Solving the Dirac Equation: Using Fourier Transform

Transcription

1 McNa Schola Reeach Jounal Volume Atcle Solvng the ac quaton: Ung oue Tanfom Vncent P. Bell mby-rddle Aeonautcal Unvety, ollow th and addtonal wok at: Recommended Ctaton Bell, Vncent P. "Solvng the ac quaton: Ung oue Tanfom," McNa Schola Reeach Jounal: Vol., Atcle. Avalable at: Th Atcle bought to you fo fee and open acce by the Jounal at Scholaly Common. It ha been accepted fo ncluon n McNa Schola Reeach Jounal by an authozed admntato of Scholaly Common. o moe nfomaton, pleae contact common@eau.edu.

2 Bell: Solvng the ac quaton: Ung oue Tanfom Vncent Bell Solvng the ac quaton: Ung oue Tanfom Vncent Bell, mby-rddle Aeonautcal Unvety Abtact When lookng at the pn of a patcle, the ac quaton ued to explan the gnfcance of the pn. Hee we olve ac quaton ung the oue Tanfom, a well a explan and defne the tem of the ac quaton. In addton, the ac Hamltonan change n Quantum Mechanc ae examned. Thee wll be a dcuon of what the alpha matce and beta matx ae made up of. The end eult defnng the ac quaton fo a patcle n an electomagnetc feld and olvng th equaton. Intoducton The ac quaton a elatvtc quantum wave equaton whch wa fomulated by Paul Aden Mauce ac n 98. The ac quaton ued fo the decpton of elementay pn ½ patcle, fo example electon. The equaton demand the extence of antpatcle and actually pedated the expemental dcovey. Th made the dcovey of the poton, the antpatcle of the electon, one of the geatet tumph of moden theoetcal phyc. The ac quaton look a follow: Whee the H H Ψ Ψ ac Hamltonan. Th Hamltonan gven by: H c ˆ P + β,, Whee the alpha matx, ˆ, and the beta matx,β, ae x matce. The ˆ cont of the Paul Matce. Paul Matce wll be dcued late. The momentum, p, a - vecto n the X, Y, Z decton, nomally een n the Catean coodnate ytem. The momentum gven by: P p, p, p [ ] x The momentum n quantum mechanc epeented a opeato and then become: y z Publhed by Scholaly Common,

3 McNa Schola Reeach Jounal, Vol. [], At. P h whee,, x y z And whee h ac contant, whch Planck contant dvded by π. o elatvtc effect to occu the patcle ha to be tavellng cloe to o a facton of the peed of lght, whch degnated by c. And, m the ma of the patcle. The momentum a vecto and the dot poduct beng done between the momentum andˆ. Alo ˆ n dmenon ut lke the momentum. Now ac Hamltonan take the fom of: H c ˆ h + β 5 6 H h c ˆ ˆ ˆ β x y z Now the teady tate ac quaton denoted by: + h c ˆ + ˆ + ˆ β Ψ 7 x y z Ψ The above equaton alo known a tme ndependent ac quaton. But the tme dependent ac quaton, the total enegy,, become an opeato n quantum mechanc. The opeato look a follow: h 8 om ung equaton 7 and 8, the tme dependent ac quaton become: + h c ˆ + ˆ + ˆ βψ x y z Ψ h 9 Snce Paul Matce and Beta Matx ˆ cont of Paul Matce, the Paul Matce ae:

4 The ˆ ha pat, a hown n equaton, and contucted fom the Paul Matce. So, the -matce ae a follow: ˆ ˆ ˆ Whee epeent the null matx that cont of a x matx wth all zeo. Th make the alpha matce a x matx, whch govened by: ˆ ˆ ˆ Smlaly to the -matce, beta matx,β, cont of the dentty matx. I I β β Snce the alpha matce and beta matx ae both x matce, theefoe the wave functon,ψ, ha to be a -component pno. Th becaue the wave functon cont of the patcle pn ½ up and down, made up of two component. But the othe two component ae made up of the ant-patcle pn, ½ up and down. The - component pno : Ψ ψ ψ ψ ψ 5 lectomagnetc eld ac quaton When a patcle n an electomagnetc feld, thee ae tem that mut be accounted fo. ue to the nteacton of the electomagnetc feld, the enegy and Bell: Solvng the ac quaton: Ung oue Tanfom Publhed by Scholaly Common,

5 McNa Schola Reeach Jounal, Vol. [], At. momentum ae changed lghtly. The genealzed ac quaton obtaned by makng the followng ubttuton: h qφ 6 P h qa 7 Whee q the chage of the patcle, φ the electc cala potental, and A the magnetc vecto potental. Th vecto ha X, Y, and Z component. The dot poduct of the A and the alpha matce ut lke the dot poduct of momentum and the alpha matce a explaned befoe. Th o that the ft alpha matx goe wth the X component of the momentum and the X component of the magnetc vecto potental. The magnetc vecto potental, A, known a: A A, A, A 8 [ ] x Wth th change the ac quaton fo a patcle n an electomagnetc feld become: [ h c ˆ cq ˆ A+ β ]Ψ 9 Ψ h qφψ quaton 9 can be changed to whee the ac quaton become: V [ hc ˆ + β ] c A Ψ Ψ+ VΨ h q φ ˆ y z Soluton Ung oue Tanfom om lookng at the teady tate tme ndependent ac quaton fo a patcle n an electomagnetc feld, you can ue oue tanfom to get the oluton. Statng wth: VΨ [ + h c ˆ β ]Ψ The - oue tanfom then become: x ψ e d x / π whee,,,

6 Next the gadent of the - oue tanfom developed nto: om pluggng equaton nto equaton, the ac quaton then can be conveted nto: [ ] ˆ c h β 5 x d e v x / π 6 Snce the wave functon the component pno, then and alo have to be component. Thee then a lot of algeba that ultmately lead to: + + c h 7 Th ytem can be mplfed down to a matx tme, whch look a follow: 8 c h om takng the nvee matx, equaton 8 become: 5 Bell: Solvng the ac quaton: Ung oue Tanfom Publhed by Scholaly Common,

7 McNa Schola Reeach Jounal, Vol. [], At. o to have an nvee, the detemnant of mut ext. If not then th oluton doe not wok. Ung the nvee - oue tanfom, the oluton can be tuned nto: ψ ψ ψ ψ x π / e d Concluon The ac quaton fo a patcle n an electomagnetc feld how u how the patcle affected by the electc feld and the magnetc feld. Alo the ac equaton tell how the feld affect the pn and the polaty of the patcle telf. om th oluton, one can model the patcle and place dffeent potental enege nto the equaton. Thee ae model that ae cuently beng bult to model how the patcle would eact n Gauan potental enegy. A emnde that the matx ha to have a detemnant fo the oue oluton to wok at all. If doe not have a detemnant, then the oluton would have to be found anothe way. But hould alway be able to have a detemnant; theefoe, a oluton wll alway be able to be obtaned. A Couple of thng that ae next ae to look at ome of the numecal oluton of, ft, the - oue oluton, then - and - numecal oluton, a well a lookng at the ac quaton n phecal coodnate ntead of Catean coodnate, a looked at hee. Acknowledgement t want to thank. Beeket Behane and. Tmothy Smth fo the gudance and tanng on the topc. Alo fo puttng n the tme to help me when I need and beng thee fo me dung th eeach expeence. I would alo lke to thank mby-rddle Aeonautcal Unvety McNa Pogam and the McNa Staff,. Manda McBde and M. Paula Reed, fo allowng me to take pat n th pecal pogam. Th pogam ha and contnue to allow me to go fathe than what I could whle n undegaduate. 6

8 Bell: Solvng the ac quaton: Ung oue Tanfom Alo thank to the othe phyc teache at mby-rddle that put n tme and/o allowed me to ue the book fo th eeach. Refeence. Gene, W., Relatvtc Quantum Mechanc. Wave quaton, Spnge; d dton, Beln, Gemany, 99.. Shu,. H., The Phyc of Atophyc Volume I: Radaton, Unvety Scence Book, Mll Valley, CA, Apl 99.. Smth, T. A., Some Note and Theoem on Moden Relatvtc Quantum Theoy, Mate The, epatment of Mathematc and Stattc, Wet loda Unvety, Penacola, L,. Publhed by Scholaly Common, 7