Algebraic treatment of the confluent Natanzon potentials

8 Pages·0.1 MB·English

by  M. T. Chefrour

February 1, 2008
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Algebraic treatment of the confluent
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2 Natanzon potentials

n
a
J
M. T. Chefrour

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D´epartement de Physique, Facult´e des Sciences,

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v Universit´e Badji Mokhtar, Annaba, Alg´erie.
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1 L. Chetouani and L. Guechi
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1 D´epartement de Physique, Facult´e des Sciences,
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2 Universit´e Mentouri, Route d’Ain El Bey,
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Constantine, Alg´erie.
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p
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Abstract
a
u
q
: Using the so(2,1) Lie algebra and the Baker, Campbell and Hausdorff
v
i formulas, the Green’s function for the class of the confluent Natanzon po-
X
tentials is constructed straightforwardly. The bound state energy spectrum
r
a is then determined. Eventually, the three-dimensional harmonic potential,
the three-dimensional Coulomb potential and the Morse potential may all be
considered as particular cases.
PACS 03.65-Quantum theory ; quantum mechanics.
PACS 03.65.Fd -Algebraic methods.
PACS 02.20.+b -Group theory.
typescript using Latex (version 2.0).
1
Inthispaper,weshallwanttopresentanalgebraictreatmentofthebound
state problem for the class of the confluent Natanzon potentials defined by
g h2 +g h+η σ h σ h2 5∆h2
2 1 1 2
V(r) = + − , (1)
R R2 − 4 R3
where∆ = σ2 4σ c andg ,g ,σ ,σ ,c andηaredimensionlessparameters.
1 2 0 1 2 1 2 0
−
The function h = h(r) is defined implicitly by the differential equation
dh . 2h
=h= , (2)
dr R(r)
q
where
R(r) = σ h2 +σ h+c . (3)
2 1 0
These potentials, originally introduced by Natanzon, are interesting in the
sense that the Schr¨odinger equation can be reduced to the confluent hyperge-
ometric formandtheyinclude, asspecial cases, theradialharmonicoscillator
for σ = c = 0, the radial Coulomb potential for σ = c = 0 and the Morse
2 0 1 0
potential for σ = σ = 0. These confluent Natanzon potentials have been
2 1
recently the object of several studies. More specifically, Natanzon1 has ob-
tained, inaSchr¨odinger equationapproach,thediscreteenergyspectrum, the
unnormalized wave functions and has derived the Green’s function. Cooper
and al2 have used the supersymmetric quantum mechanics as an algebraic
method to discuss these potentials. Furthermore, the spectrum of these po-
tentials has been re-derived by Cordero and Salam´o3 , still in a Schr¨odinger
approach, by making use a particular realization of the so(2,1) algebraic
method which is called spectrum generating algebra (SGA)4 . Finally, this
class of the confluent Natanzon potentials has been solved very recently by
Grosche5 in the framework of a Feynman path integral approach.
The Milshtein and Strakhovenko variant (MS) of the so(2,1) algebraic
approach6 providesanelegantandstraightforwardwayofsolvingtheproblem
for V(r). As we shall see, this approach consists of constructing the Green’s
function by expressing the resolvent operator for this system in the form of a
linear combination of an appropriate realization of SO(2,1) generators. This
form is then written in the Schwinger’s integral representation and the action
of the resolvent operator is obtained by means of two Baker, Campbell and
Hausdorff formulas.
LetG(r,r ;E)betheGreen’sfunctionassociatedtothepotentialsU(r) =
′
¯h2 V(r), which is solution of the differential equation
2m
h¯
(H E)G(r,r ;E) = δ(r r ), (4)
′ ′
− i −
2
where
h¯2 d2
H = +U(r) (5)
−2mdr2
is the Hamiltonian of the system and E its energy.
In terms of the function h(r) and by taking into account the relation (2),
we can write the resolvent operator H E as follows:
−
.. ...
h¯2 d2 1 h2 h
H E = + 3 2
− 2m −dr2 4  h.2 − h. 

.2  
h
+ (g σ ǫ)h2 +(g σ ǫ)h+(η 1 c ǫ) , (6)
4h2 2 − 2 1 − 1 − − 0 
h i
with E = ¯h2ǫ. 
2m
The point canonical transformation ξ = h(r) leads to
h¯
H E G(ξ,ξ ;E) = ξδ(ξ ξ ), (7)
′ ′
− i −
(cid:16) (cid:17)
f e e e
where
. .
G(ξ,ξ′;E) = h (r) h (r′)G(r,r′;E). (8)
q
Then, the dynamics of the physical system is governed by the new Hamilto-
e e
nian
h¯2 d2 η 1 c ǫ g σ ǫ
0 2 2
H = ξ + − − + − ξ , (9)
2m − dξ2 4ξ 4 !
f
and
h¯2
E = (g σ ǫ) (10)
1 1
−8m −
is the pseudo-energy. e
To see that one can introduce the three following generators
h¯2 d2 µ(µ 1) i d 1 M
T (x) = − ,T (x) = x + ,T (x) = x2,
1 −2M dx2 − x2 ! 2 −2 dx 2! 3 4h¯2
(11)
satisfying the so(2,1) Lie algebra7
[T ,T ] = iT , [T ,T ] = iT , [T ,T ] = iT , (12)
1 2 1 2 3 3 1 3 2
− − −
3
it is convenient to perform the change of variable ξ = x2 and redefine the
Green’s function as
G(ξ,ξ ;E) = √xxG(x,x;E). (13)
′ ′ ′
Under this change thee differeential equation (7)ebecomes
h¯
T (x)+2h¯2ω2T (x) E G(x,x;E) = δ(x x), (14)
1 3 ′ ′
− i −
(cid:16) (cid:17)
where e e e
h¯ 1
ω = (g σ ǫ)2 (15)
2 2
M −
with M = 4m.
e
Being a linear combination of the generators T , the operator H shows, as
i
expected, a dynamical symmetry so(2,1). By using the Schwinger’s integral
representation8 , the solution of the differential equation (14) canfbe written
as follows:
i
G(x,x;E) = ∞dSexp E +i0 S K(x,x;S), (16)
′ ′
Z0 (cid:20)h¯ (cid:16) (cid:17) (cid:21)
e e
where
iS
K(x,x;S) = exp T (x)+2h¯2ω2T (x) δ(x x). (17)
′ 1 3 ′
− h¯ −
(cid:26) h i(cid:27)
The calculation of this kernel is based upon theeuse of two Baker, Campbell-
and Hausdorff formulas7
iS
exp T +2h¯2ω2T = exp( iaT )exp( ibT )exp( icT ), (18)
1 3 3 2 1
− h¯ − − −
(cid:26) h i(cid:27)
where e
1
a = 2h¯ωtan(ωS), b = 2ln[cos(ωS)], c = tan(ωS), (19)
h¯ω
and e e e e
e
exp( iαT )exp( iβT )exp( iγT ) = exp( icT )exp(τT ), (20)
3 2 1 1 3
− − − −
with
iτ iτc c
α = , β = 2ln 1 , γ = . (21)
1 iτc/2 − 2 1 iτc/2
− (cid:18) (cid:19) −
4
It is to be noted that formulas (18) and (20) can be easily checked using the
finite representation of so(2,1) Lie algebra where the T operators are defined
i
in terms of σ Pauli matrices
i
σ iσ iσ σ +iσ
1 2 3 1 2
T1 = − , T2 = , T3 = . (22)
2√2 − 2 2√2
Here, we also have to use the Laplace transform of the Dirac distribution
M xµx1 µ i +δ M
δ(x x) = ′ − ∞ dτ exp (x2 x2)τ ;δ 0, (23)
− ′ 2h¯2 2iπ Z−i∞+δ (cid:20)4h¯2 − ′ (cid:21) ≺
in order to obtain a manageable result as follows:
1
exp( iγT )xµ = 1 iγT + ( iγT )2 +... xµ = xµ. (24)
1 1 1
− − 2! −
(cid:20) (cid:21)
Using relations (23), (18) and (20), the kernel (17) can now be written
M 1 i +δ
K(x,x;S) = x1 µexp( iaT )exp( ibT ) ∞ dτ
′ 2h¯2 ′ − − 3 − 2 2iπ Z−i∞+δ
M M
exp x2τ exp( icT )exp x2τ xµ
× −4h¯2 ′ − 1 4h¯2
(cid:18) (cid:19) (cid:18) (cid:19)
M 1 i +δ
= x1 µexp( iaT )exp( ibT ) ∞ dτ
2h¯2 ′ − − 3 − 2 2iπ Z−i∞+δ
M
exp x2τ exp( icT )exp(τT )xµ
× −4h¯2 ′ − 1 3
(cid:18) (cid:19)
M 1 i +δ
= x1 µexp( iaT )exp( ibT ) ∞ dτ
2h¯2 ′ − − 3 − 2 2iπ Z−i∞+δ
M
exp x2τ exp( iαT )exp( iβT )exp( iγT )xµ
× −4h¯2 ′ − 3 − 2 − 1
(cid:18) (cid:19)
M 1
= x1 µexp( iaT )exp( ibT )xµ
2h¯2 ′ − − 3 − 2 2iπ
i∞+δ dτexp 2M¯h2τ −x2′2 + 2−xi2τc
×Z−i∞+δ n 1 (cid:16) iτc µ+12 (cid:17)o
− 2
M (cid:16) (cid:17) iMx2
= x1 µexp( iaT )exp( ibT )xµexp
2h¯2 ′ − − 3 − 2 2h¯2c !
i∞+δ dτexp −M4¯hx2′2τ exp M¯h2xc22τ+12i/c , (25a)
×Z−i∞+δ (cid:16) 1 (cid:17)iτc µ(cid:16)+12 (cid:17)
− 2
(cid:16) (cid:17)
where formula9,10
β
β
exp( iβT2)f(x) = exp f e−2x (26)
− −4!
(cid:16) (cid:17)
5
has also been used.
The integral can be calculated thanks to the residue theorem after the
exp Mx2 1 series has been effected. Hence we obtain
¯h2c2 τ+2i/c
(cid:16) (cid:17)
M iM
K(x,x;S) = exp( iaT )exp( ibT )√xx exp (x2 +x2)
′ ih¯2c − 3 − 2 ′ 2h¯2c ′
(cid:20) (cid:21)
Mxx
′
I 1
× µ−2 ih¯2c !
M iaM b
= exp x2 exp √xx
ih¯2c (cid:18)− 4h¯2 (cid:19) −2! ′
iM Mxx
×exp(cid:20)2h¯2c (cid:16)e−bx2 +x′2(cid:17)(cid:21)Iµ−12 ih¯2ce2b′ !
Mω iMω
= √xx exp (x2 +x2)cot(ωS)
′ ′
ih¯sin(ωS) 2h¯
(cid:20) (cid:21)
e e
Mωxx′ e
I 1e , (27)
× µ−2 ih¯sin(ωS)!
e
where e
1
µ = +√η c ǫ, (28)
0
2 −
and I 1(x) is the modified Bessel function.
µ
−2
After inserting (27) into (16) and performing the integration over the
time variable S with the help of the following formula (see Gradshtein and
Ryzhik11 , p.729 , eq. (6.669.4))
e 2pq 1 √xy
∞dq − exp (x+y)cothq I
2γ
Z0 sinhq (cid:20)−2 (cid:21) sinhq!
Γ p+γ + 1
2
= M (x)W (y), (29)
Γ((cid:16)2γ +1)√x(cid:17)y −p,γ −p,γ
valid for Re p+γ + 1 > 0, Re(γ) > 0 and y > x, where M (x) and
2 p,γ
−
W (y) are(cid:16)the Whitt(cid:17)aker functions, the Green’s function (16) becomes
p,γ
−
Γ p+ µ + 1 Mω Mω
G(x,x′;E) = 2iωΓ(cid:16)(µ+212)√4x(cid:17)x′M−p,µ2−41( h¯ x′2)W−p,µ2−14( h¯ x2), (30)
e e
e
where p = E aned x > x.
′
−2¯hω
e
e
6
Eventually, by taking into account eqs. (30), (13), (8), (2) and the suc-
cessive transformations ξ = h(r) = x2, the Green’s function associated with
the confluent Natanzon potentials may be written:
1
Γ E + µ + 1 2
−2¯hω 2 4 R(r)R(r′) Mω
G(r,r′;E) = (cid:18)4iωeΓ(µ+ 21) (cid:19) qh(r)h(r′)  M−p,µ2−41( h¯ h(r′))
e e
Mω  
W−pe,µ2−41( h¯ h(r)), (31)
e
and the bound state energy eigenvalues E = ¯h2 ǫ can be obtained from the
n 2m n
poles of Γ E + µ + 1 in (31),
−2¯hω 2 4
(cid:18) (cid:19)
e
e E µ 1
+ + = n, n N (32)
0.
− 2h¯ω 2 4 − ∈
e
Upon substitution of (10), (15) and (28) into (32), we see then that
e
g σ ǫ
1 1
− +√η c ǫ = (2n+1) (33)
0
2√g σ ǫ − −
2 2
−
is an equation of fourth degree in ǫ which can be solved after some simple
n
manipulations12 .
Many particular cases of the potentials (1) arise when some of its param-
eters vanish. In the following table, we give the interesting ones, namely,
the radial harmonic oscillator, the radial Kepler-Coulomb potential and the
Morse potential
σ σ c h(r) V(r)
1 2 0
1
0 0 r2/σ1 σg11 + σg221r2 + η−r24
0 0 2r/√σ2 σg22 + 2√gσ12r + η4−r21
0 0 e r/√c0 g2e 2r/c0 + g1e r/√c0 + η
− c0 − c0 − c0
In conclusion, we have shown that the class of confluent Natanzon po-
tentials can be described by a particular realization of the so(2,1) which
enables us to work out the Green’s function. The realization of the so(2,1)
algebra used here differs substantially from that of Cordero and Salam´o3 .
The discrete spectrum is identical with the results obtained by using either
the (SGA) method3 or the Feynman path integral approach5 .
7
References
1G. N. Natanzon, Teor. Mat. Fiz. 38 (1979) 146.
2F. Cooper, J. Ginocchio and A. Kahare, Phys. Rev. D 36 (1987) 2458.
3P. Cordero and S. Salam´o, J. Phys. A: Math. Gen. 24 (1991) 5299; Con-
densed Matter Theories, 7 (1992) 49; Found. Phys.23 (1993) 675.
4P. Cordero and G. C. Ghirardi, Nuovo Cimento A 2 (1971) 217; Fortschr.
Phys. 20 (1972) 105.
5C. Grosche, J. Phys. A: Math. Gen. 29 (1996) L183.
6A. I. Milshtein and V. M. Strakhovenko, Phys. Lett. A 90 ( 1982 ) 447.
7B. G.Wybourne, Classical Groups for Physicists (Wiley, New York, 1974).
8J. Schwinger, Phys. Rev. 82 (1951) 664.
9L. Chetouani, L. Guechi and T. F. Hammann, Helv. Phys. Acta 65 (1992)
1069.
10L. Chetouani, L. Guechi and T. F. Hammann, J. Math. Phys. 33 (1992)
3410.
11I. S. Gradshtein and I. M. Ryzhik, Table of Integrals, Series and Products
( Academic, New York,1965 ).
12I. N. Bronshtein and K. A. Semendyayev, A Guide Book to Mathematics
(Harri Deutsch, Frankfurt/Main 1973).
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