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Exactly solvable potentials and
Romanovski polynomials in quantum
mechanics.
David Edwin Álvarez Castillo
July 16, 2008
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Classical Orthogonal Polynomials
•Legendre
•Laguerre
•Hermite
•Chebyshev
•Gegenbauer
Adrien Marie
Legendre
1752 - 1833
Edmond Nicolas
Laguerre
1834 - 1886
Charles
Hermite
1822-1901
•Jacobi
Pafnuty Lvovich
Chebyshev
1821 - 1894
Carl Gustav Jacobi
Leopold Bernhard
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Gegenbauer 1849 - 1903 Jacobi 1804 - 1851
Exactly solvable potentials in
quantum mechanics
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Hyperbolic Scarf Potential
Vh (z) = a2 + (b2 ¡ a2 ¡ a®)sech2 (®z) + b(2a + ®)sech(®z)t anh(®z);
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Solution
Self-adjoint form
d
(¾(x)w(x)) = ¿(x)w(x) ;
dx
in terms of the Rodrigues formula
N n dn
yn (x) =
[w(x)¾n (x)] :
w(x) dx n
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Real solutions in terms of Romanovski polynomials:
R0(1;¡ 1) (x) =
R1(1;¡ 1) (x) =
R2(1;¡ 1) (x) =
R3(1;¡ 1) (x) =
R(1;¡ 1) (x) =
4
1
¡ 1 ¡ 9x
¡ 6+ 16x + 56x2
16 + 84x ¡ 126x2 ¡ 210x3
20 ¡ 240x ¡ 360x2 + 480x3 + 360x4
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4.3 Romanovski Polynomials and
the non spherical angular functions
Consider an electron in the following
potential
V2 (µ)
V (r; µ) = V1 (r ) +
;
r2
V2 (µ) = ¡ ccot (µ) ;
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The angular equation from the SE has the
solution
Ãn = l (l + 1)¡
m
(¡ cot (µ)) =
(1 + cot (µ) 2 ) ¡
l ( l + 1)
2
e¡
l ( l + 1) t an ¡
1 (¡
(
)
cot ( µ)) R l (l + 1) + 12 ;¡ 2l (l + 1) (¡
l (l + 1) ¡ m
cot (µ)):
The total wave function is
Z lm (µ; ' ) = Ãn = l (l + 1)¡
(1 + cot (µ) 2 ) ¡
l ( l + 1)
2
e¡
im' =
(¡
cot
(µ))e
m
l ( l + 1) t an ¡
1 (¡
(
)
cot ( µ) ) R l ( l + 1) + 12 ;¡ 2l ( l + 1) (¡
l (l + 1)¡ m
cot (µ))ei m '
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Relation between the associated Legendre functions and
Romanovski polynomials if c=0 (central potential)
P m (cos(µ)) = const (1 + cot 2 (µ)) ¡
l
l
2
R ( 0;¡
m+ l
l ) (¡
cot(µ))
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Spherical Harmonics
VS
non spherical angular functions
j Y 0 (µ; ' ) j
0
j Z 0 (µ; ' ) j
0
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j Y 0 (µ; ' ) j
j Z 0 (µ; ' ) j
j Y 1 (µ; ' ) j
j Z 1 (µ; ' ) j
1
1
1
1
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j Y 0 (µ; ' ) j
j Z 0 (µ; ' ) j
j Y 1 (µ; ' ) j
j Z 1 (µ; ' ) j
2
2
2
2
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Romanovski polynomials in the
trigonometric Rosen-Morse
vt R M
1
(z) = ¡ 2bcot z + l(l + 1)
;
sin2 (z)
² n = (n + l + 1) 2 ¡
r
z= ;
d
b2
( n + l + 1) 2
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A taylor expansion shows
2b 2b
l(l + 1)
l(l + 1)
v(z) t R M ¼ ¡
+
z+
+
z2 + :::
z
3
z2
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•First term: Coulomb.
•Second term: linear confinement.
•Third term: standard centrifugal barrier.
In this sense, Rosen-Morse I can be viewed as the
image of space-like gluon propagation in coordinate
space.*
*Compean, Kirchbach (2006).
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Advantages of the RMt over the Coulomb potential +
lineal (QCD):
•Dynamical symmetry O(4),
•Exact solutions,
•Good description of nucleon’s spectrum.
Ãn
(cot ¡ 1
x) = (1 +
Cn(¡ (n+ l ) ; n2+bl ) (x)
´
x 2)¡
Rn(pn ;qn ) (x);
n+ l
2
e¡
b
n+ l
cot ¡
1 (x)
(¡
Cn
(n + l);
2b
n+ l
)
(x) ;
2b
; pn = (n + l ); n = 1; 2; :::
qn = ¡
n+ l
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Baryon resonances in the traditional quark model. Circles, bricks, and triangles stand for nucleon, ¤, and ¢ states, respect ively. Di®erent colors mark
di®erent SU(6)SF £ O(3)L multiplets. Noticet hestrong multiplet intertwining
and t he largemass separat ion insidet hemult iplets. (Courtesy M. Kirchbach)
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The nucleon excitation spectrum below 2 GeV.
(Courtesy M. Kirchbach)
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The ¢ excitation spectrum below 2 GeV. (Courtesy M. Kirchbach)
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Summary
The Romanovski polynomials appear as the solution of
the Schrodinger equation for the Hyperbolic Scarf
Potential and the Rosen-Morse trigonometric.
They define new non-spherical angular functions.
The Romanovski polynomials are the main designers of
non--spherical angular functions of a new type, which we
identified with components of the eigenvectors of the
infinite discrete unitary SU(1,1) representation,
( m 0= l ( l + 1) + 1 ) (µ; ' )g.
f D+
2
j = m+
References:
quant-ph/0603122
arXiv:0706.3897
quant-ph/0603232
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