the pauli hamiltonian · 2017. 12. 11. · hˆ so =− 2m ∂k ∂φ σi f×pˆ pauli matrices and...
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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
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