Dirac s equation We construct relativistically covariant equation that takes into account also the spin. The kinetic energy operator is

Transcription

1 Dira s equation We onstrut relativistially ovariant equation that takes into aount also the spin The kineti energy operator is H KE p Previously we derived for Pauli spin matries the relation so we an also write σ a a, H KE σ pσ p However, when the partile moves under the influene of a vetor potential these expressions differ Substituting the latter takes the form σ + i σ p p ea/ σ we have used the identities and e h σ B, σ aσ b a b + iσ a b p A i h A A p Let us suppose that for the relativistially invariant expression E / p m the operator analogy holds Here and Eop p m E op i h t i h p i h We write the operator equation into the form E op E op σ p + σ p m or i h + σ i h i h σ i h φ m φ Here φ is a two omponent wave funtion spinor We define new two omponent wave funtions φ R i h i hσ φ m φ L φ It is easy to see that these satisfy the set of simultaneous equations i hσ i h φ L mφ R i hσ i h φ R mφ L We define yet new two omponent wave funtions ψ A φ R + φ L ψ B φ R φ L These in turn satisfy the matrix equation i h x i hσ i hσ i h x ψa m ψ B ψa We now define the four omponent wave funtion ψa φ ψ R + φ L ψ B φ R φ L and the 4 4-matries γ k γ 4 iσk iσ k We end up with the Dira s equation γ + γ 4 ψ + m ix h ψ for free spin- partiles Employing the four vetor notation the equation takes the form γ µ + m ψ x µ h ψ B Note The Dira equation is in fat a set of four oupled linear differential equations The wave funtion ψ is the four omponent vetor ψ ψ ψ ψ 3 ψ 4 This kind of a four omponent objet is alled bispinor or Dira s spinor Expliitely written down the Dira equation is 4 4 µ β γ µ αβ x µ + m δ αβ ψ β h

2 Note The fat that the Dira spinor happens to have four omponents has nothing to do with our four dimensional spae-time; ψ β does not transform like a four vetor under Lorentz transformations It is easy to verify that the gamma-matries Dira matries γ µ satisfy the antiommutation rule {γ µ, γ ν } γ µ γ ν + γ ν γ µ δ µν Furthermore, every γ µ is Hermitian, and traeless, ie γ µ γ µ, Tr γ µ Let s multiply the equation γ + γ 4 ψ + m ix h ψ on both sides by the matrix γ 4 and we get hγ 4 γ i h ψ + γ 4 m ψ t Forming the Hermitean onjugate of the Dira equation γ µ + m ψ x µ h we get x k ψ γ k + x 4 ψ γ 4 + m h ψ We multiply this from right by the matrix γ 4 and end up with the adjungated equation x µ ψγµ + m h ψ Here we have used the relations x 4 it x 4 γ k γ 4 γ 4 γ k Let s multiply the original Dira equation γ µ + m ψ x µ h Denote β γ 4 σk α k iγ 4 γ k σ k whih satisfy the relations, from left with the adjungated spinor ψ and the adjungated equation x µ ψγµ + m h ψ from right with the spinor ψ and subtrat the resulting equations We then get When we now write {α k, β} β {α k, α l } δ kl H i hα + βm, the Dira equation takes the familiar form Hψ i h ψ t We define the adjungated spinor ψ like: ψ ψ γ 4 Expliitely, if ψ is a olumn vetor ψ ψ ψ ψ 3, ψ 4 then ψ and ψ are row vetors ψ ψ, ψ, ψ 3, ψ 4 ψ ψ, ψ, ψ 3, ψ 4 The quantity x µ ψγ µ ψ s µ i ψγ µ ψ ψ αψ, iψ ψ thus satisfies a ontinuity equation Aording to Green s theorem we have ψγ 4 ψ d 3 x ψ ψ d 3 x onstant, the onstant an be taken to be with a suitable normalization of ψ Beause ψγ 4 ψ ψ ψ is positively definite it an be interpreted as a probability density Then s k i ψγ k ψ ψ α k ψ an be identified as the density of the probability urrent Note It an be shown that s µ transforms like a four vetor, so the ontinuity equation is relativistially ovariant It an be proved that any sets of four matries γ µ and γ µ satisfying the antiommutation relations {γ µ, γ ν } δ µν {γ µ, γ ν} δ µν,

3 are related to eahother through a similarity transformation with a non-singular 4 4-matrix S: Sγ µ S γ µ With the help of the matries γ µ the original Dira equation an be written as S γ µs + m S Sψ x µ h Multiplying this from left with the matrix S we get γ µ + m ψ, x µ h ψ Sψ Thus Dira s equation is independent on the expliit form of the matries γ µ ; only the antiommutation of the matries is relevant If the matries γ µ are Hermitean the transformation matrix S an be taken to be unitary It is easy to show that then the probability density and urrent, for example, are independent on the representation: ψ γ µψ ψγ µ ψ Vetor potential When the system is subjeted to a vetor potential A µ A, ia, Suppose now that E m, ea m and measure the energy starting from the rest energy: We expand E ea + m E NR E m + E NR ea ENR ea + This an be taken to be the power series in v/ sine E NR ea p ea/ / mv / Taking into aount only the leading term we get σ p ea σ p ea ψ A E NR ea ψ A This an be written as p ea e h σ B + ea ψ A E NR ψ A Up to the zeroth order of v/ the omponent ψ A is thus the two omponent Shrödinger-Pauli wave funtion multiplied with the fator e imt familiar from the non-relativisti quantum mehanis The equation σ ψ A E ea + m ψ B we make the ordinary substitutions i h/x µ i h/x µ ea µ / The Dira equation takes now the form ie x µ h A µ γ µ ψ + m h ψ Assuming that A µ does not depend on time the time dependene of the spinor ψ an be written as ψ ψx, t t e iet/ h Let us write now the Dira equation for the omponents ψ A and ψ B : σ p ea ψ B E ea m ψ A σ p ea ψ A E ea + m ψ B With the help of the latter equation we eliminate ψ B from the upper equation and get σ E ea + m σ ψ A tells us that the omponent ψ B is roughly by the fator p ea/ / v/ less than ψ A Due to this, provided that E m, ψ A is known as the big and ψ B as the small omponent of the Dira wave funtion ψ We obtain relativisti orretions only when we onsider the seond or higher order terms of the expansion E ea + m Let us suppose now that The wave equation is then H A σ p This wave equation looks like a time independent Shrödinger equation for the wave funtion ψ A EHowever, ea m ψ A + E NR ea A H A ψ A E NR ψ A, ENR ea ENR ea + σ p + ea

4 evaluating orretions up to the order v/ the wave funtion ψ A is not normalized beause the probability interpretation of Dira s theory requires that ψ A ψ A + ψ B ψ B d 3 x, ψ B already of the order v/ expliitely writing down the expression for the operator H A we see that it ontains the non-hermitian term i he p the equation is not an eigenvalue equation sine H A itself ontains the term E NR Up to the order v/ the normalization ondition an now be written as ψ A + p 4m ψ A d 3 x, beause aording to the equation σ p ea ψ A E ea + m ψ B we have ψ B σ p ψ A It is worthwhile to define the new two omponent wave funtion Ψ: Ψ Ωψ A, Ω + p 8m Now Ψ is up to the order v/ normalized orretly beause Ψ Ψ d 3 x ψ A + p 4m ψ A d 3 x We multiply the equation H A ψ A E NR ψ A, on both sides with the operator and get Ω p /8m, Ω H A Ω Ψ E NR Ω Ψ Expliitely, up to the order v/ this an be written as p { p p } + ea 8m, + ea σ p E NR ea σ p Ψ E NR p 4m Ψ Writing E NR p in the form {ENR, p } we get p + ea 8m 3 + 8m {p, E NR ea } σ pe NR ea σ p Ψ E NR Ψ Beause for arbitrary operators A and B holds we an, by setting {A, B} ABA A, A, B σ p A E NR ea B, redue the equation into the form p + ea 8m 3 e hσ E p 4m e h 8m E E NR Ψ Ψ In the derivation of the equation we have employed the relations σ p, E NR ea ie hσ E σ p, ie hσ E e h E e hσ E p, the validity of whih an be verified by noting that A E E The resulting equation is a proper Shrödinger equation for a two omponent wave funtion Physial interpretation We look at the meaning of the terms in the equation p + ea 8m 3 e hσ E p 4m e h 8m E E NR Ψ Ψ The term p + ea gives the non-realtivisti Shrödinger equation The term 8m 3 is a relativisti orretion to the kineti energy as an be seen from the expansion m + p m p p 4 8m 3 +

5 e hσ E p 3 The term 4m desribes the interation between the spin of a moving eletron and eletri field Intuitively this, so alled Thomas term, is due to the fat that the moving eletron experienes an apparent magneti field E v/ If the eletri field is a entral field, ea V r, it an be written in the form e h h 4m σ E p 4m r we have substituted S hσ/ dv r dr dv dr S L, So we atually have a spin orbit interation σ x p 4 The term e h 8m E is known as the Darwin term Its meaning an be dedued when we note that E is the harge density For example, in the hydrogen atom E eδx it auses the energy shift e h 8m δx ψshrö d 3 x e h 8m ψshrö, x whih differs from zero only in the s-state