β matrices defined above in terms of these Pauli matrices as

Transcription

1 The Pauli Hamiltonian First let s define a set of x matrices called the Pauli spin matrices; i σ ; ; x σ y σ i z And note for future reference that σ x σ y σ z σ σ x + σ y + σ z 3 3 We can rewrite α & i β matrices defined above in terms of these Pauli matrices as σ x σ y σ z αx ; αy ; αz ; β σ x σ y σ z - If we then partition the four-element vector Ψ into two, two element vectors called spinors) ϕ Ψ χ where ϕ u u & χ u 3 u 4 the Dirac equation may be written as E eφ E R ) cσ i ˆp cσ i ˆp E eφ E R ) ϕ χ where E mc. We now partition the energy into the relativistic or rest mass contribution E the much smaller non-relativistic contribution E, E R E + E the Dirac equation becomes E eφ ) cσ i ˆp cσ i ˆp E + eφ + E) ϕ χ or E eφ )ϕ + cσ i ˆpχ ) χ c σ i ˆpϕ E + eφ + E From the second of these equations we can write

2 χ E + eφ + E ) cσ i ˆpϕ Note that because the rest mass energy of the electron E is 6 ~.5 ev the denominator is much larger than cp which is comparable to the kinetic energy of the electron. This means that χ is, in some sense, much smaller than ϕ. ϕ is called the large component of the wavefunction χ the small component. Inserting this expression for χ into the first results in E eφ )ϕ + cσ i ˆp E + eφ + E) cσ i ˆpϕ where we have been careful to note that φ) r is a function of r will be operated on by ˆp. If we define the function Kφ) E E + eφ + E then the equation for the spinor ϕ becomes Ĥ Dirac ϕ σ i ˆp Kφ) σ i ˆp eφr) ϕ Eϕ Note that because K φ ) depends on E this is a pseudo eigenvalue problem. Also note that up to this point in our development this equation for ϕ is exact. To proceed we note the identity So with σ i A) σ i B) Ai B) + iσ i A B) A p & B Kφ) p we have Ĥ Dirac ϕ ˆp i Kφ) ˆp i + σ i ˆp Kφ) ˆp) eφr) ϕ Eϕ To simplify the Dirac Hamiltonian we note that when the operator ˆp i Kφ) ˆp operates on an arbitrary spinor, f g it operates on each component so we can consider its effect on each spatial function independently. Consider ˆp i Kφ) ˆpf α K α f ) K f + α K i α f ) where we sum over repeated Greek indices. Now α K K φ F K α α where F is the α α component of the electric field due to the nuclear charge so ˆp i Kφ) ˆp ) f Kφ) ˆp f + i K F i ˆp ) f

3 with the same result for g the other scalar component of the spinor so ˆp i Kφ) ˆp Kφ) ˆp + i K F i ˆp ) Now consider the term σ i ˆp K ˆp First allow ˆp K ˆp ) to operate on an arbitrary function f α since since so ˆp K ˆp ) f ε α αβγ β K γ f ) ε αβγ K β γ f + β K γ f ) εαβγ β γ f is identically zero we have ˆp K ˆp K ) f ε α αβγ γ f β K ε φ K f i αβγ β γ ε φ ˆp f αβγ β γ φ F we have β β ˆp K ˆp ) i K α ε F ˆp αβγ β γ σ i ˆp K K ˆp i σ i F ˆp. i K F ˆp ) α So now the Dirac Hamiltonian operating on the two component spinor becomes ˆp Ĥ Dirac eφ + Kφ) ) ˆp + i K F i ˆp Once again we note that this is still exact, i.e., correct to all orders of V c. The first two terms constitute the Schrodinger Hamiltonian ˆp Ĥ Schrodinger eφ K σ i F ˆp The next term corrects for the variation in the mass of the electron with its speed is called the mass-velocity term Ĥ MV Kφ) ) ˆp Following this we have the Darwin term which has no classical interpretation i K F i ˆp And lastly we have the spin-orbit term

4 K σ i F ˆp Pauli Matrices Spin involves the x Pauli matrix σ so let look at some of its properties, in particular the commutation relations among its x, y,z components. Consider the commutator σ x,σ y σ σ σ σ x y y x using the definitions given above σ x σ y i i i i i iσ z while σ y σ x i i i i i iσ z so σ x,σ y σ σ σ σ iσ x y y x z In a similar fashion we find σ y,σ z iσ & σ x,σ z x iσ y These commutator s are very similar to those that define an angular momentum vector Ĵ x, Ĵ y iĵ Ŝ z plus cyclic permutations of the indicies. If we define Ŝ α σ α,α x, y,z these operators have the commutators Ŝ,Ŝy x iŝz σ so that ˆ J i.e., are therefore angular momentum operators in particular spin angular momentum. We see that since Ŝ Ŝx + Ŝy + Ŝz 3 4 it commutes with each of its components as usual we select Ŝ z & Ŝ to have simultaneous eigenfunctions. The eigenfunctions of Ŝ z are

5 & with eigenvalues ± Ŝ Ŝ We will often abbreviate & as α & β respectfully remember these are not the Dirac matrices α & β ) write Ŝ z α α & Ŝz β β Ŝ α + α 3 4 α & Ŝ β + β 3 4 β. Since F e Z r 4πε r 3 Ŝ K σ we can write σ i F ˆp Ze K 4πε r 3 m ˆ S i r ˆp Ze K 4πε r 3 m ˆ S i ˆL This is the spin-orbit term it represents the interaction of the electrons spin with the magnetic field due to the nuclear motion. Pauli Hamiltonian Correct to order V / c) We will now develop an approximate Hamiltonian correct to order V ). Lets look again at K φ ). c Classically we have c Kφ) c + eφ + E + e Z 8πε mc r + E c + r r + E c

6 Where ez 8πε mc r is approximately the size of a nuclear diameter -5 M since the rest mass energy of the electron is approximately.5 x 6 ev E is about -3.6 ev, the ratio E c important close to the nucleus. 5 :. From the plot of ) Kr we see that one can expect the effects of ) Kr to be..8.6 Kr) r/r Lets consider the mass-velocity term note that we can write c + φ+ eφ + E + p + K φ) c e E c c since the kinetic energy of the electron is considerably smaller than its rest mass we may write p Kφ) + 4m so c ˆp 4 Ĥ MV 8m 3 c 4 4 8m 3 c where 4 means we operate with twice. Now for the Darwin term

7 Since i K F i ˆp K is equal to ek c, F φ ˆp i have e K φ) c) φ i e c) φ i where we approximate can be rewritten by noting K φ ) as. Matrix elements of this operator involve ψ φ i ψ which ψ φψ ) ψ φ ψ + ψ φ i ψ + ψφ ψ ψ i ψ φ +φ ψ Since is Hermitian the left h term is cancelled by the last on the right leaving ψ φ i ψ ψ φ ψ so e c) φ since φ ez 4πε r we have e Z 8πε c) r e Zδ 3 r ) ε c) Where δ 3 r ) is the three dimensional Dirac delta function defined as δ 3 r ) δ r) where δ r) 4πr dr δ r) f r)dr f ) Now for the spin-orbit term. Ze K S ˆ i ˆL 4πε r 3 m Since K ek c e c we have Ze Ŝ i ˆL mc) 8πε r 3 And so we have the Pauli Hamiltonian Ĥ Pauli Ĥ Shrodinger + Ĥ mv + Ĥ D +