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Derivation of the S-matrix using supersymmetry
16 May 1988
PHYSICS LETTERS A
Volume 129, number 3
DERIVATION OF THE S-MATRIX USING SUPERSYMMETRY
Fred COOPER, Joseph N. GINOCCHIO and Andreas WIPF Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Received 7 March 1988; accepted for publication 22 March 1988 Communicated by D.D. Holm
Using supersymmetry and shape invariance the reflection and transmission coefficients for a large class of solvable potentials can be obtained algebraically.
The eigenspectrum and eigenstates of a class of onedimensional hamiltonians have been derived algebraically using supersymmetry [ l-3 ] and shape invariance [ 451. In this paper we show that the scattering function can be obtained algebraically as well using these two features. All one-dimensional Schrijdering equations can be factorized [ 6,7 1,
--$ +v,
H, 0:’ ’ =
=A~A@“‘=E(‘)@~‘) n
n
(x,a,)
>
7
(1)
A’=-~+W(x,u,), A=-&+W(x,u,)(2) and (3)
and a, are some parameters of the potential, n labels the states, n=O, 1, .... and n = 0 is the lowest state. The superpotential W(x, a, ) is given by w(x,u,)=--&ln@b”(x,u,).
(4)
The partner hamiltonian H2 =_4A+= - d2/dx2 + V2, v2 = W2(x, a, ) + d W(x, a, ), dx
Ehl’ =O,
EA2’ =E;:‘,
.
(6a)
In addition the eigenfunctions with the same energy are related: @;:‘, (x)= (E;:‘,
)-1’2A+@$2’(~)
.
(6b)
This degeneracy in the two spectra is due to a supersymmetry. We define a superhamiltonian H and supercharges Q and Qt:
@A’)
where
v,=w’(x,a,)--&w(x,a,),
gives the same spectrum as H, but with the ground state missing; that is,
H=[: k2],Q=[;
;I,
Q+=[;
;+I. (7)
These operators are the two-dimensional representation of the sl( 1/ 1) super algebra
[Q,Hl=[Q+,Hl=O, {Q,Q+}=H, {Q,Q)={Q+,Q+I=o.
(8)
The fact that the supercharges commute with H gives the energy degeneracy ( 6 ) . Clearly this process can be continued; a V, can be determined from V,, etc. This produces a ladder of potentials [ 4-7 1, V,,. Furthermore, if the partner potentials are “shape invariant”, i.e., V2has the same functional form as V, but different parameters except for an additive constant,
(5) v2
0375-9601/88/$ 03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
(4
a1
I=
K(x,
a2
I+
C(a,)
7
(9a)
145
Volume 129, number
PHYSICS
3
then the full ladder of potentials will be shape invariant, ~n(X,a,)=~~(x,~,)+~~“(~,)
>
(9b)
and the energy spectrum [ 431 and eigenfunctions [ 4,8] of the original potential can be determined algebraically: EL’)=
i: C(Q),
(lOa>
k=l
@;‘)(X,U,)= k~,A'(x,a,)~6"(X,u,+,).
(lob)
In this paper we shall show that, using shape invariance, we can determine the scattering solutions algebraically as well also. There are two types of scattering problems that we shall consider: ( 1) scattering from a one-dimensional potential well and (2) scattering from a spherically symmetric three-dimensional potential well. The asymptotic wavefunction for scattering from a one-dimensional potential well V, is given by CD(*)(k,x+ -_co)+eikX+R2e-ikX,
(11)
Qc2)(k, x-ba3)-+T2eikr.
(12)
Using (6b) we can determine the asymptotic behaviour of 0” ) (k, x) in terms of the asymptotic behaviour of @*) (k, x) and derive the following relation between the two transmission and reflection coefficients [ 9- 111: W,-ikT T -~ ‘W_-ik R
”
(13a)
-- W_ +ikR ‘- W_-ik ”
(13b)
where W2 = W(x+ f co). We have assumed that W: = WZ for simplicity of exposition only; this assumption implies that the potential is symmetrical asymptotically. If we also assume shape invariance, (9), then we get
(14a) (14b) 146
LETTERS
16 May 1988
A
Hence we find a recursion relation with W being the same function but with a different value of the parameters. In particular if there is some parameter set uN such that the transmission and reflection coefftcient is known, then we have T(k,ul)=
N--L W, (a,)-ik n W_ (u )_ikT(k’u,)’ n=l n
(15)
R(k’u’)=
N-’ W-(‘%~)+lk~(~,~~). n W_(a )_ik II=, n
(16)
As an example we take the PSschl-Teller potential; W(x, u,)=atanh(x),
4(&z+l)
V(x,u,)=u2,-
cosh2(x)
(17a)
’
where u,=(Y--n+1.
(17b)
If CX=N- 1, then uiy= 0 and the potential vanishes: hence the transmision coefficient is unity. Thus ( 15 ) gives T(k, a)=
r(
-ik-a) r( -ik+a+ r( -ik) r( 1 -ik)
1)
(18)
When the potential vanishes, the reflection coefftcient vanishes. From ( 16) we see that the reflection coefficient vanishes for all integer values of (Y.This means that R(k, a)=
sin(rca) Ro(k, a) .
(19)
For (Y small the potential will be small and R. (k, O)=(l/x)k(k, 0), where d is the derivative of R with respect to cr. At small (Ywe can use the Born approximation
k(
k, 0) =
J e2ikyti(y, 0) dy
(20)
and we finally get R(k, a)=
isin
sinh(xk)
T(k, a) ,
(21)
Next, we consider the spherically symmetric threedimensional Schriidinger equation:
(22)
Volume 129, number 3
PHYSICS LETTERS A
where d 1+1 Af=-d,-,fW(r,a,).
(24)
We see from the above that the partner potential will be a solution for angular momentum increased by one. The asymptotic radial wavefunction for partial wave 1 is
-(-I)
I+,,-lkr
(25)
I.
where S, is the scattering function for the Ith partial wave. Using supersymmetry and shape invariance as before, we find
S(k
a1 ) =
ik- W(c0, a, ) S/+, (S az). ik+ W( 03, a, )
(26)
For the Coulomb potential W= (21+2)-l which is shape invariant. Solving the recursion relation in (26) we derive the well-known result &+, (k) r(f+ 1+ (2ik)-‘) r( 1- (2ik)-‘) =r(1+1+(2ik)-‘)r(l+(2ik)-‘)
s (k) ’ ’
16 May 1988
which gives the scattering function for all 1 in terms of that for 1~0. The term So does not contribute to the angular distribution since 1SoI* = 1. &though the Coulomb scattering does not have the form (25 ) asymptotically, the result (27) still follows. If we identify the operators A and At with ladder operators of a potential group, then the recursion relations we have obtained can be given a group algebraic interpretation [ 12,13 1.
References [ 11 E. Witten, Nucl. Phys. B 185 ( I981 ) 5 13. [2] P. Salomonson and J.W. van Holten, Nucl. Phys. B 196 (1982) 509. [ 31 F.Cooper and B. Freedman, Ann. Phys. (NY) 146 ( 1983) 262. [ 4 ] L. Infeld and T.D. Hull, Rev. Mod. Phys. 23 ( 195 1) 2 1. [ 51 L. Gendenshtein, Pis’ma Zh. Eksp. Teor. Fiz. 38 (1983) 299 [JETPLett. 38 (1983) 3561. [6] C.V. Sukumar, J. Phys. A 18 (1985) L57. [ 71 A.A. Andrianov, N.V. Borisov and M.V. Ioffe, Phys. Lett. A 105 (1984) 19. [ 81R. Dutt, A Khare and U. Sukhatme, Am. J. Phys. (January 1988), to be published. [ 91 R. Akhoury and A. Comtet, Nucl. Phys. B 246 ( 1984) 253. [lo] D. Boyanovsky and R. Blankenbecker, Phys. Rev. D 30 (1984) 1821. [ 111C.V. Sukumar, J. Phys. A 19 (1986) 2297. [ 121 Y. Alhassid, F. Gursey and F. Iachello, Phys. Rev. Lett. 50 (1983) 873. [ 13 ] Jianshi Wu, Group theory approach to scattering, Yale University Thesis ( 1985), unpublished.
(27)
147
PHYSICS LETTERS A
Volume 129, number 3
DERIVATION OF THE S-MATRIX USING SUPERSYMMETRY
Fred COOPER, Joseph N. GINOCCHIO and Andreas WIPF Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Received 7 March 1988; accepted for publication 22 March 1988 Communicated by D.D. Holm
Using supersymmetry and shape invariance the reflection and transmission coefficients for a large class of solvable potentials can be obtained algebraically.
The eigenspectrum and eigenstates of a class of onedimensional hamiltonians have been derived algebraically using supersymmetry [ l-3 ] and shape invariance [ 451. In this paper we show that the scattering function can be obtained algebraically as well using these two features. All one-dimensional Schrijdering equations can be factorized [ 6,7 1,
--$ +v,
H, 0:’ ’ =
=A~A@“‘=E(‘)@~‘) n
n
(x,a,)
>
7
(1)
A’=-~+W(x,u,), A=-&+W(x,u,)(2) and (3)
and a, are some parameters of the potential, n labels the states, n=O, 1, .... and n = 0 is the lowest state. The superpotential W(x, a, ) is given by w(x,u,)=--&ln@b”(x,u,).
(4)
The partner hamiltonian H2 =_4A+= - d2/dx2 + V2, v2 = W2(x, a, ) + d W(x, a, ), dx
Ehl’ =O,
EA2’ =E;:‘,
.
(6a)
In addition the eigenfunctions with the same energy are related: @;:‘, (x)= (E;:‘,
)-1’2A+@$2’(~)
.
(6b)
This degeneracy in the two spectra is due to a supersymmetry. We define a superhamiltonian H and supercharges Q and Qt:
@A’)
where
v,=w’(x,a,)--&w(x,a,),
gives the same spectrum as H, but with the ground state missing; that is,
H=[: k2],Q=[;
;I,
Q+=[;
;+I. (7)
These operators are the two-dimensional representation of the sl( 1/ 1) super algebra
[Q,Hl=[Q+,Hl=O, {Q,Q+}=H, {Q,Q)={Q+,Q+I=o.
(8)
The fact that the supercharges commute with H gives the energy degeneracy ( 6 ) . Clearly this process can be continued; a V, can be determined from V,, etc. This produces a ladder of potentials [ 4-7 1, V,,. Furthermore, if the partner potentials are “shape invariant”, i.e., V2has the same functional form as V, but different parameters except for an additive constant,
(5) v2
0375-9601/88/$ 03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
(4
a1
I=
K(x,
a2
I+
C(a,)
7
(9a)
145
Volume 129, number
PHYSICS
3
then the full ladder of potentials will be shape invariant, ~n(X,a,)=~~(x,~,)+~~“(~,)
>
(9b)
and the energy spectrum [ 431 and eigenfunctions [ 4,8] of the original potential can be determined algebraically: EL’)=
i: C(Q),
(lOa>
k=l
@;‘)(X,U,)= k~,A'(x,a,)~6"(X,u,+,).
(lob)
In this paper we shall show that, using shape invariance, we can determine the scattering solutions algebraically as well also. There are two types of scattering problems that we shall consider: ( 1) scattering from a one-dimensional potential well and (2) scattering from a spherically symmetric three-dimensional potential well. The asymptotic wavefunction for scattering from a one-dimensional potential well V, is given by CD(*)(k,x+ -_co)+eikX+R2e-ikX,
(11)
Qc2)(k, x-ba3)-+T2eikr.
(12)
Using (6b) we can determine the asymptotic behaviour of 0” ) (k, x) in terms of the asymptotic behaviour of @*) (k, x) and derive the following relation between the two transmission and reflection coefficients [ 9- 111: W,-ikT T -~ ‘W_-ik R
”
(13a)
-- W_ +ikR ‘- W_-ik ”
(13b)
where W2 = W(x+ f co). We have assumed that W: = WZ for simplicity of exposition only; this assumption implies that the potential is symmetrical asymptotically. If we also assume shape invariance, (9), then we get
(14a) (14b) 146
LETTERS
16 May 1988
A
Hence we find a recursion relation with W being the same function but with a different value of the parameters. In particular if there is some parameter set uN such that the transmission and reflection coefftcient is known, then we have T(k,ul)=
N--L W, (a,)-ik n W_ (u )_ikT(k’u,)’ n=l n
(15)
R(k’u’)=
N-’ W-(‘%~)+lk~(~,~~). n W_(a )_ik II=, n
(16)
As an example we take the PSschl-Teller potential; W(x, u,)=atanh(x),
4(&z+l)
V(x,u,)=u2,-
cosh2(x)
(17a)
’
where u,=(Y--n+1.
(17b)
If CX=N- 1, then uiy= 0 and the potential vanishes: hence the transmision coefficient is unity. Thus ( 15 ) gives T(k, a)=
r(
-ik-a) r( -ik+a+ r( -ik) r( 1 -ik)
1)
(18)
When the potential vanishes, the reflection coefftcient vanishes. From ( 16) we see that the reflection coefficient vanishes for all integer values of (Y.This means that R(k, a)=
sin(rca) Ro(k, a) .
(19)
For (Y small the potential will be small and R. (k, O)=(l/x)k(k, 0), where d is the derivative of R with respect to cr. At small (Ywe can use the Born approximation
k(
k, 0) =
J e2ikyti(y, 0) dy
(20)
and we finally get R(k, a)=
isin
sinh(xk)
T(k, a) ,
(21)
Next, we consider the spherically symmetric threedimensional Schriidinger equation:
(22)
Volume 129, number 3
PHYSICS LETTERS A
where d 1+1 Af=-d,-,fW(r,a,).
(24)
We see from the above that the partner potential will be a solution for angular momentum increased by one. The asymptotic radial wavefunction for partial wave 1 is
-(-I)
I+,,-lkr
(25)
I.
where S, is the scattering function for the Ith partial wave. Using supersymmetry and shape invariance as before, we find
S(k
a1 ) =
ik- W(c0, a, ) S/+, (S az). ik+ W( 03, a, )
(26)
For the Coulomb potential W= (21+2)-l which is shape invariant. Solving the recursion relation in (26) we derive the well-known result &+, (k) r(f+ 1+ (2ik)-‘) r( 1- (2ik)-‘) =r(1+1+(2ik)-‘)r(l+(2ik)-‘)
s (k) ’ ’
16 May 1988
which gives the scattering function for all 1 in terms of that for 1~0. The term So does not contribute to the angular distribution since 1SoI* = 1. &though the Coulomb scattering does not have the form (25 ) asymptotically, the result (27) still follows. If we identify the operators A and At with ladder operators of a potential group, then the recursion relations we have obtained can be given a group algebraic interpretation [ 12,13 1.
References [ 11 E. Witten, Nucl. Phys. B 185 ( I981 ) 5 13. [2] P. Salomonson and J.W. van Holten, Nucl. Phys. B 196 (1982) 509. [ 31 F.Cooper and B. Freedman, Ann. Phys. (NY) 146 ( 1983) 262. [ 4 ] L. Infeld and T.D. Hull, Rev. Mod. Phys. 23 ( 195 1) 2 1. [ 51 L. Gendenshtein, Pis’ma Zh. Eksp. Teor. Fiz. 38 (1983) 299 [JETPLett. 38 (1983) 3561. [6] C.V. Sukumar, J. Phys. A 18 (1985) L57. [ 71 A.A. Andrianov, N.V. Borisov and M.V. Ioffe, Phys. Lett. A 105 (1984) 19. [ 81R. Dutt, A Khare and U. Sukhatme, Am. J. Phys. (January 1988), to be published. [ 91 R. Akhoury and A. Comtet, Nucl. Phys. B 246 ( 1984) 253. [lo] D. Boyanovsky and R. Blankenbecker, Phys. Rev. D 30 (1984) 1821. [ 111C.V. Sukumar, J. Phys. A 19 (1986) 2297. [ 121 Y. Alhassid, F. Gursey and F. Iachello, Phys. Rev. Lett. 50 (1983) 873. [ 13 ] Jianshi Wu, Group theory approach to scattering, Yale University Thesis ( 1985), unpublished.
(27)
147