The factorization method in its most classical form was
first used to determine the spectrum of the Hamiltonian of
the harmonic oscillator in one dimension:
1 d2
1
H= - - -2 + _x 2 •
(2.1)
2 dx
2
The method consisted of introducing the operators of
"creation" and "annihilation"
a -_ - 1
J2
*
(d-
J21 ( -
a =
+ x ) -_
d
d
"",'/2
-1e _ x'/2-----e,
J2
dx
)
dx +x = -
(2.2)
dx
J21
e
x'/2
d
dx e
_ x'/2
,
(2.3)
with the properties
a*a =H -~}
* _ H 1 =>[a,a*] = 1.
aa
-
+2
(2.4)
Ha* = a*(H + 1),
(2.5)
Ha=a(H-l).
(2.6)
leave of absence from Department of Physics, Warsaw University,
Warsaw, Poland.
3387
J. Math. Phys. 25 (12), December 1984
(2.11)
The nonexistence of any other spectrum points and eigenstates follows from the completeness of the Hermite polynomials. The above method was first employed by Dirac. 1 Its
extension for the hydrogen atom was found by Infeld and
Hull. 2 A generalized presentation is due to Plebafiski. 3 The
group theoretical meaning is owed to Moshinsky,4 Wolf,5
and other authors. Yet, there is still one aspect of the method
relatively unexplored. It can be used not only to find the
interdependence between different spectral subspaces of the
same operator but also to transform one Hamiltonian into
another.
III. MODIFIED HAMILTONIAN
Consider once more the factorized expression
Hence,
alO n
where Hn (x) are the Hermite polynomials given by the Rodriguez formula
H
+~ =
aa*.
(3.1)
Are the operators a and a* here unique? Define the new
operators
b=
_1
J2
0022-2488/84/123387-03$02.50
(~+P(X)),
dx
© 1984 American Institute of PhysiCS
(3.2)
3387
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b * = _1 ( -
..fi
and demand that H
H
+! =
~ + {3 (X)),
(3.3)
dx
+ ! be written alternatively as
bb *.
(3.4)
This leads to
1 d2
1 2
1
- T dx2 + T X + T
=
1 (
T -
d2
dx2
+
{3'
2)
+{3,
(3.5)
and so, the condition for {3 is the Ricatti equation
{3' + {32 =
1 + x 2•
(3.6)
The occurrence of the Ricatti equation in the factorization problems is a typical phenomenon. 2 ,3 In general, the
explicit solution of this type of equation is not known. This is
not the case in (3.6), where one has one particular solution
{3 = x. Hence, the general solution can be obtained putting
{3 = x + t/J (x). This yields
t/J'
+ 2tjJx +t/J 2 =~'/t/J2 + 2x(1/t/J) +
1 = 0.
ifJl = b *¢o, ifJ2 = b *¢I'"'' ifJ" = b *¢n _ I'''' (3.15)
are the eigenvectors of H' corresponding to the same eigenvaluesA n = n +!:
H'ifJn =H'b *¢n-l = b *(H + 1)¢n_1 = b *(n + !)¢n-I
= (n + !)ifJ" (n = 1,2, ... ).
(3.16)
The functions t/J" are square integrable because of the asymptotic behavior of ifJ (x) for x- ± 00. They are obviously orb *¢k- I) = (¢j_I'
thogonal,
as (ifJj,ifJk) = (b *¢j_I'
bb*¢k_I)=(¢j_I' (H+!)¢k_l) =k(¢j_I' ¢k_d=O,
for k #j. However, they do not yet span the whole of L 2(R ).
The missing element is the vector ifJo orthogonal to all of ifJ"
(n = 1,2, ..... ),
= (t/Jo,b *¢" - I) = ~(bt/>o'¢n _ I) =
(for n = 1, 2, ... )~bifJo = 0,
(t/Jo,ifJ,,)
(3.17)
and so, the "missing vector" is found from the first-order
differential equation
+ 2xy + 1 =
(3.8)
0,
whose general solution is
y = (r +
f
e-
x"
dX)~', rER.
Hence,
ifJ(x) =
e-~
r
.,
+ f~ e - x
dx'
~{3(x)=x+
e-~
----r + f~ e - .<" dx' .
(3.9)
The introduction of the operators b, b * might seem to offer
little new, as we have still bb * = aa* = H + l' However, the
commutator of band b * is not a number:
[b,b *]
= {3 '(x) = 1 + ifJ '(x).
(3.10)
Hence, the inverted product b *b is not H
defines a certain new Hamiltonian
b *b
= bb * + [b *,b ] = H + ! -
+ const,
but it
1 - ifJ ' = H' - !,
(3.11)
where
~d:
H'=H-ifJ'(x)= -
2 dx
+V(x),
(3.12)
.
(3.13)
with
V (x)
2
= -x - - d[
2
dx
e-
r
x
'
+ f~e-x"
]
dx'
If Irl >![ii, the above potential has no singUlarity and behaves like x 2 /2 for x- ± 00; and so, one obtains here a oneparameter family of self-adjoint Hamiltonians in L 2(R ). As
one can immediately see, their spectra are identical to that of
the harmonic oscillator, though their eigenvectors are different. Indeed, (3.4) and (3.11) imply
H'b * = (b *b + !)b * = b *(bb * +!)
= b *(H +
1).
(3.14)
Hence, for ¢n (n = 0, 1, ... ) being the eigenvectors of H, the
functions
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J. Math. Phys., Vol. 25, No. 12, December 1984
°
[~+ {3 (X)]ifJo =
..fi dx
~o = coe - x'/2exp(lX ifJ (x')dx ).
bifJo = _1
(3.7)
Introducing now a new function y = 1/ifJ, one ends up
with a first-order linear inhomogeneous equation
- y'
°
(3.18)
By the very definition (3.18), ifJo is another eigenvector of H'
corresponding to the eigenvalue Ao = 1:
H't/Jo = (b *b + !)ifJo = !ifJo'
(3.19)
As the system of vectors ifJO,ifJl'" is complete in L 2(R ), the
operator (3.12) is a new Hamiltonian, whose spectrum is that
of the harmonic oscillator, although the potential is not.
Since our initial Hamiltonian is parity invariant, the final
conclusion should remain valid when x_ - x. Indeed,
though each one of the potentials (3.13) is not parity invariant, the whole class is: V( - x,r) = V(x, - r) (Irl > 
