Commun. Theor. Phys. (Beijing, China) 53 (2010) pp. 242–246
c
Chinese Physical Society and IOP Publishing Ltd Vol. 53, No. 2, February 15, 2010
Solving Dirac Equation with New Ring-Shaped Non-Spherical Harmonic Oscillator
Potential
∗
HU Xian-Quan (
k
),
†
LUO Guang (
Û
1
), WU Zhi-Min (
È
¬
), NIU Lian-B in (
Ú
ë
R
),
and MA Yan (
ê
ÿ
)
College of Physics and Technology Information, Chongqing Normal University, Chongqing 400047, China
(Received January 15, 2009; revised manuscript received August 24, 2009)
Abstract A new ring-shaped non-harmonic oscillator potential is proposed. The precise bound solution of Dirac
equation with the potential is gained when the scalar potential is equal to the vector potential. The angular equation
and radial equation are obtained through the variable separation method. The results indicate that the normalized angle
wave function can be expressed with the generalized associated-Legendre polynomial, and the normalized radial wave
function can be expressed with conflu ent hypergeometric function. And then the precise energy spectrum equations are
obtained. The ground state and several low excited states of the system are solved. And those results are compared
with the non-relativistic effect energy level in Phys. Lett. A 340 (2005) 94. The positive energy states of system are
discussed and the conclusions are made properly.
PACS numbers: 03.65.Ge, 0 2.30.Jr , 31.15.-p
Key words: ring-shaped non-harmonic oscillator potential, Dirac equation, bound state, genera lize d
associated-Le gendre function
1 Introduction
In strong coupling, the relativistic effect be-
comes very important for particles moving in poten-
tial fields. And when the relativistic effect is con-
sidered, the moving particles need to be expressed
by Klein–Gordon equation or Dirac equation.
[1]
In
the past researches, the bound solution is obtained
when the potential function is Hulth´en potential,
[2−5]
Hartmann potential,
[5−9]
P¨oschl–Teller potential,
[10−11]
Wood–Sa xon potential,
[12]
Kratzer potential,
[13]
non-
reflection potential,
[14]
tan
2
(πηγ) potential,
[15]
Rosen–
Morse potential,
[16]
non-spherical harmonic oscillator
potential,
[17−19]
Makarov potential
[20]
or ring-shaped os-
cillator potential
[21]
and on condition that the scalar po-
tential is equal to or greater than the vector po tential.
Those potential have been develop e d to study the rel-
ativistic effect of the complex vibration-rotation energy
structure of multi-electron atom, multi-atom molecular,
the interaction between the ring-shaped molecules, the
distorted nucleus, the correlation states of quantum fluid
system, and the energy band structure with quantum dot
resonance tunneling effect. Most of them are non-central
potential models.
The ring-shaped molecular potentials are non-central
potentials, composed of spherical harmonic oscillator po-
tentials, and other additional potentials. Such potential
functions can be used to describ e ring-shaped molecular
(such as benzene molecular) model a nd the interaction
between the distorted nucleus. They ar e widely used
in quantum chemistry and nuclear physics. In recent
years, the relativistic effect of micro-particles in the non-
spherical harmonic oscillator potential fields has ar oused
great interest in physics, and some s ignificant res ults have
been acquired. Those researches involve typical ring-
shaped non-spherical harmonic oscillator potentials,
[22−23]
new ring-shaped coulomb potentials,
[23]
coulomb poten-
tials plus new ring-shaped potentials,
[24]
mixture poten-
tials composed of harmonic oscilla tor potentials plus neg-
ative second power function potentials,
[18−19]
Hartmann
potential plus new r ing-shaped potential,
[8−9]
etc. The
characteristics of those potential functions a re coulomb
potential plus one or two non-central potentials and then
solutions are acquired with variable separation method
and spec ial function theory. In this article, a new kind of
ring-shaped non-spherical harmonic oscillator potentials
is pr oposed. Its potential is compos ed of the superposi-
tion between the harmonic oscillator potential and nega-
tive second power function potentials, and then, plus two
terms of non-central potentials, altogether fo ur potential
functions. Its expression is
V (r, θ) =
K
2
r
2
+
A
r
2
+
β
r
2
sin
2
θ
+
γ cos
2
θ
r
2
sin
2
θ
, (1)
where K, A, β, γ are dimensionless real parameters. Tak-
ing into account the relativistic effect, the bo und state
solutions of Dirac equation according to Eq. (1) are ob-
tained by separation of va riables method and special func-
tion theory when the scalar potential is equal to the vector
potential, i.e. the angular a nd radial wave function and
the precise energy spe c trum equations of the fermions are
achieved. The ground state and s e veral low excited states
of the system are solved. And those results are compared
with the non-relativistic effect energy level in Ref. [29].
∗
Supported by the National Natural Science Foundation of China under Grant No. 60806047, and the Basic Research of Chongqing
Education Committee under Grant No. KJ060813
†
E-mail: huxquan2003@yahoo.com.cn
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