Using quantum Hamilton-Jacobi formalism of Leacock and Padgett, we show how
to obtain the exact eigenvalues for supersymmetric (SUSY) potentials.Comment: 15 pages Latex Compile twice to get cross references correct. 2
  Figures not included. Requests for figures should be sent to
  akksp@uohyd.ernet.i

Bhalla, R. S.

Kapoor, A. K.

Panigrahi, P. K.

English

arXiv

Using quantum Hamilton-Jacobi formalism of Leacock and Padgett, we show how to obtain the exact eigenvalues for supersymmetric (SUSY) potentials

Bhalla, R S

Kapoor, A K

Panigrahi, P K

CERN Document Server

MRI-PHYS-95/11
Energy Eigenvalues For Supersymmetric Potentials via Quantum
Hamilton-Jacobi Formalism
R.S. Bhalla
1
, A.K. Kapoor
2
Mehta Research Institute, 10 Kasturba Gandhi Marg,
Allahabad-211 002, India
and
School of Physics, University of Hyderabad,
Hyderabad-500 034, India.
and
P. K. Panigrahi
3
School of Physics, University of Hyderabad,
Hyderabad-500 034, India.
ABSTRACT
Using quantum Hamilton-Jacobi formalism of Leacock and Padgett, we show
how to obtain the exact eigenvalues for supersymmetric (SUSY) potentials.
July 1995
1
email: bhalla@mri.ernet.in
2
email: akksp@uohyd.ernet.in
3
email: panisp@uohyd.ernet.in
1. Introduction
In classical mechanics, the Hamilton-Jacobi formalism is used to arrive at the
frequencies of periodic systems, through the action variable, without having to ob-
tain the complete solution of the equation of motion [1]. Analogously, quantum
Hamilton-Jacobi (QHJ) formalism has been developed [2] where the quantum ac-
tion variable yields the exact energy spectra of the bound state problems, avoiding
an explicit solution of the corresponding Schrodinger equation. The QHJ equation
is equivalent to the Schrodinger equation and is written in terms of the quantum
momentum function (QMF), p(x;E), which is the logarithmic derivative of the wave
function. The quantization of the energy E arises from a suitable contour integral
of the QMF, equated to integral multiple of h. In this paper we apply this formal-
ism as developed by Leacock and Padgett to obtain exact energy eigenvalues for
potentials which exhibit supersymmetry (SUSY) and shape invariance (SI) [3]. It
is well known that for potentials having the above two properties, one can nd out
the eigenvalues algebraically.
Our motivation in carrying out this explicit computation has been (a) to gain a
better understanding of the working of the QHJ formalism and (b) to see what kind
of singularities are possible for the QMF which play a signicant role, as will be seen
later, in determining the eigenvalues. It is hoped that the singularity structure of
QMF will enable us to analyse exactness or nonexactness of the well known SUSY
WKB formula [3,4].
The quantum Hamilton-Jacobi equation for a given potential V (x) is given by
h
i
@
2
W (x;E)
@x
2
+
 
@W (x;E)
@x
!
2
= E   V (x)
 p
2
c
(x;E); (1)
where we have set 2m = 1 and where, W (x;E) is the quantum Hamilton's charac-
2
teristic function. The quantum momentum function p(x;E) is dened as
p(x;E) 
@W (x;E)
@x
: (2)
The QMF is required to satisfy the boundary condition
lim
h!0
p(x;E)! p
c
(x;E):
Here p
c
(x;E) is the classical momentum function, and is dened in such a way that
it has the value +
q
E   V (x), just below the branch cut which joins the two turning
points.
Explicitly, the QMF is related to the wavefunction by
p =  ih
1
 
@ 
@x
:
For the n-th energy level,  has n zeros between the classical turning points. These
zeros of the wave function correspond to simple poles of the momentum function
with residue given by h=i for each of these zeros. The location of these poles is
energy dependent and we shall refer to them as the moving poles. Other poles
whose location does not depend on energy will be refered to as the xed poles of
QMF. Let C be a contour enclosing the moving poles between the classical turning
points. The integral
J(E) 
1
2
I
C
p(x;E)dx; (3)
called the quantum action variable is obviously J(E) is equal to nh. Thus
J(E) = nh; (4)
when inverted for E, gives the exact energy eigenvalues.
In this paper we show how the QHJ method gives exact eigenvalues for SUSY
potentials. SUSY quantum mechanics (QM) has been studied extensively in the
3
last decade. For a review of SUSY in QM, the reader is referred to recent review
article by Cooper, Khare and Sukhatme [3]. In SUSY QM one considers a pair of
potentials dened by (2m = 1).
V

= !
2
(x) h
d
dx
!(x); (5)
where !(x) is called the super-potential. The corresponding Hamiltonians
H

= p
2
+ V

(x);
can be factorized as
H
 
= A
y
A (6)
H
+
= AA
y
; (7)
where
A = h
d
dx
+ !(x) (8)
A
y
=  h
d
dx
+ !(x): (9)
Whenever the function exp ( 
R
x
!(y)dy) is square integrable, it represents the
ground state wave function of H
 
. Let  
0
denote this ground state wave function.
Then we have
!(x) =  h
1
 
0
@ 
0
@x
: (10)
The eigenvalues and eigenfunctions for the two Hamiltonians H

are related. Using
the intertwining relations AH
 
= H
+
A and A
y
H
+
= H
 
A
y
, it can be easily shown
that E
( )
n+1
= E
(+)
n
(apart from the ground state wavefunction satisfying A 
( )
0
= 0).
Hence E
( )
n+1
and E
(+)
n
are the energy eigenvalues for H
 
and H
+
respectively. It
has been known that in addition to SUSY, if a potential has a discrete reparam-
eterization invariance called shape invarience, the corresponding eigenvalues and
4
eigenfunctions can be explicitly obtained algebraically. In the next section we give
the details of the steps in QHJ formalism required to solve one of these potentials.
In Sec. 3. we shall take up the solution of a class of SUSY potentials.
2. Eckart Potential
In case of SI potentials both V
+
and V
 
are of the same functional form, albeit with
dierent values of the parameter. In the following, we will consider only the V
 
and
apply the QHJ formalism to obtain the corresponding eigenvalues. In this section
we shall obtain the energy eigenvalues for bound states of the Eckart potential given
by
V (x) = A
2
+
B
2
A
2
+A(A  h)cosech
2
x  2Bcothx : (11)
(x  0)
The corresponding superpotential is given by
!(x) =  Acothx+
B
A
; (12)
where A and B are constants and the quantum Hamilton- Jacobi equation is given
by
p
2
(x;E)  ih
@p(x;E)
@x
= E  A
2
 
B
2
A
2
 A(A  h)cosech
2
x+ 2Bcothx : (13)
A simple way to obtain the eigenvalues of Eckart hamiltonian is to make use of the
transformation
y = cothx: (14)
Under this mapping p(x;E) becomes a function of y when x is expressed in terms
of y. We shall continue to use p to denote the function of y so obtained. Thus
5
p(y;E) means the QMF p(x;E), expressed in terms of the variable y. This will be
understood for all change of variables to be considerd below.
The quantization condition (4) then becomes
I
C
1

1
2
I
C
1
p(y;E)dy
1   y
2
= nh; (15)
where C
1
is the image of the contour C of (3), enclosing the turning point, under
the mapping x! y = cothx: The QHJ equation written in terms of y is given as
p
2
  ih
dp
dy
(1 + y)(1  y) = E  A
2
 
B
2
A
2
 A(A  h)(y + 1)(y   1) + 2By (16)
Note that the above mapping y = cothr introduces additional singularities in the
integrand at y 1. The contour integral I
C
1
is calculated by deforming the contour,
(see g 1.) so as to enclose all the singular points of the integrand. The poles
of QMF can be located using available results [5] on complex zeroes of solutions of
linear dierential equations. They are easily found by inspection for all the problems
of interest in this paper. For evaluation of the integral (15) consider the contour
integral I
 
R
for a circle  
R
of radius R which is taken to be large enough so that
outside  
R
, p(y) has no singularities. The singular points of the integrand are the
poles at y = 1 and the moving poles enclosed inside C
1
. Therefore, we have
1
2
I
 
R
pdy
1  y
2
=
1
2
 
I
C
1
pdy
1  y
2
+
I

1
pdy
1   y
2
+
I

2
pdy
1  y
2
!
: (17)
Denoting the integrals in (17) as I
 
R
; I
C
1
; I

1
and I

2
(g.1), we rewrite (17) as
I
 
R
= I
C
1
+ I

1
+ I

2
: (18)
The contour integral I
 
R
is calculated by one more change of variable to z = 1=y.
In terms of the variable z the integral I
 
R
becomes
I
 
R
=
I
 
R
pdy
1   y
2
(19)
6
=I

0
pdz
1   z
2
(20)
 I

0
(21)
where 
0
is a small circle in the z-plane enclosing only one singular point z = 0. It
is worth reminding that both the contours are in the anticlockwise direction and the
singularity at y!1 is mapped to the singularity at z = 0.
Therefore (15), (18) and (21) give
nh =
1
2
I
C
1
pdy
1  y
2
=
1
2
 
I

0
pdz
1  z
2
 
I

1
pdy
1  y
2
 
I

2
pdy
1  y
2
!
: (22)
The calculation of various integrals requires behavior of the momentum function
near the singular points. In particular, we need the value of the function at y = 1
and its residue at z = 0. These are calculated by substituting appropriate Taylor
or Laurent series expansion of p in the QHJ equation and solving for the rst few
coecients in the series expansion. For example, for calculation of the contour
integral around y = 1, the series expansion of p(y;E) around y = 1 is used.
The integrand suggests that we need to calculate only the coecient of the constant
term. We illustrate this for computing the integrals I

1
; for this purpose we at rst
expand p(y;E) as
p(y;E) = 
0
+ 
1
(y   1) + 
2
(y   1)
2
+      
and substite the expansion in (16). Comparing the terms independent of y on both
sides gives

2
0
= E  A
2
 
B
2
A
2
+ 2B (23)
Let 
0
be the constant term in the expansion of p(y;E) in powers of (y + 1). Then

0
is similarly determined and is given by

2
0
= E  A
2
 
B
2
A
2
  2B (24)
7
Note that, there are two roots for 
0
and 
0
, corresponding to the two signs of the
square of the right hand side in (23)and (24). Similarly for obtaining the integral
I

0
, we need to compute the residue of p(z;E) at z = 0. We expand p(z;E) as
p(z;E) = b
1
=z + a
0
+ a
1
z +    (25)
and substitute (25) the QHJ equation written in terms of the variable z
p
2
  ih
 
1   z
2
z
2
!
dp
dz
= E  A
2
 
B
2
A
2
 A(A  h)
 
1   z
2
z
2
!
+
2B
z
: (26)
Comparing the coecients of 1=z
2
on the two sides gives,
b
1
=
 ih i(h  2A)
2
(27)
We select the correct root for 
0
; 
0
and b
1
from (27), (23) and (24) by comparing
them with the answer for E = 0. It is straight forward to obtain the value of b
1
,

0
and 
0
for E = 0, by recalling that for zero energy the QMF is related to the
superpotential by p(x;E = 0) = i!(x). Writing the superpotential !(x) in terms of
z, we get
!(z) =  
A
z
+
B
A
: (28)
The residue of !(z) at z = 0 is  A. Comparing this answer with value of b
1
, we see
that the correct choice of b
1
is given by b
1
=  iA for all E. Similarly one looks for
the coecient of expansion of w in powers of y  1 and this is compared with the
values of 
0
and 
0
respectively for E = 0. It is found that these roots have relative
opposite signs and are given by

0
=  
s
E  A
2
 
B
2
A
2
+ 2B (29)

0
=
s
E  A
2
 
B
2
A
2
  2B (30)
It may be noted that the correct sign of the residues calculated can also be xed
by taking h! 0 and looking at the behaviour of p
c
near the point of interest. This
8
procedure, as originally suggested by Leacock and Padgett, is a bit complicated.
For SUSY potentials under consideration in this article we have found it useful to
follow the alternate procedure as explained above.
The contour integrals I

0
, I

1
and I

2
are, therefore, computed to be
I

1
=
i
0
2
; I

2
=
i
0
2
; and I

0
=
A

:
Thus the energy eigenvalues are obtained from
I
C
1
= nh =
A

 
i
0
2
 
i
0
2
(31)
which on further simplication gives
E
n
= A
2
+
B
2
A
2
 
B
2
(nh +A)
2
  (nh +A)
2
; (32)
One can arrive at the above energy eigenvalues using other mappings. Use of a
dierent mapping will be considered in the next section and we shall then show how
to obtain the bound state eigenvalues for other SUSY potentials.
9
3. Other SUSY potentials
In this section we will show how the QHJ method could be used for other SUSY
potentials. We will use y = exp(ix) mapping for the SUSY potentials involving
trigonometric functions. The remaining potentials involving hyperbolic functions
y = exp(x) will be used. The treatment of each of these cases runs parallel to the
treatment given to the Eckart potential in the previous section except for a new point
which requires special attention. The use of the mapping y = exp(x) gives rise to
extra energy dependent poles in p(x;E) in the non-classical region. Computation of
the exact energy eigen-values requires the knowledge of an integral along a contour
enclosing these energy dependent poles. In all the cases investigated in this paper
this integral can be related to the integral around the contour which encloses poles
in the classical region on the real axis.
In this section at rst we shall work out the eigenvalues for the Eckart potential
again, using the mapping y = exp(x), but concentrating only on the new points as
compared to the treatment in the previous section.
The corresponding super potential for the Eckart potential written in terms of the
variable y = exp(x) is
!(y) =  A
 
y
2
+ 1
y
2
  1
!
+
B
A
; (33)
and the QHJ equation is
p
2
  ihy
dp
dy
= E  A
2
 
B
2
A
2
 
4A(A  h)y
2
(y
2
  1)
2
+
2B(y
2
+ 1)
(y
2
  1)
: (34)
Now the equation
E  A
2
 
B
2
A
2
 
4A(A  h)y
2
(y
2
  1)
2
+
2B(y
2
+ 1)
(y
2
  1)
= 0; (35)
has four solutions for the turning points. These are shown as A, B, A
0
B
0
(see Fig.
2). The moving poles of p are (i) on the real axis and between A and B and (ii)
10
on the real axis between A
0
and B
0
. It should however be noted, that two of these
turning points A
0
and B
0
are in the non-classical region. We note that the symmetry
y!  y in (35) interchanges A with A
0
and B with B
0
. This symmetry implies
I
C
1
= I
C
2
; (36)
where C
1
and C
2
are contours enclosing A;B; and A
0
; B
0
respectively as shown in
Fig. 2. Next note that now y = 0 is a pole of the integrand in the action integral
(see (38) below). Also p(y) has poles at y = 1 and y =  1 because the right hand
side of QHJ is singular at y = 1 and y =  1: Introducing I
 
R
for the large circle  
R
enclosing all the singular points we arrive at
I
 
R
= I
C
1
+ I
C
2
+ I
1
+ I

2
+ I

3
: (37)
Here I

1
; I

2
; I

3
are integrals along contours 
1
; 
2
; 
3
which enclose the singular
points y = 1; y =  1 and y = 0 respectively. Therefore, using (36), the quantization
condition
I
C
1

1
2
I
C
1
dy
y
p(y;E) = nh: (38)
becomes
I
 
R
 
X
p
I

p
= 2nh: (39)
The rest of the calculation is same as in Sec. 2. and one easily arrives at
E
n
= A
2
+
B
2
A
2
 
B
2
(nh +A)
2
  (nh +A)
2
: (40)
The calculation of eigenvalues for other SUSY potentials proceeds in a similar fash-
ion. The results are summarized in Table I and Table II. The range of the variable x
is  1 < x <1, unless indicated otherwise. Expressions listed in the coloumn four
are I

p
are the values for a contour 
p
enclosing only the singular point indicated
in the third column of the table. The value of I

p
for the pole at 1 stands for
11
the value I
 
R
. For the xed poles at y = 0 and 1 the square roots in the residue
are found to have relatively opposite signs, on comparision with the coecients of
corresponding terms in the expansion of !(x) for E = 0. The eigen-values listed in
the last column are obtained using (39).
4. Conclusion:
In conclusion, we have explicitly worked out the eigenvalues for the SUSY potentials
using the QHJ method. Apart from checking the correctness of the formalism, this
exercise provides insight into solvability of these potentials. The main eort involved
in use of this scheme lies in selecting the correct root for the residues needed. This
problem was solved here by comparing the answers obtained from the QHJ for
E = 0 with that obtained from the superpotential. In general the choice of the
correct root for the residue can be made by using the boundary condition on QMF,
viz., p! p
c
in the limit h! 0 and where the branch of p
c
is selected in such a way
that it corresponds to the positive sign just below the cut on the real axis joining
the physical turning points.
The Leacock-Padgett method is a powerful method for obtaining the eigenvalues
analytically as well as numerically and can be applied to other potentials. It is
worth pointing out that for the potentials considered in this article, the SUSY-
WKB approximation gives exact answer. Since QHJ as shown here also gives exact
answer, it is natural to enquire as to the relation between these two approaches.This
will throw light on the question of exactness of the SUSY WKB formula. This
investigation will be reported elsewhere.
Acknowledgements : We acknowledge useful discussions with Profs. A. Khare,
V. Srinivasan, Pankaj Sharan, U. Sukhatme and S. Chaturvedi and Drs. C. Na-
graj Kumar and R. Adhikari. Two of us (R.S.B. and A.K.K.) would like to thank
12
Director, Mehta Research Institute for hospitality and support during their stay.
References
1. H. Goldstein, Classical Mechanics (Addison-Wesley, New York,1950).
2. R.A. Leacock and M.J. Padgett, Phys. Rev. D28, 2491 (1983)
R.A.Leacock and M.J. Padgett, Phys. Rev. Lett. 50, 3 (1983).
3. F. Cooper, A. Khare and U. Sukhatme Phys. Reports 251, 267 (1995).
4. A. Comtet, A. Bandrauk and D.K. Campbell, Phys. Lett. B150, 159 (1985).
5. E. L. Ince , Ordinary Dierential Equations, Dover Publcations N.Y (1956).
13
Table 1 : Trignometric Potentials.
For all the potentials listed in this table the mapping y = exp(ix) is used.
Name of Potential Location of I

p
Eigen
potential xed poles value
0
p
E +A
2
Scarf I  A
2
+ (A
2
+B
2
+Ah) i  A+B (A+ nh)
2
(Trignometric) sec
2
x B(2A+ h)  A
2
sec x tan x  i  (A+B)
( =2  x  =2)
1  
p
E +A
2
0  
q
E +A
2
 B
2
=A
2
+ 2iB
Rosen- A(A  h)cosec
2
x 1 A +B
2
=A
2
Morse-I  A
2
+B
2
=A
2
A
2
  (A+ nh)
2
(trignometric) +2B cot x  1 A  B
2
=(A+ nh)
2
(0  x  )
1
q
E +A
2
 B
2
=A
2
  2iB
Table 2 : Hyperbolic Potentials.
For all the potentials listed in this table the mapping y = exp(x) is used.
Name of Potential Location of I

p
Eigen
potential xed poles value
0  i
p
E  A
2
Scarf II A
2
+ (B
2
 A
2
 Ah) i iB  A
(hyperbolic) sech
2
x +B(2A+ h) A
2
  (A  nh)
2
sechx tanhx  i  iB  A
1 i
p
E  A
2
0  i
q
E  A
2
 B
2
=A
2
+ 2B
Rosen - A
2
+B
2
=A
2
i  A A
2
+B
2
=A
2
Morse II  A(A+ h) sech
2
x  (A  nh)
2
(Hyperbolic) +2Btanhx  i  A  B
2
=(A  nh)
2
1 i
q
E  A
2
 B
2
=A
2
  2B
0 i
q
E  A
2
 B
2
=A
2
  2B
Eckart A
2
+B
2
=A
2
1 A A
2
+B
2
=A
2
(Hyperbolic) +A(A  h) cosech
2
x  (A+ nh)
2
 2Bcothx  1 A  B
2
=(A+ nh)
2
(x  0)
1  i
q
E  A
2
 B
2
=A
2
+ 2B
0  i
p
E  A
2
Generalised A
2
+ (B
2
+A
2
+Ah) 1  A+B
Poschl- cosech
2
x B(2A+ h) A
2
  (A  nh)
2
Teller cothx cosechx  1  (A+B)
(x  0)
1 i
p
E  A
2


Energy Eigenvalues For Supersymmetric Potentials via Quantum Hamilton-Jacobi Formalism

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