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Notizen 651

Elementary Derivation

of the Dirac Equation. IX

H a n s S a l l h o f e r

Z. Naturforsch. 40a, 651 - 6 5 2 (1985);

received May 9, 1985

The transversality in the electrodynamic hydrogen field is discussed.

In [1] the a t t e m p t is m a d e to e n s u r e the o b l i g a t o r y transversality of t h e e l e c t r o d y n a m i c h y d r o g e n field t h r o u g h the r e q u i r e m e n t that t h e i n n e r p r o d u c t of the field with centrally directed vectors s h o u l d vanish because of its central s y m m e t r y . A n o t h e r p r o b l e m there was t h e e s t a b l i s h m e n t of t h e field c o m p o n e n t s h a v i n g t h e s a m e p h a s e velocity.

In contrast to this we will here a t t e m p t to e n s u r e the transversality f r o m the b e g i n n i n g t h r o u g h a s u i t a b l e ansatz. a n d to obtain t h e e q u a l i t y of t h e velocities of the field c o m p o n e n t s by c o m p l e t e l y e x h a u s t i n g the full p o t e n t i a l of t h e solutions.

As is well k n o w n , cov.ariant D i r a c - l i k e electro- d y n a m i c s

rot H = -E, c rot £ = - — / / .

c

d i v £ = 0 , (1) d i v / / = 0 .

E _L grad e, H _L grad //,

can, a c c o r d i n g to D a Silveira [2], w i t h the h e l p of the relation

( <t • V)(a-A) = \-A+itJ- (\xA) be b r o u g h t into t h e f o r m

£ ll © \ 0

(2)

tl) a a (D V -

//1 I 5 (ct) E _L grad e, H _L grad //.

i(o-E)

(o-H) = 0 ,

(3)

Reprint requests to Dr. H. Sallhofer, Fischerstrasse 12, A-5280 Braunau. Austria.

T h e n (3), or (1) respectively, has, a c c o r d i n g to [3] (6), t h e two complex solutions

V = [iE,, / ( £ , + / E2), H3, ( / / , + / H2)],

E X grad e, H _L grad / / ; (4)

«?=[/(£, - iE2),-iE3, ( / / , - iH2),-H3],

E _L grad e, H I . grad /.i. (5) T h e relations (4) and (5) can be rewritten as f o u r

systems of equations, an e x a m p l e of w h i c h is t h e system for the £ - f i e l d

y, = /£3, =iEl-E2, DE = i ei(p,

0 = E\ cos cp + E2 sin cp + E3 cot .9. (6) Analogous systems are o b t a i n e d for t h e t h r e e re-

m a i n i n g fields H. E and H. T h e solution of these f o u r systems yields the e l e c t r o m a g n e t i c fields (E, H) and ( £ , H) as follows:

+ V, cot ,9 + i*P2 sin <p

H

DE

+ / cot ,9 - i V2 cos (p DE

/ lF3 cot ,9 + sin <p

- W

(7)

DH

- V3 cot .9 - y4 cos (p Da

and

E = iiF] sin ( p - V 2 cot ,9

H

DE

- / cos ip + i*P2 cot ,9 DE

sin + / V4 cot ,9 DH

{V3 cos cp + cot ,9

D H - n (8)

0340-4811 / 85 / 0600-0649 $ 01.30/0. - Please order a reprint rather than making your own copy.

This work has been digitalized and published in 2013 by Verlag Zeitschrift für Naturforschung in cooperation with the Max Planck Society for the Advancement of Science under a Creative Commons Attribution-NoDerivs 3.0 Germany License.

On 01.01.2015 it is planned to change the License Conditions (the removal of the Creative Commons License condition “no derivative works”). This is to allow reuse in the area of future scientific usage.

Dieses Werk wurde im Jahr 2013 vom Verlag Zeitschrift für Naturforschung in Zusammenarbeit mit der Max-Planck-Gesellschaft zur Förderung der Wissenschaften e.V. digitalisiert und unter folgender Lizenz veröffentlicht:

Creative Commons Namensnennung-Keine Bearbeitung 3.0 Deutschland Lizenz.

Zum 01.01.2015 ist eine Anpassung der Lizenzbedingungen (Entfall der Creative Commons Lizenzbedingung „Keine Bearbeitung“) beabsichtigt, um eine Nachnutzung auch im Rahmen zukünftiger wissenschaftlicher Nutzungsformen zu ermöglichen.

(2)

652 Notizen Since V is d e t e r m i n e d u p to a complex constant f a c t o r only, t h e general solution is of the form

E

= g

E

+ g

E

and

H

= g

H

+ g

H

(9) {g, g : a r b i t r a r i l y complex).

E a n d H are now necessarily transverse and possess t h e h y d r o g e n f r e q u e n c i e s if e and // are specialized to the h y d r o g e n values according to [4] (12). T h e

solutions consist of a g g r e g a t e s w h i c h in turn are each c o m p o s e d of an e v e n - n u m b e r e d and an o d d - n u m b e r e d c o m p o n e n t of f . This m e a n s that, in each c o m p o n e n t of E and / / , a m p l i t u d e s with the azi- m u t h a l a n g u l a r f u n c t i o n s eimq> a n d ei(m+])(p a r e b r o u g h t t o g e t h e r . H e r e n o t only the equality of t h e velocity of t h e c o m p o n e n t s suggests itself, but also that the q u a n t u m n u m b e r m be half-integer.

[1] H. Sallhofer, Z. Naturforsch. 39 a, 692 (1984).

[2] Da Silveira. Z. Naturforsch. 34 a, 646 (1979).

[3] H. Sallhofer, Z. Naturforsch. 40 a, 94 (1985).

[4] H. Sallhofer, Z. Naturforsch. 33a, 1378 (1978).

Corrigendum zu [3]:

The superscript k in {Vk is to be replaced by h (for hermitian conjugate) everywhere, yielding V'1. In (4), (5), (7), (8), (10). and (15) "replace y<*> by y(4\ yk by y4, and dk by S4. The right hand side of (6) is to be understood as a matrix also.

N a c h d r u c k — a u c h auszugsweise — nur mit schriftlicher G e n e h m i g u n g des Verlages gestattet Verantwortlich für den Inhalt: A. KLEMM

Satz und D r u c k : K o n r a d Triltsch. W ü r z b u r g

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