) +m 0 c2 β K Ψ k (4)

Transcription

1 i ħ Ψ t = c ħ i α ( The Nature of the Dirac Equatio by evi Gibso May 18, 211 Itroductio The Dirac Equatio 1 is a staple of relativistic quatum theory ad is widely applied to objects such as electros ad protos. Ψ ) +m c2 β Ψ (1) The α ad β are 4 4 matrices, the Ψ is a 4-compoet colum matrix ad m is the rest mass. The motivatio behid this piece is to explore this equatio, what does it say ad mea. Reworkig the equatio It ca be show that the solutios to (1) must also satisfy the lei-gordo equatio 2 3, which is a scalar equatio. 2 Ψ 1 Ψ c 2 2 t ( = m c 2 2 ħ ) Ψ (2) I this light the basis set to the solutio of (1) ca be expressed as the product of a scalar solutio to the lei-gordo equatio, call it Ψ k, alog with a colum matrix cosistig of costats, call it. = k (3) Puttig (3) ito (1) ad utilizig the assumptio that is costat. i ħ ( Ψ ) k = cħ α t i ( ( Ψ ) k ) +m c2 β ( Ψ k ) i ħ Ψ k = c ħ t i α ( Ψ k ) +m c2 β Ψ k (4) We ca reduce this dow to a scalar equatio by performig a matrix multiplicatio with the traspose of the complex cojugate of. This is to allow for potetially complex elemets. T i ħ Ψ k = c ħ t i T α ( Ψ k ) +m c2 T β Ψ k i ħ Ψ k t = ( c 2 with 2 T T α ) ( ħ i Ψ k ) +m ( 1 c2 β ) 2 T Ψ k (5a) (5b) 1.

2 The portios of (5a) cotaied i the paretheses are i fact scalars. With the appropriate substitutios this ca be re-writte as: i ħ k t = V ħ i k m c2 k (6a) V = c 2 ( T α ) (6b) = 1 2 T (6c) At this poit the matrix equatio (1) has bee trasformed ito scalar equatio (6a). As stated above Ψ k must be a solutio to the lei-gordo equatio, therefore equatio (6a) ca be viewed as a boudary coditio to be satisfied o (2). But what does mea? Thus far there are two thigs that ca be deduced. First, it is eeded to iterface the scalar Ψ k ito (1). Secod, it iflueces the boudary coditio (6a). Yet there is still more to be leared. Coectig to relativity Before explorig some o-quatum mechaical physics eeds to be explored. The mass-shell relatioship is 4 E 2 =c 2 p 2 m c 2 2 (7) This ca be liearized usig the stadard equatios for relativistic eergy ad mometum 5. m c 2 E=c 2 m u p m 2 c 4 E= u p m c 2 1 For eergy, mometum eigestates states (6a) ad (8) are compatible with two assumptios V u 1 (8) (9) 2.

3 The ature of the solutio So what is the sigificace of the elemets of? To determie this substitute the stadard expressios for α ad β ito (6a) ad (6b). This gives V 1 2 R 1 R 2 (1a) V 2 2 R 1 R 2 (1b) V 3 2 I 1 I 2 (1c) = = 1 2 (1d) The coclusios are iterestig. Because of (1d), states where 1 ad 2 are the oly ozero elemets represet positive eergy while states where 1 ad 2 are the oly ozero elemets represets egative eergy. A examiatio of equatios (1a) to (1c) shows that states with oly oe ozero elemet of will represet a state at rest. I order for there to be motio oe eeds to have either 1 or 2 as a o-zero elemets ad either or as a o-zero elemets. Because motio i a particular directio, ala (1), does ot deped o a sigle elemet of, it caot represet a vector, at least ot i space-time. 3.

4 The state of the Uio A iterestig cosequece arises from the third bullet poit above. Cosider three scearios below. matrix Γ Eergies 1 = 1 E = m c 2 = 4-1 E = -m c 2 1 = 4-1 Γ 1 E m c 2 Table 1. A state with both 1 ad elemets will be i motio ad so will have a positive eergy greater tha its rest eergy or a egative eergy less tha its egative rest eergy. A coceptual coflict arises if we evisio each elemet of as a separate state. The reaso for the coflict is that if 1 ad represet states the the third state i the above table should costitute a mixture of positive ad eergy states ad so havig a eergy E m c 2 (11) should be possible. However it is ot. It is therefore logically icosistet to assig a state to each elemet. All that oe ca say is that a particular cofiguratio of leads to a particular state. Spi I the literature the first ad third elemets of Ψ are assiged to the spi up states while the balace belog to spi dow. However, as ca be see from the above argumets, this is a poor iterpretatio. The most that ca be said is that a state with the oly ozero elemets beig 1 ad ca be assiged the label of spi-up. Likewise states with oly ozero elemets beig 2 ad ca be called spi-dow. However simply classifyig other states as a mixture of spi-up ad spi dow is problematic. The difficulty lies i ot beig able to separate ay Ψ ito spi-up ad spi-dow states. 4.

5 The reaso is that while ca be so divided the associated Ψ k from (3) caot, for whe oe divides the associated Ψ k would have to chage So what ca be said about spi accordig to Dirac? Accordig to the paradigm, the Dirac equatio shows spi as a itrisic property 6. However the derivatio itself does ot iclude ay iformatio about ay of the itrisic properties of the particle i questio. Moreover this solutio is said to be the solutio for matter with two spi states, called spi ½ particles, while the lei-gordo equatio is oly valid for particles without spi. Yet as was see above the elemets of Ψ caot be assiged to specific states. Thus the questio of what (1) says about spi is still left ope ad vaque. Coectig the Dirac ad lei-gordo equatios via (8) shows the uderlyig physics behid the derivatios. Ideed both are extesios of (7). So whereas the lei-gordo is a secod order scalar equatio the Dirac is a first order matrix equatio. Neither equatio cotais either extra iformatio or a deficit of iformatio ad so the two tell the same story, thus demostratig a weakess i the orthodox iterpretatio of the elemets of Ψ. Works Cited Paul A. Tipler ad Ralph A. Llewelly Moder Physics 4 th ed. p evi Gibso A ew statistical view for elemetary matter (upublished) 6. G.E. Uhlebeck ad S. Goudsmit, Naturwisseschafte 47 (1925) 953 Cotact iformatio kevi.gibso@asu.edu kevilg@mesacc.edu 5.