The Drac Equaton for a One-electron atom In ths secton we wll derve the Drac equaton for a one-electron atom. Accordng to Ensten the energy of a artcle wth rest mass m movng wth a velocty V s gven by E R = m R c = mc 1 V c where m s the relatvstc mass and R c s the seed of lght. Accordngly the energy of an electron of mass m and charge e movng n the electrc feld of an nfntely heavy nucleus of atomc number Z s ez = ) where φ) r = 4πε r s the electrostatc otental at the oston of ER mrc eφ r the electron due to the nuclear charge. If we defne the lnear momentum as = m RV the term R mcmay be rewrtten as ) the electron n a one electron atom becomes ) ER mc c eφ r = + ). mc = mc + c so the energy of We can magne transformng ths classcal energy exresson nto a quantum mechancal Hamltonan by relacng the classcal momentum wth ts oerator equvalent ˆ = and wrtng the wave-equaton as ĤΨ= EΨ where ) ˆ ˆ φ ) H = mc + c e r. The roblem here s what s one to make of the square root of an oerator? Drac suggested that one attemt to extract the square root formally and wrte mc ) + ˆ c = c βmc + α ˆ ) where β & α are to be determned so that ths s an equalty. If we square both sdes we have m c 4 + c ˆ = β m c 4 + c α ˆ) + βmc α ˆ + α ˆβmc R
where we have been careful not to assume that β & α commute. For ths to be an equalty requres β = α x = α y = α z = 1 βα + α β = and α α j + α j α = where = x, y, or, z. These condtons are not ossble wth ordnary numbers and Drac showed that the smlest reresentaton of β & α s as 4x4 matrces: β = 1 1 1 1 ; α x = 1 1 1 1 ; α y = ; α z = 1 1 1 1 + + Note that these four matrces are Hermtan,.e., β = β & α = α Drac s wave equaton s then βmc + cα ˆ eφr)1 )Ψ = E R Ψ Where 1 s a 4x4 unt matrx and the wave functon Ψ s a four element column vector Ψ =. Lets take a moment to famlarze our selves wth ths notaton. Consder for examle the exresson α ˆΨ = α x + α y ˆ y + α z ˆ z )Ψ and the term α x Ψ. If we allow to oerate frst we have Ψ = and then followed by α x
α x Ψ = 1 1 1 1 = or Obvously we could allow α x to oerate frst and then have oerate on the result. Dong the same thng for the remanng two comonents results n cα ˆΨ = cˆx + cˆ y + cˆ z = c ˆ y + ˆ z + ˆ y ˆ z ˆ y + ˆ z + ˆ y ˆ z the term βmc Ψ becomes 1 βmc = mc 1 1 1 = mc and then eφr)ψ = eφr)ψ = eφr). Addng these, lettng E = mc and usng the exlct oerator for the lnear momentum, ˆ = = four couled dfferental equatons x + y + z results n the
c E E + eφ R ) + z + x y = c E E + eφ R ) z + x + y = c E + E + eφ R ) + z + x y = c E + E + eφ R ) z + x + y = Angular Momentum and the Drac Hamltonan The frst queston to be answered s does the orbtal angular momentum oerator commute wth the Drac Hamltonan as t does wth the Schrodnger Hamltonan? We show below that t does not and n fact L, ˆ Ĥ Drac = c α ˆ ). Subsequent to ths we show that there s an oerator Ŝ, whch we call sn and has the roerty of beng an angular momentum oerator, whch has the commutator S, ˆ Ĥ Drac = c α ˆ ) so that the total angular momentum Ĵ = ˆL + Ŝ commutes wth the Drac Hamltonan, J, ˆ Ĥ Drac = and so the egenfunctons of Ĥ Drac are smultaneously egenfunctons of J ˆ & Ĵ. Lets begn wth the orbtal angular momentum and evaluate z L, ˆ Ĥ Drac = ˆL,βmc + cα ˆ eφr)1 = c ˆL, α ˆ where we recognze that ˆL commutes wth β because t sn t a functon of any satal varables and wth φr) because ts shercally symmetrc. So L, ˆ α ˆ = Consder, α ˆ, α ˆ ˆx +,α x + ˆL y, α ˆ ŷ +,α y ˆ y + and snce ˆL commutes wth α we have ˆL z, α ˆ ẑ,α z ˆ z
, α ˆ α x Now, and, ˆ y and so, ˆ z, + α y yˆ zˆ, ˆ z y y yˆ zˆ, ˆ z y z, ˆ y + α z yˆ, ˆ z y, ˆ z y, ˆ y ˆ = ˆ z z zˆ, ˆ y z z, ˆ z ˆ y = ˆ y, α ˆ α ˆ α ˆ = α ˆ α ˆ y z z y y z z y and by symmetry ˆLy, α ˆ α ˆ ) = α ˆ ) x ) and ˆLz, α ˆ y α ˆ ) z so L, ˆ α ˆ = α ˆ ) and fnally L, ˆ Ĥ Drac = c α ˆ ) as requred. Now for the sn comonent. We begn by notng that the 4 by 4 matrces β &α are block dagonal and can be wrtten as α x = σ x σ x ; α = σ y y σ y ; α = σ z z σ z ; β = 1 x -1 x where σ x = 1 1 ; σ y = ; σ z = 1 1 and 1 = 1 x 1 σ x,σ y,& σ z are called the Paul Sn matrces. Note that σ x ) = σ y ) = σ z ) = 1 x = 1 1 and therefore σ = σ x + σ y + σ z = 3 1 1 =31 x Now consder the commutator
σ,σ x y σ σ x y σ y σ x From the defntons σ x σ y = 1 1 = = 1 1 = σ z whle σ y σ x = 1 1 = = 1 1 = σ z and so σ x,σ y σ σ x y σ y σ x = σ z In a smlar fashon we fnd σ y,σ z σ x & σ z,σ x σ y Defne a vector S so S = ˆx + S y ŷ + S z ẑ where = σ x σ x ; S = y σ y σ y ; S = z σ z σ z From above these sn comonents have the characterstc angular momentum commutators S x,s y S, S,S z y z S &,S z x S y Now lets show that they commute wth the Drac Hamltonan. Frst S, Ĥ Drac S,βmc + cα ˆ eφr)1 c S, α ˆ Then consder, α ˆ and snce,α x S,α x x +,α y &,α y σ α α σ x y y x ˆ + S y,α x z ˆ z ) = α z &,α z σ α α σ x z z x ) = α y
we have ) x, α ˆ α ˆ α ˆ = α ˆ z y y z Snce there s nothng secal about x we have S, α ˆ α ˆ ) and so L, ˆ α ˆ + S, α ˆ Ĵ, α ˆ ) α ˆ ) = = + α ˆ and so we can fnd smultaneous egenfunctons of Ĵ and Ĵα A full dscusson of the Drac equatons s gven subsequently. However there s a frst order aroxmaton to the full Drac Hamltonan called the Paul Hamltonan that s very useful for low Z atoms and much easer to use. Addtonally t rovdes some nsght nto the nature of the relatvstc correctons to the Schrodnger equaton so we wll develo ths aroxmaton frst.