A search on Dirac equation

Abstract

The solutions, in terms of orthogonal polynomials, of Dirac equation with analytically solvable potentials are investigated within a novel formalism by transforming the relativistic equation into a Schrodinger like one. Earlier results are discussed in a unified framework and certain solutions of a large class of potentials are given. Comment: 9 pages

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arXiv:quant-ph/0702151v2 16 Feb 2007
A search on Dirac equation
M Ko¸ cak and B G¨ on¨ ul
Department of Engineering Physics, Faculty of Engineering,
University of Gaziantep, 27310 Gaziantep -Turkey
The solutions, in terms of orthogonal polynomials, of Dirac equation with an-
alytically solvable potentials are investigated within a novel formalism by trans-
forming the relativistic equation into a Schr¨ odinger like one. Earlier results are
discussed in a unified framework and certain solutions of a large class of poten-
tials are given.
Keywords:Dirac equation
PACS No: 03.65.Fd
1 Introduction
A new algebraic technique for solving Schr¨ odinger and Klein-Gordon equations
[1], and the related other works therein, has been introduced recently and used
to search many interesting problems in different disciplines of physics. These
works have clarified the power of the suggested model when compared the results
obtained with those provided by other analytical methods in the literature. Nev-
ertheless, this formalism involves a deficiency in its present form which requires,
to express the excited state wave functions, the application of linear operators
on the ground state wave function appeared automatically in the mathematical
framework. This is indeed a cumbersome procedure though it provides explicit
expressions for the state functions having non-zero angular momenta.
To remove this drawback inherent in the formalism used in our previous works
[1], we suggest here an alternative scheme, unifying the spirit of the two theoret-
ical models [1, 2], to work out relativistic/non-relativistic quantum mechanical
problems analytically in a unified framework. This is the main motivation be-
hind the work presented in this article which in particular focuses on the solution
of Dirac equation since recently considerable attention has been paid to exactly
solvable Dirac equations.
The arrangement of this article is as follows. In the next section, a brief
introduction of the usual Dirac formalism and its treatment within the frame of
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new scheme are presented. Third section involves application results. Finally,
the results are summarized in the concluding section.
2 Theoretical Consideration
To proceed, let us first remind the mathematical frame of Dirac formalism which
is discussed briefly in the following section. Section 2.2 then illustrates the for-
malism of the new model.
2.1 Background on Dirac equation
Dirac equation for scalar and vector potentials is given by [3] (¯ h = c = 1)
HΨ = {ˆ α.ˆ p + β(m + VS) + VV}Ψ (1)
where ˆ p is the momentum operator, m is the rest mass of the particle, VS and
VV are scalar and vector potentials respectively and ˆ α, β are Pauli matrices.
To separate angular part of Eq. (1) from the radial part one follows
Ψℓ
jm=
?
iGℓj
rϕℓ
Fℓj
r
jm
ˆ σ.ˆ r
rϕℓ
jm
?
(2)
where ˆ σ represents the Pauli spin matrices while Gℓjand Fℓjare the radial com-
ponents
?
G−
j
j = ℓ − 1/2
and ϕℓ
?
ϕ−
jm
j = ℓ − 1/2
Then, using the standard relations
Gℓj=
G+
j
j = ℓ + 1/2
?
,Fℓj=
?
F+
j
F−
j
j = ℓ + 1/2
j = ℓ − 1/2
?
jmis the angular part of the wave function
ϕℓ
jm=
ϕ+
jm
j = ℓ + 1/2
?
.
ˆ σ.ˆ pg(r)
r
ϕℓ
jm= −i
r(dg
dr+kg
r)ˆ σ.ˆ r
r2ϕℓ
jm
(3)
and
ˆ σ.ˆ pˆ σ.ˆ r
r2
g(r)
r
ϕℓ
jm= −i
r(dg
dr−kg
r)ϕℓ
jm
(4)
where
k =
?
−(ℓ + 1) = −(j + 1/2)
+ℓ = +(j + 1/2)
Hence, the radial equations appear as
j = ℓ + 1/2
j = ℓ − 1/2
?
and g(r) is an arbitrary function.
−dF(r)
dr
+k
rF(r) = (ε − m − VS− VV)G(r) (5)
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dG(r)
dr
+k
rG(r) = (ε + m + VS− VV)F(r),(6)
where ε is the total relativistic energy of the system. From (5) and (6), omitting
the derivatives of VSand VV, together with the elimination of one radial compo-
nent (F(r)), one obtains a Schr¨ odinger like equation for the other component of
the relativistic wave function
?
−d2
dr2+k(k + 1)
r2
+ (V2
S− V2
V) + (2mVS+ 2εVV)
?
G = (ε2− m2)G.(7)
2.2Formalism
Exact solutions of systems in physics has a great importance. To provide such
solutions recently a new method has been carried out successfully in our earlier
works [1], which unfortunately has a considerable algebraic difficulty in the cal-
culation of excited state functions. To overcome this tedious procedure in the
calculations, we propose here to use a new scheme involving orthogonal polyno-
mials in order to express all bound state wave functions in an explicit form.
For this purpose, we start from Eq.(7) which can be defined as
G′′(r)
G(r)
= V (r) − E , (8)
where V (r) =k(k+1)
known, the solution of (8) generally takes the form
r2
+ (V2
S− V2
V) + (2mVS+ 2εVV) and E = ε2− m2. As is well
G(r) = f(r)F[s(r)]. (9)
The substitution of (9) into (8) yields the second-order differential equation
?f′′
f
+F′′s′2
F
+s′′F′
F
+2F′s′f′
Ff
?
= V − E ,(10)
and rearranging (10) for a more useful form, one gets
F′′+
?s′′
s′2+ 2f′
s′f
?
F′+
?f′′
s′2f+E − V
s′2
?
F = 0 .(11)
Eq. (11) is in the form of the most familiar second-order differential equations
to the hypergeometric type [2],
F′′(s) +τ(s)
σ(s)F′(s) +˜ σ(s)
σ2(s)F(s) = 0 , (12)
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where σ and ˜ σ are at most second degree polynomials, and τ is a first degree
polynomial. The form of
F(s) [2]. From (12), it follows that
τ(s)
σ(s)and
˜ σ(s)
σ2(s)is well defined for any special function
s′′
s′2+ 2f′
s′f=τ(s)
σ(s),
f′′
fs′2+E − V
s′2
=
˜ σ
σ2.(13)
From the previous works in [1], the energy and potential terms in (13) can be
decomposed in two pieces, which provides a clear understanding for the individual
contributions of the F and f terms to the whole of the solutions, such that
E − V = (Ef+ EF) − (Vf+ VF). Therefore, the second equality in (13) is
transformed to a couple of equation
f′′
f
= Vf− Ef,−˜ σ
σ2s′2= VF− EF ,(14)
where f can be expressed in an explicit form due to the first part in (13)
f(r) = (s′)−1/2exp
?1
2
?s(r)τ(s)
σ(s)ds
?
.(15)
Since the corresponding σ, ˜ σ and τ terms are well known for a given polynomial
(F), the transformation function (s) in (14), and afterwards f in (15), are easily
defined. So, from (9), the corresponding total wave function is readily obtained
for the whole spectrum.
The potential and total energy terms for Dirac equation in this case
f′′
f
= Vf− Ef,Vf= 2mVS+ 2εVV+k(k + 1)
r2
, (16)
−˜ σ
σ2s′2= VF− EF,VF= V2
S− V2
V,(17)
and
Ef+ EF= ε2− m2.(18)
To understand how efficiently this method works, some physically possible
potentials are solved in the following section to obtain their eigenvalues and
eigenfunctions, within the frame of the present formalism.
3 Application
As illustrative examples, here we deal with the two well known problems of the
literature: Dirac oscillator and Dirac-Coulomb problem. The other solutions are
shown in Table 1.
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In order to get exact solutions for the present consideration, one needs to
start with choosing a physically plausible equal magnitudes for the vector and
scalar potentials. Otherwise, the system considered becomes quasi-exactly solv-
able which is out of the scope of the present article.
Setting VV = VS = ar2for the relativistic treatment of oscillator problem
right hand sides of equations (16) and (17) gives
Vf= 2a(m + ε)r2+k(k + 1)
r2
,VF= 0 , (19)
Concentrating on the generalized Laguerre polynomials Lα
fluent hypergeometric functions, one sees that [2]
n(s) related to con-
σ = s ,τ = α + 1 − s ,˜ σ = nσ . (20)
Keeping in mind that the right-hand sides of (14) provide a three dimensional har-
monic oscillator potential, one obviously realizes thats′2
s =1
s= 2w and consequently
2ωr2. Then, substituting (20) into (15) it is not hard to see that
f = Crα+1/2e
−ωr2
4
,(21)
where C =
1
√2(ω
2)
α
2. This makes possible to predict Vfand Efas
f′′
f
= Vf− Ef,Ef= (α + 1)ω ,
Vf=1
4ω2r2+(α − 1/2)(α + 1/2)
r2
, (22)
where α = −(k + 1/2) = ℓ +1
one should consider Eq. (17). After some simple algebra we find
2for the case j = ℓ +1
2. To find also VF and EF,
VF= 0 ,EF= 2nω .(23)
Thus, in the non-relativistic limit, the full energy spectrum and wave functions
for the system of interest are gives as
E = Ef+ EF= (α + 1 + 2n)ω = (2n + ℓ + 3/2)ω ,
Ψ = fF = Cs
(ℓ+1)
2 e−s
2L
(ℓ+1
n
2)
(s) . (24)
Finally, the relativistic energy of Dirac oscillator reads
ε2= m2+ (2n + ℓ + 3/2)ω. (25)
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