ThePauliHamiltonian

Firstlet’sdefineasetof2x2matricescalledthePaulispinmatrices;

0 1 0 1 0 ; ;

1 0 0 0 1x y z

ii

−⎛ ⎞ ⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠ ⎝ ⎠

σ σ σ

Andnoteforfuturereferencethat

σ x

2 = σ y2 = σ z

2 = 1 = 1 00 1

⎛

⎝⎜⎞

⎠⎟

and

σ 2 = σ x

2 +σ y2 +σ z

2 = 3 1 00 1

⎛

⎝⎜⎞

⎠⎟ =31

Wecanrewrite &iα β matricesdefinedaboveintermsofthesePaulimatricesas

; ; ; =yx zx y z

yx z

⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞= = =⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠⎝ ⎠ ⎝ ⎠

σσ σα α α β

σσ σ00 0 1 0

00 0 0 -1

Ifwethenpartitionthefour-elementvectorΨ intotwo,twoelementvectors(calledspinors)

⎛ ⎞= ⎜ ⎟⎝ ⎠

ϕΨ

χwhere

ϕ =u1

u2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

& χ =u3

u4

⎛

⎝⎜⎜

⎞

⎠⎟⎟theDiracequationmaybewrittenas

E0 − eφ − ER( )1 cσ ip

cσ ip −E0 − eφ − ER( )1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

ϕχ

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 0

where E0 = mc2 .Wenowpartitiontheenergyintotherelativisticorrestmasscontribution E0

andthemuchsmallernon-relativisticcontribution E , ER = E0 + E theDiracequationbecomes

−E − eφ( )1 cσ ip

cσ ip − 2E0 + eφ + E( )1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

ϕχ

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 0

or

−E − eφ( )ϕ + cσ ipχ = 0

andcσ ipϕ − 2E0 + eφ + E( ) χ = 0

Fromthesecondoftheseequationswecanwrite

χ = 2E0 + eφ + E( )−1cσ ipϕ

NotethatbecausetherestmassenergyoftheelectronE0 is6~ 0.5 10 eV× thedenominatoris

muchlargerthancpwhichiscomparabletothekineticenergyoftheelectron.Thismeansthat χ is,

insomesense,muchsmallerthanϕ .ϕ iscalledthelargecomponentofthewavefunctionand χ the

smallcomponent.

Insertingthisexpressionfor χ intothefirstresultsin

−E − eφ( )ϕ + cσ ip 2E0 + eφ + E( )−1

cσ ipϕ = 0

wherewehavebeencarefultonotethat ( )rφ isafunctionof r andwillbeoperatedonby p .Ifwe

definethefunction K(φ) =

2E0

2E0 + eφ + Ethentheequationforthespinorϕ becomes

HDiracϕ =

σ ip K(φ)

2mσ ip − eφ(r)1

⎛⎝⎜

⎞⎠⎟ϕ = Eϕ

Notethatbecause ( )K φ dependson E thisisapseudoeigenvalueproblem.Alsonotethatuptothis

pointinourdevelopmentthisequationforϕ isexact.Toproceedwenotetheidentity

(σ iA)(σ iB) = (

A iB)1+ i

σ i (A×B)

Sowith A = p &

B = K(φ) p wehave

HDiracϕ = 1

2mp i K(φ) p1+ i

2mσ i ( p × K(φ) p)− eφ(r)

⎛⎝⎜

⎞⎠⎟ϕ = Eϕ

TosimplifytheDiracHamiltonianwenotethatwhentheoperator p i K(φ) p operatesonan

arbitraryspinor,

fg

⎛

⎝⎜⎜

⎞

⎠⎟⎟itoperatesoneachcomponentandsowecanconsideritseffectoneach

spatialfunctionindependently.Consider

p i K(φ) pf = −2∇α (K∇α f ) = −2 K∇2 f +∇α K i∇α f( )

wherewesumoverrepeatedGreekindices.

Now ∇α K = ∂K

∂φ∇αφ = − Fα

∂K∂φ

where Fα istheα componentoftheelectricfieldduetothe

nuclearchargeandso

p i K(φ) p( ) f = K(φ) p2 f + i ∂K∂φF ip( ) f

withthesameresultfor g theotherscalarcomponentofthespinorandso

p i K(φ) p = K(φ) p2 + i ∂K∂φF ip( )

Nowconsidertheterm σ ip × Kp

Firstallow p × Kp( )

αtooperateonanarbitraryfunction f

p × Kp( )

αf = −2εαβγ∇β K∇γ f( ) = −2εαβγ K∇β∇γ f +∇β K∇γ f( )

andsince fαβγ β γε ∇ ∇ isidenticallyzerowehave

p × Kp( )α

f = −2εαβγ∇γ f∇β K = −2 ∂K∂φ

εαβγ∇βφ∇γ f = −i ∂K∂φ

εαβγ∇βφ pγ f

andsince Fβ βφ∇ = − wehave

p × Kp( )α= i ∂K

∂φεαβγ Fβ pγ = i ∂K

∂φF × p( )

α

andso

σ ip × Kp = i ∂K

∂φσ iF × p .

SonowtheDiracHamiltonianoperatingonthetwocomponentspinorbecomes

HDirac =

p2

2m− eφ + K(φ)−1( )

p2

2m+ i

2m∂K∂φF ip −

2m∂K∂φσ iF × p

Onceagainwenotethatthisisstillexact,i.e.,correcttoallordersofV c .

ThefirsttwotermsconstitutetheSchrodingerHamiltonian

HSchrodinger =

p2

2m− eφ

Thenexttermcorrectsforthevariationinthemassoftheelectronwithitsspeedandiscalledthe

mass-velocityterm

H MV = K(φ)−1( )

p2

2m

FollowingthiswehavetheDarwintermwhichhasnoclassicalinterpretation

HDarwin =

i2m

∂K∂φF ip

Andlastlywehavethespin-orbitterm

HSO = −

2m∂K∂φσ iF × p

PauliMatricesandSpin

HSO involvesthe2x2Paulimatrix σ soletlookatsomeofitsproperties,inparticularthe

commutationrelationsamongits x, y, z components.Considerthecommutator

σ x ,σ y⎡⎣ ⎤⎦ = σ xσ y −σ yσ x

andusingthedefinitionsgivenabove

σ xσ y =

0 11 0

⎛

⎝⎜⎞

⎠⎟0 −ii 0

⎛

⎝⎜⎞

⎠⎟= i 0

0 −i

⎛

⎝⎜⎞

⎠⎟= i 1 0

0 −1⎛

⎝⎜⎞

⎠⎟= iσ z

while

σ yσ x =

0 −ii 0

⎛

⎝⎜⎞

⎠⎟0 11 0

⎛

⎝⎜⎞

⎠⎟= −i 0

0 i

⎛

⎝⎜⎞

⎠⎟= −i 1 0

0 −1⎛

⎝⎜⎞

⎠⎟= −iσ z

so

σ x ,σ y⎡⎣ ⎤⎦ = σ xσ y −σ yσ x = 2iσ z

Inasimilarfashionwefind

σ y ,σ z⎡⎣ ⎤⎦ = 2iσ x & σ z ,σ x⎡⎣ ⎤⎦ = 2iσ y

Thesecommutator’sareverysimilartothosethatdefineanangularmomentumvector J i.e.,

J x , J y⎡⎣ ⎤⎦ = iJ z pluscyclicpermutationsoftheindicies.Ifwedefine

S =

2σ sothat

Sα =

2σα ,α = x, y, z theseoperatorshavethecommutators

Sx , Sy⎡⎣ ⎤⎦ = iSz

andarethereforeangularmomentumoperatorsandinparticularspinangularmomentum.Wesee

thatsince S 2 = Sx

2 + Sy2 + Sz

2 = 342 1 itcommuteswitheachofitscomponentsandasusualwe

select Sz & S 2 tohavesimultaneouseigenfunctions.Theeigenfunctionsof

Sz =

2

1 00 −1

⎛

⎝⎜⎞

⎠⎟are

10

⎛

⎝⎜⎞

⎠⎟& 0

1⎛

⎝⎜⎞

⎠⎟witheigenvalues

±

2

and

S 2 1

0⎛

⎝⎜⎞

⎠⎟= 32

410

⎛

⎝⎜⎞

⎠⎟= 1

212+1

⎛⎝⎜

⎞⎠⎟2 1

0⎛

⎝⎜⎞

⎠⎟

and

S 2 0

1⎛

⎝⎜⎞

⎠⎟= 32

401

⎛

⎝⎜⎞

⎠⎟= 1

212+1

⎛⎝⎜

⎞⎠⎟2 0

1⎛

⎝⎜⎞

⎠⎟

Wewilloftenabbreviate

10

⎛

⎝⎜⎞

⎠⎟& 0

1⎛

⎝⎜⎞

⎠⎟as α & β respectfully(rememberthesearenottheDirac

matrices α & β )andwrite

Szα =

2α & Szβ = −

2β and

S 2α = 1

212+1

⎛⎝⎜

⎞⎠⎟2α = 32

4α & S 2β = 1

212+1

⎛⎝⎜

⎞⎠⎟2β = 32

4β .

Since

F = e2Z r

4πε0r3 and

S =

2σ wecanwrite

HSO = −

2m∂K∂φσ iF × p = − Ze

4πε0r3m

∂K∂φS ir × p = − Ze

4πε0r3m

∂K∂φS iL

Thisisthespin-orbittermanditrepresentstheinteractionoftheelectronsspinwiththemagnetic

fieldduetothenuclearmotion.

PauliHamiltonianCorrecttoorder (V / c)2

WewillnowdevelopanapproximateHamiltoniancorrecttoorder ( )2Vc .Letslookagainat ( )K φ .

Classicallywehave

K(φ) = 2mc2

2mc2 + eφ + E= 1

1+ e2Z8πε0mc2r

+ E2mc2

= 1

1+r0

r+ E

2mc2

Where2

0 208e Zrmcπε

= isapproximatelythesizeofanucleardiameter≈10-15M

andsincetherestmassenergyoftheelectronisapproximately0.5x106evandEisabout-13.6eV,

theratio 52 10

2Emc

−: .Fromtheplotof ( )K r weseethatonecanexpecttheeffectsof ( )K r tobe

importantclosetothenucleus.

Letsconsiderthemass-velocitytermandnotethatwecanwrite

2

22

2 2

2 1 1( )22 1 12 2

mcK e E p mmc e Emc mc

φ φφ+= = =++ + +

andsincethekineticenergyoftheelectronisconsiderablysmallerthanitsrestmasswemaywrite

K(φ) 1−

p2

4m2c2 + so

H MV = −

p4

8m3c2 = − 4∇4

8m3c2

where ∇4 meansweoperatewith ∇

2 twice.NowfortheDarwinterm

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

K(r)

r/r0

HDarwin =

i2m

∂K∂φF ip

SinceK∂∂φ

isequalto2

22eKmc

− , F = −∇φ and

p = −i∇ have

HDarwin =

2e2K 2(φ)(2mc)2 ∇φ i∇ = 2e

(2mc)2 ∇φ i∇

whereweapproximate 2 ( )K φ as1.Matrixelementsofthisoperatorinvolve ψ ∇φ i∇ψ which

canberewrittenbynoting

ψ ∇2 φψ = ψ ∇ i ψ∇φ +φ∇ψ( ) = ψ ∇2φ ψ + 2 ψ ∇φ i∇ψ + ψφ ∇2 ψ

Since 2∇ isHermitianthelefthandtermiscancelledbythelastontherightleaving

ψ ∇φ i∇ψ = − 1

2ψ ∇2φ ψ andso

HDarwin =

−2e2(2mc)2 ∇

2φ andsince04

eZr

φπε

= wehave

HDarwin =

−2e2Z8πε0(2mc)2 ∇

2 1r=

2e2Zδ 3(r )2ε0(2mc)2

Where δ3(r ) isthethreedimensionalDiracdeltafunctiondefinedas

δ 3 r( ) = δ (r)

4πr 2 where δ (r)

0

∞

∫ dr = 1 and δ (r)

0

∞

∫ f (r)dr = f (0)

Nowforthespin-orbitterm.

HSO = − Ze

4πε0r3m

∂K∂φS iL

Since

∂K∂φ

= − eK 2

2mc2 ≈ − e2mc2 wehave

HSO = − Ze2

(mc)28πε0r3

S iL

AndsowehavethePauliHamiltonian

HPauli = HShrodinger0 + Hmv + HD + HSO