Exact solutions of Dirac equation
S V K U L K A R N I and L K S H A R M A
Department of Applied Physics, Government Engineering College, Jabalpur 482 011 MS received 9 November 1978; revised 30 January 1979
Abstract. In this paper Dirac equation for two electromagnetic potentials viz vector potential and scalar potential have been solved. These solutions of the Dirac equa- tion are written in terms of known solutions of the SchrOdinger equation. The pre- sentation is within the two-component relativistic description. Mainly the bound state solutions have been obtained.
Keywords. Two component equation; bound state solution; K.ratzer's potential;
Dirac equation.
1. Introduction
Recent investigation on the intense electron beams ( R a n d e r et al 1970) and the pre- sence o f other configurations o f parallel electric and magnetic fields (Occhionero and Demianski 1969) in astrophysical problems where relativistic particles are likely to exist (Datlowe and M e y e r 1970) have led to the necessity f o r a clear understanding o f a relativistic q u a n t u m particle in external electromagnetic fields. This can be readily accomplished i f we know the exact solutions. A careful literature survey reveals that very few solvable configurations have been found since the f o r m u l a t i o n o f the Dirac equation. T h e i m p o r t a n t amongst them are: C o u l o m b potential (Dirac 1928), a constant magnetic field (Rabi 1928), a constant electric field (Sauter 1931), the field o f a plane wave (Volkov 1935), the field o f a plane wave with a constant magnetic field parallel to the direction o f propagation o f plane wave ( R e d m o n d 1965), four cases in which the electromagnetic potentials assume p a r t i c u l a r functional depend- e n c e on the space coordinates (Stanciu 1966) and the one where the electric and magnetic fields are crossed (Larn 1970). Recently a few m o r e exact solutions o f the Dirac equations have been found. Franklin and M a r b u r g e r (1975) have derived a new class o f exact solutions which is applicable to variety o f fields, including electro- magnetic waves f o r which the length o f the wave four-vector is non-zero. Critchtield (1976) has generalised the D i r a c equation in a central field to include the scalar poten- tial p r o p o r t i o n a l to r and r -1. Barut and Kraus (1976) have solved the Dirac equa- tion with the C o u l o m b potential plus the additional interaction due to the anomalous magnetic m o m e n t o f the electron in the Coulomb field.
T h e aim o f this p a p e r is to present the exact solutions for the electromagnetic potentials which assume particular functional dependance o n the space co-ordinate.
Uncoupling o f the four equations implied by the Dirae e q u a t i o n is effected through the introduction o f a two c o m p o n e n t spinor. The Dirac equation in the Pauli repre- sentation is
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[Ys (P+eA)~ + iy,
(IV+eV)--im] ~F
= 0 , (1)whore yj=p~o, )'4=Pa, p's and o's have tbeir usual meanings.
The rolativistically invariant two-component equation for an electron in the external electromagnetic field is written as (Foynman and Gell-Mann 1958)
[(P+eA) ~" + m 2 + e a o'- ( H + i E ) ] ~' =
(W+ev) ~ ~F.
(2) It is sufficient to solve this two component equation. The four component spinors which are solutions of the Dirac equation (I) are generated from the solutions of(2) by~b~ = ( [ , . ( P + e A ) + ( W + e v ) + m ] ; ) .
(3) [o" ( P + e A ) +(W+ev)--m]
Sections 2 and 3 deal with the solution o f two component equation (2) for the particular configurations of the vector and scalar potentials.
2. Magnetic field solution
It is observed that the two component equation (2)becomes very simple when the scalar field V is zero and if we choose the magnetic field to have only a z-component depending only on one co-ordinate, say y. This external field along with the asymmetric choice o f the gauge, Ay----Az = 0 , transforms (2) into
[P,2+ P ~ + [(Px+eAx(y)]2+m'+esH~(y)] ~F,X,= Bra~FXj.
(4) The spin index s assumes the values + 1 and --1 corresponding to the spinorsX l = [ 0 [ andX-1---1011 'respectiv°ly"
In (4), the variables Px and P, are constants o f motion and hence can be taken as constants. After suppressing piano wave dependance on x and z, q~, is only a func- tion o f y and then (4) is in the canonical form o f one-dimensional Schr6dinger equa- tion. With the proper choice of Ax(y), (4) can be made equivalent to Schr6dinger equation with the solvable potential.
We make the following choice o f the vector potential
A~O, ) = _ HolY. (5)
The corresponding magnetic field is given by
H,(r) = Hdy', (6)
where H, is a positive quantity.
With this choice, (4) takes the form
[p)2 _2et~oPx + erie (etI°q-S)y, J] ~,x s = (W~--Px2--P~2--m z) ~,x,. (7)
It is seen that (7) is formally equivalent to a one-dimensional Schr6dinger equation with
K.ratzer (1920)
potential.[ dd-~z-- "~ + 272 y
with the notations:
W, satisfies the differential equation
A(A--1)] ? jr, =0, (8)
~2=P x2 q- P~2 q- m ~-- W ~,
y2=eHoP,, (9)
and A given by the equation
~(~-- l) = eHo(eHoq-s ). (10)
We take
only the positive solution o f this equation,i.e.
~t = ½ q- [erie(ell o q-s) q- ¼] 1/2, (11)
and exclude the negative solution as it will lead to a physically unacceptable solution for (8). The differential equation 1,8) has an irregular singularity at y = co where its normalisable solutions in bound state behaves as exp(--Ey). It further has a singularity at y = 0 where ~ , oc y~. Therefore, it is reasonable to set
, :ya(e-,Y) • fs(Y). (12)
With this substitution (8) leads to
YA"(Y) q-(2~t--2 ~Y)A'(Y) q-(27~--2a)0A(Y) =0. (13)
This can be further transformed into Kummer's equation with the change o f variable z=2~y. The equation thus obtained is
zf'(z)+(2X--z)ff(z)q-(~'--?t)f(z) =0. (14)
The solution o f (13) is given in terms o f confluent hypor-goometric function i.e.,
FI( 7~ )
A(Y)---1 A - - - - , 2h; 2~y (15)
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and thus, solution (12) becomes~ , = ya(e-~Y)tF 1 ~ - - y ,
23.; 2Ey . (16)The regularity conditions imply that bound states exist only if
A . . . . n; n=O, 1, 2 . . .
7 ~ ~ > 0 (17)The energy eigenvalues are obtained from condition (16) and are given by:
. / . , ~ \ 2
W~.s=Px'-k-Pz~+m"-- t n ~ ) ,
(18)where ~, and A are given by (9) and (11).
The scattering solutions are not of much interest. However, the solutions obtained for pure imaginary value of ~ are in the non-rdativistic form o f the Coulomb scatter- ing. To show this we first obtain the solutions in terms of Whittakar functions.
To transform (14) into Whittakar equation, we set A(z) = z "-;t exp (z/2)
F,(z)
A - - v ' / , : ½Kq-/* (19)
2a --- 1+2/,
( 1
and get F~" (z) + -- a + F~(z) = 0. (20)
Z Z 2 ]
The general solution (12) then becomes
[ (,' ,)
v~ = (2,)-a P(2,yl~V, • exp ( - , y ) ~Fo - 7 + 2 1 , - ~, + 1; - g - y
4-Q(--2q)-:"/~ • exp ( q ) ~F o - £ , 1--2;~+ _~ ; . (21)
For pure imaginary value o f
~=--i~,
(21) takes the form:V , = ( - - 2 i ~ ) -a
(--2iny)-Cli~
exp( q-i~y) zF a °r2A, l q - ~ ;
+ Q (2i~7y)+'/'1l~
exp(--i~Ty) zFo -]~--,
1--23. 177; ~ (22)For studying the scattering, we find the behaviour of ~ , at y =--I-oo,
~F~ = (--2i~)-~[P(--2i~Ty)-y'/i~ exp (q-i~Ty)q-Q(2i~y)y*/i~
y - , i c e
exp (--i~y)]. (23)
Setting ~ ___ y + (y~/~2) log y (24)
gives (y)C/i,1 = exp [--ir/(~:--y)]
and (y)~'li,1 = exp [(iT (~--Y)]
Thus (23) takes the form
R/, = (--2in) -a [P exp (i7/~:) (--2i~) 'y'/i~
y ----> Z~ ¢~
÷ Q exp ( - - i ~ O (2i~7)~'~1i'I]. (25)
The asymptotic solution (25) has evidently the form of Coulomb scattering solution.
3. The electric field solution
For the external field configuration and a scalar potential depending upon only one direction, say z, (2) can be written in the following form:
[p,a + m 2 + ise Ez] lFs X, = ( W + eV) 2 ~FsX,. (26)
Comparison o f (26) with (4) shows that they are equivalent. Taking the scalar potential in the form
V ( z ) = - - Eo/z (27)
(26) is transformed to
P " + -~ (e" E°'--ise E°) + l ( 2 e z W E ° ) ~ , Z,, =
• [ I V ' - - P ~ ' - - P , ' - - m ' ] ~F, X~. (28)
After relabelling the variables and the constants, this equation becomes identical to (8). So the scalar potential (27) gives rise to an equation which is formally equivalent to a one-dimensional Schr6dinger equation with Kratzer's potential.
4 8 0 S V Kulkarni a n d L K S h a r m a References
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