Select/Special Topics in Atomic Physics · 2017-08-04 · contravar ian t xxx x xyz ct ... Time evolution of state vector…. 22 2 22 2 2 22 2 m pp c m v E p.p m c c

P. C. Deshmukh Department of PhysicsIndian Institute of Technology MadrasChennai 600036

[email protected] Unit 3(i) Lecture 13

Relativistic Quantum Mechanics of the

Hydrogen Atom - 1References: RQM by Bjorken and Drell

and RQM by Greiner

Select/Special Topics in Atomic Physics

August‐September 2012 PCD STiAP Unit 3 1


( )4 1 4 1

2

4 4i c mc e

t × ××

∂ψ = α π + β + φ ψ

∂i

2 2 2 24 4

2 2 2 2

00

ii

i× ×

×× ×

⎡ ⎤σα = ⎢ ⎥σ⎣ ⎦

2 2 2 2

2 2 2 2

1 00 1

× ×

× ×

⎡ ⎤β = ⎢ ⎥−⎣ ⎦

Electron spin requires two components,- but Dirac equation admits a wavefunction with 4-components.

Anti-matter / negative energy solutions

2 2

14spin orbit

e VHm c r r−

∂= σ ⋅

∂Some questions about relativistic effects in atomic physics that may have concerned you:

Where does this come from?

Dirac equation

What is all this about?

ep Ac

⎛ ⎞π = −⎜ ⎟⎝ ⎠

Unit 3


Hydrogen Atom

Unit 3(i)

First: Transition from Non-relativistic

to Relativistic Dynamical Variables

Then: quantization!


Galilean relativity

'(( )) cr u tr tt = +( )r t

OI

O’

XI

X’

Y’

Z’

ZI

YI

'( )r t '

= + cd rdt

rd

udt' I cO O u t=F

F’

'cudr r

tdtdd

− =

August‐September 2012 PCD STiAP Unit 3

: constant velocity

cu

4

Aarav baby on treadmill.flv

( http://www.youtube.com/watch?v=odJqqXdqJe8 )


http://www.youtube.com/watch?v=odJqqXdqJe8

What is the velocity of the oncoming car?

……… relative to whom?

What would happen if

the object of your

observations is LIGHT?

'cudr r

tdtdd

− =


z'e

“Light (EM waves) travels at the constant speed in all inertial frames of references”

0 0

1c= μ ε

v

xe x'e

y'e

ze

ye

~~~~

The rocket frame moves toward the right at a constant velocity where 0<f<1.

OIInertial frame

RL

COUNTER-INTUITIVE ?


f c×

7

X X’

Y Y’

ZZ’

v

F F’

Origins O and O’ of the two frames F and F’coincide at t=0 and t’=0.

2 2

x'= (x vt) x= (x'+vt')y'=y y=y'z'=z z=z'

vx vx't'= t t= t'+c c

−

⎛ ⎞ ⎛ ⎞−⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

γ γ

γ γ

2

2

2

1=v1-

1 1

: 1 as v 0.

=−

→→

c

Note

γ

βγ

Lorentz transformations transform the space-time coordinates of ONE EVENT.


2

2

1

' 1-

ΔΔ = ≥ Δ

−

= ≤

t

L l l

τ τβ

β

v/c 1= ⟨β


Time dilation and Lorentz contraction

Neither ‘time’ intervals, nor ‘space’ intervals is invariant / absolute.

Simultaneity is not absolute.

Then what is invariant?

Non-relativisticSchrodinger equation:

( )2

2

iH V(r )

m

− ∇= +

H Eψ = ψ

distance not Lorentz invariant: →

How can we get a relativistic description of temporal evolution of the state of a system?

‘Events’ take place in 4-dimensional ‘MINKOWSKI’ space-time continuum


We use 4D vectors to describe events0 1 2 3

subscripts: covaria

tn

x x x x

x y z

ct

τ

↓

0 1 2 3

superscripts: contravariant

x x x x

x y z

ct

τ

↓

( )( )2ds dx dxμμ=

( )( ) ( )( )( )( ) ( )( )

2 0 10 1

2 32 3 +

ds dx dx dx dx

dx dx dx dx

= + +

+

‘interval’ / ‘distance’ between two ‘events’


0

10 1 2 3

2

3

11

11

dxdxdx dx dx dxdxdx

⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥⎡ ⎤= ⎣ ⎦ ⎢ ⎥⎢ ⎥⎣

⎢ ⎥−⎢ ⎥⎢ ⎥−⎢ ⎥−⎣ ⎦ ⎦

0 1 2 3

covariant

x x x x

x y zτ

0 1 2 3

contravarian

t

x x x x

x y zτ


( ) ( ) ( ) ( )2 2 2 22 0 1 2 3ds dx dx dx dx= − − −

( ) ( )2 gds dx dxμυ

υμ=

0

10 1 2 3

2

3

dxdxdx dx dx dxdxdx

⎡ ⎤⎢ ⎥−⎢ ⎥⎡ ⎤= ⎣ ⎦ ⎢ ⎥−⎢ ⎥−⎣ ⎦

+ - - - pseudo-Euclidian

a g aνμ μν=

( )( ) ( )( ) ( )( ) ( )( )2 0 1 2 30 1 2 3ds dx dx dx dx dx dx dx dx= + + +



( )( ) ( )( ) ( )( ) ( )( )2 0 0 1 1 2 2 3 3ds dx dx dx dx dx dx dx dx= − − −

a g aνμ μν=

( ) ( ) ( ) ( )2 2 2 22ds d dx dy dz+ − − −= τ

( ) ( ) ( ) ( )2 2 2 22d d dx dy dzξ = + + +τ +

( )( )2d dx dx : Euclideanμ μξ =

+ - - - PSEUDO-EUCLIDEAN

All physical phenomena take place in the 4D Pseudo-Euclidean Space.

0 1 2 3

contravar

ian

t

x x x x

x y zctτ

+ + + + Euclidean

0 1 2 3

covariant

x x x x

x y zτ


Signature + - - - PSEUDO-EUCLIDEAN

X X’

Y Y’

ZZ’

v

F F’

Origins O and O’ of the two frames F and F’ coincide at t=0 and t’=0.

2 2

' ( v ) ( ' v ')' '' '

v v '' '

= − = += == =

⎛ ⎞ ⎛ ⎞= − = +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x x t x x ty y y yz z z z

x xt t t tc c

γ γ

γ γ

2 2

2

1 111-

: 1 0.

= =−

→→

vc

Noteas v

γβ

γ


( ) ( ) ( ) ( )2 2 2 22ds' d ' dx ' dy ' dz '= τ − − −

{ } ( ) ( )2

2 2 222

vxt (x vt)c

ds' dy ' dd d z 'c⎧ ⎫⎛ ⎞γ − γ −⎨ ⎬⎜ ⎟⎝ ⎠

− −⎩ ⎭

= −

14

( )22 invariance criterionds ds' :=( ) ( ) ( ) ( )2 2 2 22ds d dx dy dz= + τ − − −

22

2 2 2

2

1v v1-

= =−c

cc

γ

{ } ( ) ( )2

2 2 222

vxt (x vt)c

ds' dy ' dd d z'c⎧ ⎫⎛ ⎞γ − γ −⎨ ⎬⎜ ⎟⎝ ⎠

− −⎩ ⎭

= −

{ } ( ) ( )2

2 2 22 2 2 22

vdxdt dx vdtc

ds' dy ' dc z '⎧ ⎫γ − γ −⎨ ⎬⎩ ⎭

= − − −

{ }

( ) ( )

2 22 2 2 2 2 2

2 42

2

2

2

vdtdxdt 2 dx 2 vdt+vc

v dxc dx dtds'

d dz

c

y ' '

⎧ ⎫γ − + γ= −

− −

−⎨ ⎬⎩ ⎭

( ) ( )

2 22 2 2 2 2 2 2 2 2 2 2

2

24

2

2

dt 2 vdtdx dx 2 vdt v

v dxc c dx dtds'

dy ' d

c

z '

⎧ ⎫γ − γ + γ − γ + γ − γ⎨ ⎬=

−

⎩ ⎭

−


( ) ( )2 2

2 22 2 2 2 2 2 2 2 2 2 24dt dx vv dxds' c c dt dy ' dz '

c= γ + γ − γ − γ − −

( ) ( )2 2

2 22 2 2 2 2 2 2 2 2 2 24dt v dxv dxds' c dt c dy ' dz '

c= γ − γ + γ − γ − −

( ) ( ) ( )2

2 22 2 2 2 2 2 22v 1 dxvds' c dt dy ' dz '

c⎛ ⎞

= γ − + γ − − −⎜ ⎟⎝ ⎠

12 22

2 2 2

v1v

−⎛ ⎞

= − =⎜ ⎟ −⎝ ⎠

cc c

γ

( ) ( ) ( )2 2 2

2 22 2 2 2 22 2 2 2 2v 1 dx

v vc c vds' c dt dy ' dz '

c c c⎛ ⎞

= − + − − −⎜ ⎟− − ⎝ ⎠

( ) ( )2 22 2 2 2dxds' c dt dy ' dz '= − − −

2 2 2 2 2 2dxds' d dy dz ds= τ − − − =


INVARIANCE

2

2

of dx

ds dx dx dx g dx

ds' dx ' dx ' dx ' g dx '

NORM

μ μ υμ μυ

μ μ υμ μυ

μ

=

= =

=

=

=

is a measure of INVARIANT INTERVAL

( ) ( ) ( ) ( )2 2 2 22ds d dx dy dz= τ − − −

-may be positive : “Time like”- zero : “Light like”- negative: “Space like”


( )op op

op op op op

QUANTIZATION: q q p p

F(q,p) F F q ,p

→ →

→ → →

expressed as

drp mv mdt

= =Space

Time

Lorentz contraction

Time dilationproper length proper velocity proper time

drvd

STRut pshot

= η = =

( ) 2

2

dr dr 1 ; 1dttd 1

;vc

η = = γ γ =

−

γ ≥

γ

Refer: Module 6: Special Theory of RelativityNPTEL course: Special/Select Topics on Classical MechanicsNPTEL http://nptel.iitm.ac.in/courses/115106068/Youtube http://www.youtube.com/playlist?p=PLB368471AD70B8A6B


( ) 2

2

dr dr 1 ; 1dttd 1

;vc

η = = γ γ =

−

γ ≥

γ

gives 3 of the 4-component '4-vector' for 'proper velocity' with =0,1,2,3:

μ

η

η μ

( )( )0 0

0 d ctdx dx cd t dt dt

η = = γ = γ = γγ

( )( )1 1

1x

d xdx dx vd t dt dt

η = = γ = γ = γγ

( ) ( )2 3

2 3 and y zdx dxv v

d t d tη = = γ η = = γ

γ γ

proper velocity

:μη


( ) 2

2

dr dr 1 ; 1dttd 1

;vc

η = = γ γ =

−

γ ≥

γ

0 cη = γ

1xvη = γ

2

3

y

z

v

v

η = γ

η = γ

( ) ( ) ( ) c, c, v c,vμη = γ η = γ γ = γProper velocity

Scalar product

2 2 2 20 1 2 3

gμ μ νμ μνη η = η η

= η − η − η − η

( )

( ) ( )

2 2 2 2 2 2 2

22 2 2

2 2

c v c v

c c v cc v

μμη η = γ − γ = γ −

= − =−

Manifestly invariant in all inertial frames

2 cμμη η =


0 0p m m c= η = γ 1 1xp m m v= η = γ

( ) ( ) ( ) p m mc,m mc, v m c,vμ μ= η = γ η = γ γ = γ

Proper Momentum = mass x proper velocity

Scalar product:2 2 2 20 1 2 3

p p p g p

p p p p

μ μ νμ μν=

= − − −( )

( ) ( )

2 2 2 2 2 2 2 2 2 2

2 22 2 2 2

2 2

mp p c m v m c v

m c c v m cc v

μμ = γ − γ = γ −

= − =− Manifestly

invariant in all inertial frames

2 2 p p m cμμ =

2 2yp m m v= η = γ 3 3

zp m m v= η = γ2

2

1

1 vc

γ =

−

22 2 2 2 2 2

2 m Ep p c m v p.pc

μμ = γ − γ = −


Quantization!

Equation of Motion ….

Time evolution of state vector….

2 2 2 2 2 2

22 2

2

m p p c m v

E p.p m cc

μμ = γ − γ

= − =


DYNAMICAL VARIABLES

m ?

Appears in both of the TWO most famous equations

Relativistic Energy

?F ma=2 ?E mc=

2E mc= γ2

2

1

1 vc

γ =

−12 2

2 221 vE mc mc

c

−⎛ ⎞

= γ = −⎜ ⎟⎝ ⎠

22 22 2

2 2

1 1 11 2 212 2

v vE mc mc ....c ! c

⎛ ⎞⎛ ⎞+⎜ ⎟⎜ ⎟⎛ ⎞⎝ ⎠⎜ ⎟= γ = + + +⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎝ ⎠


m ?

2

2

1

1 vc

γ =

−

12 2

2 221 vE mc mc

c

−⎛ ⎞

= γ = −⎜ ⎟⎝ ⎠

4 62 2 2

2 4

1 3 52 8 16

v vE mc mc mv m m ....c c

= γ = + + + +

22 22 2

2 2

1 1 11 2 212 2

v vE mc mc ....c ! c

⎛ ⎞⎛ ⎞+⎜ ⎟⎜ ⎟⎛ ⎞⎝ ⎠⎜ ⎟= γ = + + +⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎝ ⎠

constantrest energy

NR K.E.

2E mc= γRelativistic Energy

Note! We measure only changes in K.E.


4 62 2 2

2 4

1 3 52 8 16

v vE mc mc mv m m ....c c

= γ = + + + +

constantrest energy NR K.E.

2E mc= γRelativistic Energy

Relativistic Kinetic Energy2 2 2 1relativisticT mc mc mc ( )= γ − = γ −

2 2

2

20

1

1

00

? forphoton

mv c

E mc mcvc =

=

⎡ ⎤⎢ ⎥⎢ ⎥= γ =⎢ ⎥

−⎢ ⎥⎣ ⎦

= for Photon

?


The question of the rest mass of the photon is connected with how well do we know the inverse square law.

Note that Coulomb.

1( ) ~ ; or ( ) ~ ?

μ−

r cheV r V r

r r

0 μ → ⇒Inverse force requires:

so that the force would vary as: 2

1( ) ~ ,

1 .

V rr

r


Electrostatic potential

Photon: massless particle

26

2 2

2

2 photon 0 and

1 00

1form v c

E mc mcvc

= =

⎡ ⎤⎢ ⎥⎢ ⎥= γ = =⎢ ⎥

−⎢ ⎥⎣ ⎦

222 2

2 holdsEm c p p pc

μμ= = − ←2

2

20 Em p E pcc

= ⇒ = ⇒ =For

massless photons

h cE pc c h h= = = = νλ λ


Defined by de Brogliewavelength

Some books define

RELATIVISTIC MASS

as2

2

1

1rel rest restm m m

vc

= γ =

−2 restE m c= γ

( ) ( ) ( ) QUANTIZATION!

Next :p m mc,m mc, v m c,vμ μ= η = γ η = γ γ = γ


; op opq q p p i→ → = − ∇

( )QUANTIZATION of p m m c,vμ μ= η = γ

p i i i ,x ct

μ μ

μ

∂ ∂⎛ ⎞= ∂ = ≡ −∇⎜ ⎟∂ ∂⎝ ⎠

p i i i ,x ctμ μ μ

∂ ∂⎛ ⎞= ∂ = ≡ ∇⎜ ⎟∂ ∂⎝ ⎠


( )

( )

1 2 3

0

;

1

op opq q p ,p ,p p p i

Ep mc i ic c t ct

→ ≡ → = − ∇

∂ ∂= γ = → =

∂ ∂


p i i i ,x ct

μ μ

μ

∂ ∂⎛ ⎞= ∂ = ≡ −∇⎜ ⎟∂ ∂⎝ ⎠

p i i i ,x ctμ μ μ

∂ ∂⎛ ⎞= ∂ = ≡ ∇⎜ ⎟∂ ∂⎝ ⎠


2 2 0p p m cμμ⇒ − =

p i i i ,x ct

p i i i ,x ct

μ μ

μ

μ μ μ

∂ ∂⎛ ⎞= ∂ = = −∇⎜ ⎟∂ ∂⎝ ⎠

∂ ∂⎛ ⎞= ∂ = = ∇⎜ ⎟∂ ∂⎝ ⎠

QUANTIZATION

operatorsOperate on state vector, wavefunction

2 2 0op

p p m cμμ⎡ ⎤− ψ =⎣ ⎦

0

p=

E p ic ct

i

∂= =

∂− ∇

2 2 2 2 2 2

22 2

2

m p p c m v

E p.p m cc

μμ = γ − γ

= − =


2 2 0p p m cμμ − =

2 2 2 4E p pc m c= +i

QUANTIZATION!

2 2 2 2 2 2

22 2

2

m p p c m v

E p.p m cc

μμ = γ − γ

= − =

22 2

2 0E p p m cc

− − =i

( )22

2 2

1 0mcc t

⎡ ⎤∂ ⎛ ⎞− + ∇ ∇ − ψ =⎢ ⎥⎜ ⎟∂ ⎝ ⎠⎢ ⎥⎣ ⎦i

( )( )

0

1 2 3

1

op

Ep mc i ic c t ct

p ,p ,p p p i

∂ ∂= γ = =

∂ ∂

≡ = − ∇

→

→


Klein-Gordon equation

Quantization!

Equation of Motion ….

Time evolution of state vector

Questions? [email protected]

2 2 2 2 2 2

22 2

2

mp p c m v

E p.p m cc

μμ = γ − γ

= − =



[email protected]

Unit 3(ii) Lecture 14


Hydrogen Atom - 2




2 2 0p p m cμμ − =

2 2 2 4E p pc m c= +i

QUANTIZATION!

2 2 2 2 2 2

22 2

2

m p p c m v

E p.p m cc

μμ = γ − γ

= − =

22 2

2 0E p p m cc

− − =i

( )22

2 2

1 0mcc t

⎡ ⎤∂ ⎛ ⎞− + ∇ ∇ − ψ =⎢ ⎥⎜ ⎟∂ ⎝ ⎠⎢ ⎥⎣ ⎦i

( )( )

0

1 2 3

1

op

Ep mc i ic c t ct

p ,p ,p p p i

∂ ∂= γ = =

∂ ∂

≡ = − ∇

→

→


( )22

2 2

1 0mcc t

⎡ ⎤∂ ⎛ ⎞− + ∇ ∇ − ψ =⎢ ⎥⎜ ⎟∂ ⎝ ⎠⎢ ⎥⎣ ⎦i

2

0mc⎡ ⎤⎛ ⎞+ ψ =⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

( )2

2 2

1c t x xμ

μ

⎡ ⎤∂ ∂ ∂= − ∇ ∇ =⎢ ⎥∂ ∂ ∂⎣ ⎦

i D’Alembertian Operator

Klein-Gordon equation

KG Eq. indefinite probability density


1928: Dirac Relativistic QMProc. Roc. Soc. (Lond.) A117 610 (1928)Proc. Roc. Soc. (Lond.) A118 351 (1928)

P.A.M.Dirac: Principles of Quantum Mechanics


Dirac equation:- fundamental role in accounting for

atomic properties and processes.2 2 0p p m cμ

μ − =( )

( )

0

1 2 3

1

op

Ep mc i ic c t ct

p ,p ,p p p i

∂ ∂= γ = =

∂ ∂

≡ = − ∇

→

→

How would one get1st ORDER TIME DERIVATIVE?

37

http://physicsworld.com/cws/article/print/1705/1/pw1102091

2 2 0p p m cμμ − = ( )

( )

0

1 2 3

1

op

Ep mc i ic c t ct

p ,p ,p p p i

∂ ∂= γ = =

∂ ∂

≡ = − ∇

→

→

How would one get 1st ORDER TIME DERIVATIVE?

( )

0 1 2 3 2 20 1 2 3

0 2 20

20 2 2

0

0

0

p p p p p p p p m c

p p p.p m c

p p.p m c

+ + + − =

− − =

− − = ( )20 2 2

0

0

p

p m c

=

− =

IF

THEN

( )( )0 0 0factorize : p mc p mc+ − = ( )( )

0

0

0

0

p mc

p mc

+ =

− =Either or Both


How would one factorize

2 2 0p p m cμμ − =

( )( )

0

1 2 3

1

op

Ep mc i ic c t ct

p ,p ,p p p i

∂ ∂= γ = =

∂ ∂

≡ = − ∇

→

→

( )20 2 2 0p p.p m c− − =

0p ≠WHEN

( )( )2 2 p p m c p mc p mcμ κ λμ κ λ− = β + γ −

Explore ? ?


( )( )0 0 0factorize : p mc p mc+ − = ( )20 2 2

0

0

p

p m c

=

− =

WHEN:

40

2 2 0p p m cμμ − = ( )

( )

0

1 2 3

1

op

Ep mc i ic c t ct

p ,p ,p p p i

∂ ∂= γ = =

∂ ∂

≡ = − ∇

→

→

( )( )2 2 p p m c p mc p mcμ κ λμ κ λ− = β + γ −

Explore ? ?

{ }{ }

0 1 2 3

0 1 2 3

, , ,

, , ,

κ

λ

β ≡ β β β β

γ ≡ γ γ γ γ

8 coefficients to be determined

( )( )2 2

2 2

p p m c p mc p mc

p p mc p mc p m c

μ κ λμ κ λ

κ λ κ λκ λ κ λ

− = β + γ −

= β γ − β + γ −


( )( )2 2

2 2

p p m c p mc p mc

p p mc p mc p m c

μ κ λμ κ λ

κ λ κ λκ λ κ λ

− = β + γ −

= β γ − β + γ −

( ) p p p p mc pμ κ λ κ κμ κ λ κ= β γ − β − γ

p p p p mc p mc pμ κ λ κ κμ κ λ κ κ= β γ − β + γ

lhs: quadratic in momentum

quadratic in momentum

Linear term in momentum

κ κ⇒ β = γ4 terms 16 terms

p p p pμ κ λμ κ λ⇒ = γ γ

≡


p p p pμ κ λμ κ λ= γ γ

0 1 2 30 1 2 3p p p p p p p p p pμ λ λ λ λ

μ λ λ λ λ= γ γ + γ γ + γ γ + γ γ

0 0 1 0 2 0 3 00 0 1 0 2 0 3 0

0 1 1 1 2 1 3 10 1 1 1 2 1 3 1

0 2 1 2 2 2 3 20 2 1 2 2 2 3 2

0 3 1 3 2 3 3 30 3 1 3 2 3 3 3

+

+

+

p p p p p p p p p p

p p p p p p p p

p p p p p p p p

p p p p p p p p

μμ = γ γ + γ γ + γ γ + γ γ

γ γ + γ γ + γ γ + γ γ

γ γ + γ γ + γ γ + γ γ

γ γ + γ γ + γ γ + γ γ

0123

λ =λ =λ =λ =


0 0 1 0 2 0 3 00 0 1 0 2 0 3 0

0 1 1 1 2 1 3 10 1 1 1 2 1 3 1

0 2 1 2 2 2 3 20 2 1 2 2 2 3 2

0 3 1 3 2 3 3 30 3 1 3 2 3 3 3

+

+

+

p p p p p p p p p p

p p p p p p p p

p p p p p p p p

p p p p p p p p

μμ = γ γ + γ γ + γ γ + γ γ

γ γ + γ γ + γ γ + γ γ

γ γ + γ γ + γ γ + γ γ

γ γ + γ γ + γ γ + γ γ

i j j ipp p p= we can combine the terms having common, equal, factors


( ) ( )

( ) ( )

( ) ( )

0 0 1 1 2 2 3 30 0 1 1 2 2 3 3

0 1 1 0 0 2 2 00 1 0 2

0 3 3 0 1 2 2 10 3 1 2

1 3 3 1 2 3 3 21 3 2 3

+

p p p p p p p p p p

p p p p

p p p p

p p p p

μμ = γ γ + γ γ + γ γ + γ γ

γ γ + γ γ + γ γ + γ γ

+ γ γ + γ γ + γ γ + γ γ

+ γ γ + γ γ + γ γ + γ γ

( ) ( ) ( ) ( )2 2 2 20 1 2 3lhs : p p p p p pμμ = − − −


( ) ( )

( ) ( )

( ) ( )

0 0 1 1 2 2 3 30 0 1 1 2 2 3 3

0 1 1 0 0 2 2 00 1 0 2

0 3 3 0 1 2 2 10 3 1 2

1 3 3 1 2 3 3 21 3 2 3

+

p p p p p p p p p p

p p p p

p p p p

p p p p

μμ = γ γ + γ γ + γ γ + γ γ

γ γ + γ γ + γ γ + γ γ

+ γ γ + γ γ + γ γ + γ γ

+ γ γ + γ γ + γ γ + γ γ

( ) ( ) ( ) ( )2 2 2 20 1 2 3lhs : p p p p p pμμ = − − −


0

1 0 0 00 1 0 00 0 1 00 0 0 1

⎡ ⎤⎢ ⎥⎢ ⎥γ =⎢ ⎥−⎢ ⎥−⎣ ⎦

1

0 0 0 10 0 1 00 1 0 01 0 0 0

⎡ ⎤⎢ ⎥⎢ ⎥γ =⎢ ⎥−⎢ ⎥−⎣ ⎦

3

0 0 1 00 0 0 11 0 0 0

0 1 0 0

⎡ ⎤⎢ ⎥−⎢ ⎥γ =⎢ ⎥−⎢ ⎥⎣ ⎦

2

0 0 00 0 00 0 0

0 0 0

ii

ii

−⎡ ⎤⎢ ⎥⎢ ⎥γ =⎢ ⎥⎢ ⎥−⎣ ⎦

2 21 ×

2 21 ×−

12 2×σ

12 2×−σ

22 2×σ

22 2×−σ

32 2×σ

32 2×−σ


2 2

1 01

0 1×

⎡ ⎤= ⎢ ⎥⎣ ⎦

12 2

0 11 0×

⎡ ⎤σ = ⎢ ⎥

⎣ ⎦

22 2

00i

i×

−⎡ ⎤σ = ⎢ ⎥

⎣ ⎦3

2 2

1 00 1×

⎡ ⎤−σ = ⎢ ⎥−⎣ ⎦

2 2 2 20

2 2 2 2

1 00 1

× ×

× ×

⎡ ⎤γ = ⎢ ⎥−⎣ ⎦

2 2 2 24 4

2 2 2 2

00

ii

i× ×

×× ×

⎡ ⎤σγ = ⎢ ⎥−σ⎣ ⎦

1 2 3i , ,=


4 4 matrices×

( )( )2 20 p p m c p mc p mcμ κ λμ κ λ= − = γ + γ −

Factorization employing the

matrices is possible!4 4×γ

( ) 0p mcκκγ + = ( ) 0p mcκ

κγ − =

( ) 0p mcκκγ − = p i i i ,

x ctκ κ κ

∂ ∂⎛ ⎞= ∂ = = ∇⎜ ⎟∂ ∂⎝ ⎠

( ) 0i mcκκγ ∂ − ψ = ( )

4 1 4 14 4 0i mc

×

κκ ××

γ ∂ − ψ =

The Dirac Equation can be solved for the Coulomb potential exactly. ( ) 0p mc− ψ =


( ) 0p mcκκγ − =

( )0 1 2 30 1 2 3 0p p p p mcγ + γ + γ + γ − =

1 2 32 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

0 1 2 31 2 32 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

2 2 2 2

2 2 2 2

1 0 0 0 00 1 0 0 0

1 00

0 1

p p p p

mc

× × × × × × × ×

× × × × × × × ×

× ×

× ×

⎡ ⎤ ⎡ ⎤ ⎡ ⎤σ σ σ⎡ ⎤+ + +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥− −σ −σ −σ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

⎡ ⎤− =⎢ ⎥

⎣ ⎦

0 1 2 3

1 0 0 0 0 0 0 1 0 0 0 0 0 1 00 1 0 0 0 0 1 0 0 0 0 0 0 0 10 0 1 0 0 1 0 0 0 0 0 1 0 0 00 0 0 1 1 0 0 0 0 0 0 0 1 0 0

1 0 0 00 1 0 0

00 0 1 00 0 0 1

ii

p p p pi

i

mc

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥+ + +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥− − −⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥− − −⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

⎡ ⎤⎢ ⎥⎢ ⎥− =⎢ ⎥⎢ ⎥⎣ ⎦


( ) 0p mcκκγ − =

( )( )

0 3 1 2 1

0 1 2 3 2

3 1 2 0 3

1 2 3 0 4

00

00

0

p mc p p ip up mc p ip p u

p p ip (p mc) up ip p (p mc) u

− −⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥− + −⎢ ⎥ ⎢ ⎥ =⎢ ⎥ ⎢ ⎥− − − − +⎢ ⎥ ⎢ ⎥− + − +⎢ ⎥ ⎣ ⎦⎣ ⎦

0 1 21

2

3

43

1 0 0 0 0 0 0 1 0 0 00 1 0 0 0 0 1 0 0 0 00 0 1 0 0 1 0 0 0 0 00 0 0 1 1 0 0 0 0 0 0

0 0 1 0 1 0 0 00 0 0 1 0 1 0 01 0 0 0 0 0 1 0

0 1 0 0 0 0 0 1

ii

p p pi u

i uuu

p mc

⎧ ⎫−⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎪ ⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎪ ⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥+ +⎪ ⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥− − ⎡⎪ ⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢− − −⎪⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎪ ⎢⎨ ⎬

⎡ ⎤ ⎡ ⎤⎪ ⎪⎢ ⎥ ⎢ ⎥⎪ ⎪− ⎣⎢ ⎥ ⎢ ⎥⎪ ⎪+ −⎢ ⎥ ⎢ ⎥−⎪ ⎪⎢ ⎥ ⎢ ⎥⎪ ⎪⎣ ⎦ ⎣ ⎦⎩ ⎭

0

⎤⎥⎥ =

⎢ ⎥⎢ ⎥

⎦


( )4 1 4 14 4

0i mc×

κκ ××

γ ∂ − ψ =

Dirac equation introduces first-order time derivative operator,

p i i i ,x ctκ κ κ

∂ ∂⎛ ⎞= ∂ = = ∇⎜ ⎟∂ ∂⎝ ⎠

- but also note the fact that the momentum operators are only linear.


2 2

1 01

0 1×

⎡ ⎤= ⎢ ⎥⎣ ⎦

12 2

0 11 0×

⎡ ⎤σ = ⎢ ⎥

⎣ ⎦

22 2

00i

i×

−⎡ ⎤σ = ⎢ ⎥

⎣ ⎦3

2 2

1 00 1×

⎡ ⎤σ = ⎢ ⎥−⎣ ⎦

2 2 2 20

2 2 2 2

1 00 1

× ×

× ×

⎡ ⎤γ = ⎢ ⎥−⎣ ⎦

2 2 2 24 4

2 2 2 2

00

ii

i× ×

×× ×

⎡ ⎤σγ = ⎢ ⎥−σ⎣ ⎦1 2 3i , ,=

{ } { } { }0 1 2 3 0, , , , ,μγ = γ γ γ γ = γ γ = β βα

0 1 , −β = γ α = β γ

2 2 2 2

2 2 2 2

1 00 1

× ×

× ×

⎡ ⎤β = ⎢ ⎥−⎣ ⎦

2 2 2 24 4

2 2 2 2

00

ii

i× ×

×× ×

⎡ ⎤σα = ⎢ ⎥σ⎣ ⎦

PAULI Representation

DIRAC SPACE & ‘PAULI SPACE’August‐September 2012 PCD STiAP Unit 3 52

{ } { } { }0 1 2 3 0 : 4 matrices, , , , ,μγ = γ γ γ γ = γ γ = β βα

( )2

4 41 1 2 3 4 : 1 matrix, , ,μ×γ = ∀μ =

16 linearly independent Matrices built from Dirac matrices

; : 6 matricesi μ νμνσ = γ γ μ ≠ ν

2 2 2 25 0 1 2 3

2 2 2 2

0 11 0

= : 1 matrixi × ×

× ×

⎡ ⎤γ = γ γ γ γ ⎢ ⎥

⎣ ⎦5 0 1 2 3 : : 4 matrices, , ,μγ γ μ =

16 matrices constitute the “CLIFFORD ALGEBRA”

S

V

T

P

A

(S)

(V)

(T)

(P)

(A)

Γ

Γ

Γ

Γ

Γ

ψΓΨ


0†ψ = ψ γHow does transform?ψΓψ

( )4 1 4 14 4

0i mc×

κκ ××

γ ∂ − ψ =

Electromagnetic potential:

Dirac equation:

{ } { } { }0 1 2 3 0A A ,A ,A ,A A ,A ,Aμ ≡ ≡ ≡ φ

212

qansatz : L mv q (r,t) v A(r,t)c

= − φ + i

0d L Ldt q q

∂ ∂− =

∂ ∂

0

also similar equations for y & zx

d L L ;dt v x

∂ ∂− =

∂ ∂


( ) ( )2 2 212 x y z x x y y z z

qL m v v v q (r,t) v A (r,t) v A (r,t) v A (r,t)c

= + + − φ + + +

0x

d L Ldt v x

∂ ∂− =

∂ ∂

212

qL mv q (r,t) v A(r,t)c

= − φ + i

x xL qmv A (r,t)x c∂

= +∂

x x x xx

A (r,t) A (r,t) A (r,t) A (r,t)d L q dx dy dzmvdt x c x dt y dt z dt t

⎧ ⎫∂ ∂ ∂ ∂∂= + + + +⎨ ⎬∂ ∂ ∂ ∂ ∂⎩ ⎭

xx

dA (r,t)d L qmvdt x c dt

∂= +

∂

yx zx y z

A (r,t)A (r,t) A (r,t)L (r,t) q q qq v v vx x c x c x c x

∂∂ ∂∂ ∂φ= − + + +

∂ ∂ ∂ ∂ ∂


x

d L Ldt v x

∂ ∂=

∂ ∂


( ) ( )2 2 212 x y z x x y y z z


= + + − φ + + +

x x x xx

A (r,t) A (r,t) A (r,t) A (r,t)d L q dx dy dzmvdt x c x dt y dt z dt t

⎧ ⎫∂ ∂ ∂ ∂∂= + + + +⎨ ⎬∂ ∂ ∂ ∂ ∂⎩ ⎭

yx zx y z

A (r,t)A (r,t) A (r,t)L (r,t) q q qq v v vx x c x c x c x

∂∂ ∂∂ ∂φ= − + + +

∂ ∂ ∂ ∂ ∂

x x x xx

yx zx y z

A (r,t) A (r,t) A (r,t) A (r,t)q dx dy dzmvc x dt y dt z dt t

A (r,t)A (r,t) A (r,t)(r,t) q q qq v v vx c x c x c x

⎧ ⎫∂ ∂ ∂ ∂+ + + + =⎨ ⎬∂ ∂ ∂ ∂⎩ ⎭

∂∂ ∂∂φ= − + + +

∂ ∂ ∂ ∂

1 xx

yx z xy z

A (r,t)(r,t)mv qx c t

A (r,t)A (r,t) A (r,t) A (r,t)q qv vc y x c x z

∂∂φ⎧ ⎫= − +⎨ ⎬∂ ∂⎩ ⎭∂⎧ ⎫∂ ∂ ∂⎧ ⎫− − + −⎨ ⎬ ⎨ ⎬∂ ∂ ∂ ∂⎩ ⎭⎩ ⎭

x y z

x y z

y yz z x xx y z

ˆ ˆ ê e e

B Ax y z

A A A

A AA A A Aˆ ˆ ê e ey z x z x y

⎡ ⎤⎢ ⎥∂ ∂ ∂⎢ ⎥= ∇× = ⎢ ⎥∂ ∂ ∂

⎢ ⎥⎢ ⎥⎣ ⎦∂ ∂⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂⎛ ⎞= − + − + −⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠ ⎝ ⎠

x y z

x y z

y yz z x x

ˆ ˆ ê e e

v B v v v

A AA A A Ay z x z x y

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥× = ⎢ ⎥⎢ ⎥∂ ∂⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂⎛ ⎞⎢ ⎥− − −⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎢ ⎥⎝ ⎠⎝ ⎠ ⎝ ⎠⎣ ⎦

( ) y x z xy zx

A A A Av B v vx y x z

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


( ) ( ) y x z xy zx x

A A A Av v v vx y x z

B A∂⎛ ⎞∂ ∂ ∂⎛ ⎞× = × = − − −⎜ ⎟ ⎜ ⎟∂ ∂

∇×∂ ∂⎝ ⎠⎝ ⎠

( )1

1

xx x

x

A (r,t)(r,t) qmv q v Bx c t c

q E v Bc

∂∂φ⎧ ⎫= − + − ×⎨ ⎬∂ ∂⎩ ⎭⎛ ⎞= + ×⎜ ⎟⎝ ⎠

1LorentzF mv q E v B

c⎛ ⎞= = + ×⎜ ⎟⎝ ⎠

AEt

B A

∂= −∇φ −

∂= ∇×

Our ansatz for the Lagrangian is correct!


1 xx

yx z xy z

A (r,t)(r,t)mv qx c t

A (r,t)A (r,t) A (r,t) A (r,t)q qv vc y x c x z

∂∂φ⎧ ⎫= − +⎨ ⎬∂ ∂⎩ ⎭∂⎧ ⎫∂ ∂ ∂⎧ ⎫− − + −⎨ ⎬ ⎨ ⎬∂ ∂ ∂ ∂⎩ ⎭⎩ ⎭

( ) ( )2 2 212 x y z x x y y z z


= + + − φ + + +

212

qL mv q (r,t) v A(r,t)c

= − φ + i

x x xL qp mv A (r,t)x c∂

→ = +∂

qp mv A(r,t)c

→ +ep mv A(r,t)c

→ −

e ep p A i Ac c

μ μ μ μ μ μ⎛ ⎞→ π = − → ∂ −⎜ ⎟⎝ ⎠

Quantization( )0 0

generalized energy momentum 4-vector

ep mc , p Ac

μ ⎧ ⎫⎛ ⎞π = π = = γ π = −⎨ ⎬⎜ ⎟⎝ ⎠⎩ ⎭


Generalized momentum

e ep p A i Ac c

μ μ μ μ μ→ − → ∂ −

e ep p A i Ac cκ κ κ κ κ→ − → ∂ −

( )4 1 4 14 4

0i mc×

κκ ××

γ ∂ − ψ =

( )( )4 1 4 14 4

0i mc×

κκ ××

γ ∂ − ψ =

4 1 4 14 4

0ei A mcc ×

κκ κ ×

×

⎛ ⎞⎛ ⎞γ ∂ − − ψ =⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

4 1 4 14 4

0ei A mcc ×

κ κκ κ ×

×

⎛ ⎞γ ∂ − γ − ψ =⎜ ⎟⎝ ⎠


( )( )

4 1

0 1 2 30 1 2 3

0 1 2 30 1 2 3 4 1

4 4

0

i

e A A A Ac

mc× ×

×

⎛ ⎞γ ∂ + γ ∂ + γ ∂ + γ ∂⎜ ⎟⎜ ⎟

− γ + γ + γ + γ ψ =⎜ ⎟⎜ ⎟

−⎜ ⎟⎝ ⎠

4 1 4 14 4

0ei A mcc ×

κ κκ κ ×

×

⎛ ⎞γ ∂ − γ − ψ =⎜ ⎟⎝ ⎠

( )4 1

1 2 3

0 1 2 34 1

4 4

0

x y z

x y z

ict x x xe A A A Ac

mc× ×

×

⎛ ⎞∂ ∂ ∂ ∂⎛ ⎞β + βα + βα + βα⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠⎜ ⎟⎜ ⎟

− β − βα − βα − βα ψ =⎜ ⎟⎜ ⎟

−⎜ ⎟⎜ ⎟⎝ ⎠

{ } { }0, ,μγ = γ γ = β βα

00

11

22

33

AAAAAAAA

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥−⎣ ⎦⎣ ⎦


( )4 1

1 2 3

0 1 2 34 1

4 4

0

x y z

x y z

ict x x xe A A A Ac

mc× ×

×

⎛ ⎞∂ ∂ ∂ ∂⎛ ⎞β + βα + βα + βα⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠⎜ ⎟⎜ ⎟

− β − βα −βα −βα ψ =⎜ ⎟⎜ ⎟

−⎜ ⎟⎜ ⎟⎝ ⎠

4 1 4 14 4

0e ei i A mcct c c × ×

×

∂⎛ ⎞β + βα ∇ − βφ + βα − ψ =⎜ ⎟∂⎝ ⎠i i

4 1 4 14 4

0e ei p A mcct c c × ×

×

∂⎛ ⎞β − βα − βφ + βα − ψ =⎜ ⎟∂⎝ ⎠i i

4 1

4 1

2

4 4

ei c p A mc et c×

×

×

∂ψ ⎛ ⎞⎛ ⎞= α − + β + φ ψ⎜ ⎟⎜ ⎟∂ ⎝ ⎠⎝ ⎠i

Dirac equation in ‘standard representation’


( )4 1 4 1

2

4 4i c mc e

t × ××

∂ψ = α π + β + φ ψ

∂i


( )0 0

4

ep mc , p Ac

generalized energy momentum - vector

μ ⎧ ⎫⎛ ⎞π = π = = γ π = −⎨ ⎬⎜ ⎟⎝ ⎠⎩ ⎭

( )2 1 2 12

4 42 1 2 1

i c mc et

× ×

×× ×

ϕ ϕ⎛ ⎞ ⎛ ⎞∂= α π + β + φ⎜ ⎟ ⎜ ⎟χ χ∂ ⎝ ⎠ ⎝ ⎠

i

4 1

2 1

2 1×

×

×

ϕ⎛ ⎞ψ = ⎜ ⎟χ⎝ ⎠

2 2 2 24 4

2 2 2 2

00

× ××

× ×

σ⎡ ⎤α = ⎢ ⎥σ⎣ ⎦

2 2 2 24 4

2 2 2 2

1 00 1

× ××

× ×

⎡ ⎤β = ⎢ ⎥−⎣ ⎦


( )4 1 4 1

2

4 4i c mc e

t × ××

∂ψ = α π + β + φ ψ

∂i

Dirac equation in ‘standard representation’( )0 0p mc ,

ep Ac

μ

⎧ ⎫π = = γ⎪ ⎪

π = ⎨ ⎬⎛ ⎞π = −⎪ ⎪⎜ ⎟⎝ ⎠⎩ ⎭

2 2 2 24 4

2 2 2 2

1 00 1

× ××

× ×

⎡ ⎤β = ⎢ ⎥−⎣ ⎦


( )4 1 4 1

2

4 4i mc

t × ××

∂ψ = β ψ

∂

For a free electron at rest:

Solutions:

2 2

1 2

1 00 10 00 0

mc mci t i t

e e− −

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ψ = ψ =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

2 2

3 4

0 00 01 00 1

mc mci t i t

e e+ +

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ψ = ψ =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

Positive energy Negative energy

Resulting from 2 2 2 2 4H p c m c= +

Electron spin requires two components,- but Dirac equation admits

4-component wavefunctionDirac equation – admits ‘negative energy’ solutions-Positrons (anti particles)Carl D Anderson (1932)

All elementary particles in nature that obey Fermi statistics have two components and spin ½ (QFT)

4-component wavefunctionMulticomponent wavefunction : non-zero spin

2 2 4 2

2 4 2

E m c p.pc

E m c p.pc

= +

= ± +


Emission of a positron with energy E is the same as the absorption of an electron of energy -E.

Where is all the anti-matter?


4-component wavefunctionDirac equation – admits ‘negative energy’ solutions


2 2 4 2

2 4 2

E m c p.pc

E m c p.pc

= +

= ± +


[1] Reduction to 2-component ‘Pauli’ relation.

[2] Foldy Wouthuysen Transformation: “…… as far as it goes……..” – Trigg’s QM

2s = σ

2g =

Wolfgang Pauli1900 - 1958



2 2 4 2

2 4 2

E m c p.pc

E m c p.pc

= +

= ± +


4-component wavefunction

( )4 1 4 1

2

4 4i c mc e

t × ××

∂ψ = α π + β + φ ψ

∂i

Paul Adrien Maurice Dirac1902-1984



[email protected]

Unit 3(iii) Lecture 15


Hydrogen Atom


Reduction of Dirac Eq. to two-component formPAULI Equation

4 1

2 1

2 1×

×

×

ϕ⎛ ⎞ψ = ⎜ ⎟χ⎝ ⎠

Foldy-Wouthuysen Transformations - 1



4-component wavefunctionDirac equation – admits ‘negative energy’ solutions


2 2 4 2

2 4 2

E m c p.pc

E m c p.pc

= +

= ± +



[2] Foldy Wouthuysen Transformation: “…… as far as it goes……..” – Trigg’s QM

( )2 1 2 12

4 42 1 2 1

i c mc et

× ×

×× ×

ϕ ϕ⎛ ⎞ ⎛ ⎞∂= α π + β + φ⎜ ⎟ ⎜ ⎟χ χ∂ ⎝ ⎠ ⎝ ⎠

i

4 1

2 1

2 1×

×

×

ϕ⎛ ⎞ψ = ⎜ ⎟χ⎝ ⎠

2 2 2 2 2 2 2 24 4 4 4

2 2 2 2 2 2 2 2

0 1 0

0 0 1× × × ×

× ×× × × ×

σ⎡ ⎤ ⎡ ⎤α = β =⎢ ⎥ ⎢ ⎥σ −⎣ ⎦ ⎣ ⎦

2 2 2 2

2 2 2 2

2 2 2 2 2 12

2 2 2 2 2 1

2 2 2 2

2 2 2 2 4 4

00

1 0

0 1

1 00 1

c

r.h.s. mc

e

× ×

× ×

× × ×

× × ×

× ×

× × ×

⎛ ⎞σ⎡ ⎤π⎜ ⎟⎢ ⎥σ⎣ ⎦⎜ ⎟

⎜ ⎟ ϕ⎡ ⎤ ⎛ ⎞⎜ ⎟= + ⎜ ⎟⎢ ⎥− χ⎜ ⎟⎣ ⎦ ⎝ ⎠⎜ ⎟

⎡ ⎤⎜ ⎟+ φ⎢ ⎥⎜ ⎟⎣ ⎦⎝ ⎠

i Vector algebra!

Matrix algebra

QM, operator algebra


2 2 2 2

2 2 2 22 1 2 1

2 1 2 12 2 2 2 2 2 2 22

2 2 2 2 2 2 2 2 4 4

0 11 0

1 0 1 0

0 1 0 1

ci

tmc e

× ×

× ×× ×

× ×× × × ×

× × × × ×

⎛ ⎞⎡ ⎤σ π⎜ ⎟⎢ ⎥ϕ ϕ⎛ ⎞ ⎛ ⎞∂ ⎣ ⎦⎜ ⎟=⎜ ⎟ ⎜ ⎟⎜ ⎟χ χ∂ ⎡ ⎤ ⎡ ⎤⎝ ⎠ ⎝ ⎠⎜ ⎟+ + φ⎢ ⎥ ⎢ ⎥⎜ ⎟−⎣ ⎦ ⎣ ⎦⎝ ⎠

i

2 1 2 1 2 1 2 122 2

2 1 2 1 2 1 2 1

i c mc et

× × × ××

× × × ×

ϕ χ ϕ ϕ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂= σ π + + φ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟χ ϕ −χ χ∂ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

i

[ ] 21 i c mc et∂ϕ = σ πχ + ϕ + φϕ

∂i

[ ] 22 i c mc et∂χ = σ πϕ − χ + φχ

∂i



( )4 1 4 1

2

4 4i c mc e

t × ××

∂ψ = α π + β + φ ψ

∂i 4 1

2 1

2 1×

×

×

ϕ⎛ ⎞ψ = ⎜ ⎟χ⎝ ⎠

2i c mc et∂ϕ = σ πχ + ϕ + φϕ

∂i

2i c mc et∂χ = σ πϕ − χ + φχ

∂i

02

0 where Ei t

e E mc−ϕ ϕ⎡ ⎤ ⎡ ⎤

= =⎢ ⎥ ⎢ ⎥χ χ⎣ ⎦ ⎣ ⎦

slowly varying functions of time, :ϕ χ

20E i c mc e

tϕ ϕ χ ϕ ϕ⎡ ⎤ ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂

+ = σ π + + φ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥χ ∂ χ ϕ −χ χ⎣ ⎦ ⎣ ⎦ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠i

Stationary state solution

Recall! Stationary state solution to the time-dependent Schrodinger equation results in time-independent equation.


slowly varying functions of time, :ϕ χ

20E i c mc e

tϕ ϕ χ ϕ ϕ⎡ ⎤ ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂

+ = σ π + + φ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥χ ∂ χ ϕ −χ χ⎣ ⎦ ⎣ ⎦ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠i

0 0E i c E et

ϕ ϕ χ ϕ ϕ⎡ ⎤ ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂+ = σ π + + φ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥χ ∂ χ ϕ −χ χ⎣ ⎦ ⎣ ⎦ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

i

0

02i c e E

tϕ χ ϕ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ ⎡ ⎤∂

+ = σ π + φ −⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥∂ χ ϕ χ χ⎣ ⎦ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦i


2 02i c e mc

tϕ χ ϕ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ ⎡ ⎤∂

+ = σ π + φ −⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥∂ χ ϕ χ χ⎣ ⎦ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦i

22i c e mct∂

+ χ = σ πϕ + φχ − χ∂

i2nd Eq.

20 2c mc≈ σ πϕ − χi

2mcσ πϕ ≈ χi2mcσ π

⇒ ϕ ≈ χi : small

: largeχ

⇒ϕ

2i c e

t mc∂ σ π

+ ϕ ≈ σ π ϕ + φϕ∂

ii1st Eq.


slowly varying functions of time, :ϕ χ22since e mcφ

2 i e

t m∂ σ π σ π⎡ ⎤+ ϕ + φ ϕ⎢ ⎥∂ ⎣ ⎦

i i

a b a b i a bσ σ = + σ ×i i i i

iσ π σ π = π π + σ π× πi i i ie ep A p Ac c

⎛ ⎞ ⎛ ⎞π× π = − × −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠


( )e p A A pc

π× π = − × + ×


( )p A A p ?× + × =

( )p A A p f(r ) i Af(r ) A f(r )⎡ ⎤× + × = − ∇× + ×∇⎣ ⎦

( ) ( )p A A p f(r ) i A A f(r )× + × = − ∇× + ×∇

( ) ( )p A A p f(r ) i A f(r ) A f(r ) A f(r )⎡ ⎤× + × = − ∇× − ×∇ + ×∇⎣ ⎦

( ) ( ) ( )p A A p f(r ) i A f(r ) p A f(r )× + × = − ∇× = ×

( ) ( )p A A p p A× + × ≡ ×( ) ( )A A A∇× + ×∇ ≡ ∇×

( )e p A A pc

π× π = − × + ×

( ) ( )e e iep A i A Bc c c

π× π ≡ − × ≡ − − ∇× =

2 i e

t m∂ σ π σ π⎡ ⎤+ ϕ + φ ϕ⎢ ⎥∂ ⎣ ⎦

i i

a b a b i a bσ σ = + σ ×i i i i

iσ π σ π = π π + σ π× πi i i iie ieA Bc c

π× π = ∇× =

( )1 12 2

e Bm m c

⎛ ⎞σ π σ π = π π − σ⎜ ⎟⎝ ⎠

i i i i

2

2 2ei B e

t m mc⎡ ⎤∂ π

+ ϕ − σ + φ ϕ⎢ ⎥∂ ⎣ ⎦i

Pauli eq. for the ‘large’ component


iei Bc

⎛ ⎞σ π σ π = π π + σ ⎜ ⎟⎝ ⎠

i i i i

2

2 2ei B e

t m mc⎡ ⎤∂ π

+ ϕ = − σ + φ ϕ⎢ ⎥∂ ⎣ ⎦i

( )

2

2 22

2

2 2

2 2 2

e ep A p Ac c

m mp e eA p p A Am mc mc

⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟π ⎝ ⎠ ⎝ ⎠=

= − + +

i

i i

( )2 2

2 2 2p ie A A

m m mcπ

= + ∇ +∇i i

( )( )

2

A A f A f Af

A f A f A f

A f

∇ +∇ = ∇ +∇

= ∇ + ∇ + ∇

= ∇

i i i i

i i i

i

120

now, for uniform ,B

A r B

A

A B

= − ×

∇ =

∇× =

i

( )2 2

22 2 2

p ie Am m mcπ

= + ∇i2

2p e A pm mc

= − i

Pauli eq. for the large component

2 12 2p e r B pm mc

⎛ ⎞= − − ×⎜ ⎟⎝ ⎠

i


2 2 12 2 2

p e r B pm m mcπ ⎛ ⎞= − − ×⎜ ⎟

⎝ ⎠i

2 2 12 2 2

p e r B pm m mcπ

= + × i2 2 1

2 2 2p e B r p

m m mcπ

= − × i

2 2 21 12 2 2 2 2

p e p eB r p Bm m mc m mcπ

= − × = −i i

2 2 12 2 2

p e Bm m mcπ

= − i


2

2 2ei B e

t m mc⎡ ⎤∂ π

+ ϕ = − σ + φ ϕ⎢ ⎥∂ ⎣ ⎦i

Pauli eq. for the large component

2 2 12 2 2 2

and p e B sm m mcπ

= − = σi

2 12 2p e ei B s B e

t m mc mc⎡ ⎤∂

+ ϕ = − − + φ ϕ⎢ ⎥∂ ⎣ ⎦i i

( )2

22 2p ei s B e

t m mc⎡ ⎤∂

+ ϕ = − + + φ ϕ⎢ ⎥∂ ⎣ ⎦i

⇒


22e s Bmc

i


ei

r p= ×

1 îA Ac

μ =

r

p

2e vi eT r

= =π ( )21 1

2 21 1

2 2 2

v evrˆ ê r A Ac r c

e e ev r L Lc c m mc

⎛ ⎞μ = π =⎜ ⎟π⎝ ⎠

= − × = − = −

12B Le ; gmc

μ = =

s s Bsgμ = − μ

2L L Be L Lgmc

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

L sB

g L g s+μ = −μ 1

2L

s

gg

=

=2B

s+μ = −μ

μ


Gyromagnetic ratio; g=2g value: measure of MAGNETIC MOMENT- Dimensionless magnetic moment

( )2

22 2p ei s B e

t m mc⎡ ⎤∂

+ ϕ = − + + φ ϕ⎢ ⎥∂ ⎣ ⎦i

( )

2

22 2

2 2

B

B

s

emce s e smc mc

+μ = −μ

μ =

+μ = − = − +

4 1

4 1

2

4 4

ei c p A mc et c×

×

×

∂ψ ⎛ ⎞⎛ ⎞= α − + β + φ ψ⎜ ⎟⎜ ⎟∂ ⎝ ⎠⎝ ⎠i


( )2 1 2

2 2p ei s B e

t m mc⎡ ⎤∂

+ ϕ = − + + φ ϕ⎢ ⎥∂ ⎣ ⎦i

4 1

2 1

2 1×

×

×

ϕ⎛ ⎞ψ = ⎜ ⎟χ⎝ ⎠

02

0 where Ei t

e E mc−ϕ ϕ⎡ ⎤ ⎡ ⎤

= =⎢ ⎥ ⎢ ⎥χ χ⎣ ⎦ ⎣ ⎦

Pauli2-component

2-components: required for the two spin degrees of freedom for the electron with j=½

To determine expectation values to order , one has to employ all the four components

2

2

vc


4 1

4 1

2

4 4

ei c p A mc et c×

×

×

∂ψ ⎛ ⎞⎛ ⎞= α − + β + φ ψ⎜ ⎟⎜ ⎟∂ ⎝ ⎠⎝ ⎠i

2 2 2 24 4

2 2 2 2

00

× ××

× ×

σ⎡ ⎤α = ⎢ ⎥σ⎣ ⎦

2 2 2 24 4

2 2 2 2

1 00 1

× ××

× ×

⎡ ⎤β = ⎢ ⎥−⎣ ⎦

‘Odd’ operators scramble the upper two components with the lower two components.

Dirac equation describes both electrons (positive energy states) and the positrons (the "negative energy' states).

Also, it describes both the spin orientations, hence the solution has four components.


2 2 2 4E p pc m c= +i 2 2 4

2 2 4

E p pc m c

E p pc m c

= + +

= − +

i

i0

0

E : empty

E : occupied

⟩

⟨

in vacuum

Fully occupied and fully unoccupied states are not observable

Particle-AntiParticle annihilation

2hν Singly occupied particle, or anti-particle state is observable.


Canonical transformations of the wavefunction – i.e. changes in the representation of the wavefunction

mix/unmix particle / anti-particle components.

Admixture of particle and anti-particle states?T.D.Newton & E.P.Wigner (1949, Rev. Mod.Phys.)

Interference between positive and negative energy components produces fluctuations (at ~1021Hz: zitterbewegung) of the position of an electron.


Electron charge is smeared out over a distance of the Compton wavelength of the electron.

The problems of generalizing the classical ‘position’ to relativistic quantum theory

- several contributors:

Dirac,

Pryce,

Newton and Wigner,

Foldy and Wouthuysen


"Zitterbewegung" is eliminated when you take expectation values for wave-packets made up completely positive energy states (or completely negative energy states), as achieved by the Foldy - Wouthuysen transformation.

The Zitterbewegung is caused by interference between the positive and negative energy components of the wave packet.• It vanishes if the wave packet is a superposition of only positive or only negative energy solutions.


L. L. Foldy and S. A. Wouthuysen, Phys Rev., 78, 29(1950)J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, McGraw-Hill (1964).



The Foldy-Wouthuysen

transformations also enable us cast

the Dirac theory in a form which

displays the different interaction terms

between an electron and an applied

field in easily interpretable form.Ref.: Bjorken & Drell, RQM, Ch.4


Non-Relativistic limit of the Dirac equation for an electron in an external potential is not reached simply by

The limiting process is attained by carrying out a unitary transformation which block diagonalizes the Dirac Hamiltonian and separates the positive and negative energy part of its spectrum.

This unitary transformation, is the Foldy–Wouthuysen transformation, and it is equivalent to a change of picture.

c →∞


( ) 2 1 2 12

4 42 1 2 1

mc c e it

× ×

×× ×

ϕ ϕ⎛ ⎞ ⎛ ⎞∂β + α π + φ =⎜ ⎟ ⎜ ⎟χ χ∂⎝ ⎠ ⎝ ⎠

i

Foldy-Wouthuysen transformations:

- Systematic procedure to go from 4-component Dirac theory to a 2-component theory.

H it

∂ψψ =

∂

H it

∂ψψ =

∂'H' ' i

t∂ψ

ψ =∂

FW ''H'' '' it

∂ψψ =

∂'''H''' ''' i

t∂ψ

ψ =∂

- Transform the relativistic equations in such a way that the odd operators play an ignorable role inthe transformed representation.

( )2

4 4 DiracH mc c

×= β + θ + ε



4 1

4 1

2

4 4

ei c p A mc et c×

×

×

∂ψ ⎛ ⎞⎛ ⎞= α − + β + φ ψ⎜ ⎟⎜ ⎟∂ ⎝ ⎠⎝ ⎠i

2 2 2 2 2 2 2 24 4 4 4

2 2 2 2 2 2 2 2

0 1 0

0 0 1× × × ×

× ×× × × ×

σ⎡ ⎤ ⎡ ⎤α = β =⎢ ⎥ ⎢ ⎥σ −⎣ ⎦ ⎣ ⎦

( )4 1

4 1

2

4 4Free electron: i c p mc

t×

××

∂ψ= α + β ψ

∂i

iS

' iS iSop op op

' ee e−

ψ → ψ = ψ

Ω → Ω = Ω

iS iS iS iSH' e He i e et

+ − + −∂= −

∂

2i pS p

mc m− ⎛ ⎞= βα⋅ ω⎜ ⎟

⎝ ⎠

For free electron, consider time-independent

Hermitian operator:

2s = σ2g = Wolfgang Pauli

1900 - 1958



2 2 4 2

2 4 2

E m c p.pc

E m c p.pc

= +

= ± +


4-component wavefunction

( )4 1 4 1

2

4 4i c mc e

t × ××

∂ψ = α π + β + φ ψ

∂i

Paul Adrien Maurice Dirac1902-1984

( )2 1 2

2 2p ei s B e

t m mc⎡ ⎤∂

+ ϕ = − + + φ ϕ⎢ ⎥∂ ⎣ ⎦i



[email protected]

Unit 3(iv) Lecture 16


Hydrogen Atom



4 1

2 1

2 1×

×

×

ϕ⎛ ⎞ψ = ⎜ ⎟χ⎝ ⎠



H it

∂ψψ =

∂

iS' eψ → ψ = ψ


( ) 2 1 2 12

4 42 1 2 1

mc c e it

× ×

×× ×

ϕ ϕ⎛ ⎞ ⎛ ⎞∂β + α π + φ =⎜ ⎟ ⎜ ⎟χ χ∂⎝ ⎠ ⎝ ⎠

i

97

iS iS'i e H i e 't t

+ −⎡ ⎤∂ψ ∂⎛ ⎞= − ψ⎜ ⎟⎢ ⎥∂ ∂⎝ ⎠⎣ ⎦


+ − + −∂⇒ = −

∂

'i H' 't

∂ψ= ψ

∂

OBJECTIVE: odd operators play an ignorable role.


( )4 1

4 1

2


t×

××

∂ψ= α + β ψ

∂i

iS iSH' e He+ −=2i pS p

mc m− ⎛ ⎞= βα⋅ ω⎜ ⎟

⎝ ⎠

( )2iS iScH' e ep mc+ −α + β= i

( )2iS iS c p me cH' e+ − βα= β +i

( )( )

1 2

2

iS iS

iS iS

H' e e

e

c p mc

c p mc e

+ −

+ −

−

β

=

= β

β β α +

α +

i

i

12 2

iiiS

p pp pmc m mc me e e

⎡ ⎤ ⎡ ⎤− −⎛ ⎞ ⎛ ⎞βα⋅ ω βα⋅ ω⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠

−⎦ ⎦− ⎣ ⎣β = β = β

( ) 2 1 2 12

4 42 1 2 1

mc c e it

× ×

×× ×

ϕ ϕ⎛ ⎞ ⎛ ⎞∂β + α π + φ =⎜ ⎟ ⎜ ⎟χ χ∂⎝ ⎠ ⎝ ⎠

i


( )4 1

4 1

2


t×

××

∂ψ= α + β ψ

∂i

2i pS p

mc m− ⎛ ⎞= βα⋅ ω⎜ ⎟

⎝ ⎠1

2 2ii

iSp pp p

mc m mc me e e⎡ ⎤ ⎡ ⎤− −⎛ ⎞ ⎛ ⎞βα⋅ ω βα⋅ ω⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥

⎝ ⎠ ⎝ ⎠−

⎦ ⎦− ⎣ ⎣β = β = β

0

n

x

n

xen!

∞

=

= ∑

( )

0 0

1 12 2

nnnn

iS

n n

p pp pmc m mc me

n! n!

∞ ∞−

= =

⎡ ⎤ ⎡ ⎤− −⎛ ⎞ ⎛ ⎞ ⎛ ⎞βα ⋅ ω βα ⋅ ω⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝β

β = β =⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦∑ ∑

( )2iS iS c p me cH' e+ − βα= β +i


( ) ( ) ( )Now: 1n nnp pβα ⋅ βα ⋅β = − β

( ) ( )

0

1 12

nnn

iS

n

n ppe

n!mc m∞

−

=

⎡ ⎤⎛ ⎞ ⎛ ⎞− βα ⋅ ω⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦β

β = ∑

( ) ( ) ( )

0

1 1

12

nn

iS

n

nnn pp

men!

c m∞−

=

⎡ ⎤⎛ ⎞ ⎛ ⎞− βα ⋅ ω⎜ ⎟ ⎜ ⎟⎢ ⎥⎝− β

β ⎝ ⎠⎣ ⎦= ⎠∑

( )0

1 2

∞−

=

βα ⋅β = β

⎡ ⎤⎛ ⎞ ⎛ ⎞ω⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦∑

niS

n

nn

en

pm!

pmc

iS iSe e− +β = β

0

n

x

n

xen!

∞

=

= ∑

2i pS p

mc m− ⎛ ⎞= βα⋅ ω⎜ ⎟

⎝ ⎠

0 0 =

12 2

n n

iS

n n

n i p pi pe

n! nm c m

!

pmc m∞ ∞

= =

⎡ ⎤ ⎡ ⎤− ⎛ ⎞ ⎛ ⎞βα ⋅ ω βα ⋅ ω⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦= ∑ ∑likewise


( )2iS iS c p me cH' e+ − βα= β +i iS iSe e− +β = β

( )2 2i S c p' mcH e+ β= β α +i

( )( )

2 2 2

2 2

i S

i S

c p mcH' e

c p mce

+

+

β α += β

βα +=

i

i0

12

=

n

iS

n

pc

!

pe

nm m∞

=

⎡ ⎤⎛ ⎞βα ⋅ ω⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦∑

( )0

2

1n

n

ppmc m c p mcH'

n!

∞

=

⎧ ⎫⎪ ⎪⎪ ⎪= β⎨ ⎬

⎡ ⎤⎛ ⎞βα ⋅ ω

⎪ ⎪⎪ ⎪⎩ ⎭

⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦ α +∑ i


( )0

2

1n

n

ppmc m c p mcH'

n!

∞

=

⎧ ⎫⎪ ⎪⎪ ⎪= β⎨ ⎬

⎡ ⎤⎛ ⎞βα ⋅ ω

⎪ ⎪⎪ ⎪⎩ ⎭

⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦ α +∑ i

( ) ( )0

21 n nn

n

p p c p mcm

'! m

Hnc

∞

=

βα ⋅ ⎡ ⎤⎛ ⎞ ⎛ ⎞ω α +⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣

⎧ ⎫⎪ ⎪= β⎨ ⎬⎪⎩ ⎭⎦ ⎪∑ i

( )

22

23 43 4

12

1 13 4

1 p p p pm mc ! m

c p mcp p p p

mc ! m

mcH'

.....mc ! m

⎡ ⎤βα ⋅ βα ⋅⎛ ⎞⎛ ⎞ ⎛ ⎞ω + ω⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎣ ⎦ α +⎡ ⎤ ⎡ ⎤βα ⋅ βα ⋅⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ω ω⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥

⎧ ⎫+⎪ ⎪

⎪ ⎪= β⎨ ⎬⎪ ⎪+ +⎪ ⎪⎩ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎭⎦ ⎣

i


( )

22

23 43 4

12

1 13 4

1 p p p pm mc ! m

c p mcp p p p

mc ! m

mcH'

.....mc ! m

⎡ ⎤βα ⋅ βα ⋅⎛ ⎞⎛ ⎞ ⎛ ⎞ω + ω⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎣ ⎦ α +⎡ ⎤ ⎡ ⎤βα ⋅ βα ⋅⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ω ω⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥

⎧ ⎫+⎪ ⎪

⎪ ⎪= β⎨ ⎬⎪ ⎪+ +⎪ ⎪⎩ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎭⎦ ⎣

i

Γ = βα ⋅NOW, let : p

2 2 2 2 2 2 2 22

2 2 2 2 2 2 2 2

22 2 2 24 4

2 2 2 2

0 00 0

01

0

p pp p

p pp

p p

× × × ×

× × × ×

× ××

× ×

−σ ⋅ −σ ⋅⎡ ⎤ ⎡ ⎤Γ = ⎢ ⎥ ⎢ ⎥σ ⋅ σ ⋅⎣ ⎦ ⎣ ⎦

−σ ⋅ σ ⋅⎡ ⎤= = − ×⎢ ⎥−σ ⋅ σ ⋅⎣ ⎦

2 2 2 2 2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2 2 2 2 2

1 0 0 00 1 0 0

× × × × × ×

× × × × × ×

σ σ⎡ ⎤ ⎡ ⎤ ⎡ ⎤βα = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥− σ −σ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2

0 00 0

pp

p× × × ×

× × × ×

σ σ ⋅⎡ ⎤⎡ ⎤Γ = ⋅ = ⎢ ⎥⎢ ⎥−σ −σ ⋅⎣ ⎦ ⎣ ⎦


( )

22

23 43 4

12

1 13 4

1 p p p pm mc ! m

c p mcp p p p

mc ! m

mcH'

...m ! m

.c

⎡ ⎤βα ⋅ βα ⋅⎛ ⎞⎛ ⎞ ⎛ ⎞ω + ω⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎣ ⎦ α +⎡ ⎤ ⎡ ⎤βα ⋅ βα ⋅⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ω ω⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎝ ⎠

⎡ ⎤+⎢ ⎥

⎢ ⎥= β⎢ ⎥⎢ ⎥+ +⎢ ⎥⎣ ⎦⎝ ⎠⎣ ⎦ ⎣ ⎦

i

( ) 222 24 41 ×Γ = βα ⋅ = − × = −p p p

( )

( )( ) ( )( )

222

23 43 4

22 2

1 12

1 1 13 4

1

1

p p ppm mc ! m

c p mcp pp p p

mc ! m m

mc

c ! m

H'

.....

⎡ ⎤βα ⋅ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ω + − ω⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦ α +⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞βα ⋅ − ω − ω⎜ ⎟ ⎜ ⎟ ⎜

⎧ ⎫+⎪ ⎪

⎪ ⎪= β⎨ ⎬⎪ ⎪+ +⎪ ⎪⎩ ⎭

⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦

i

( ) ( )

( )( )

2 42 422 2

233

3

1 1 1 12 4

1

1 13

.....H'

pp mc

p pp pmc ! m mc ! m

c p mcp p p pp .....

m mc p ! m

⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ − ω − ω⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦ α +⎡ ⎤⎛ ⎞βα ⋅ βα ⋅⎛ ⎞ ⎛ ⎞ ⎛ ⎞ω − ω +⎜ ⎟⎜ ⎟ ⎜

⎧ ⎫+⎪ ⎪

⎪ ⎪= β⎨ ⎬⎪ ⎪+ +⎪ ⎪⎩

⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎣ ⎭⎦

i

Even powers

Odd powers


( )

( ) ( )

( )

2 2 4 6

0

1 2 1 3 5 7

1

11

2 2 4 6

12 1 3 5 7

n n

n

n n

n

cos

sin

....n! ! ! !

....n ! ! ! !

∞

=

− −∞

=

⎡ ⎤− ξ ξ ξ ξ⎢ ⎥ = − + − +⎢ ⎥⎣ ⎦⎡ ⎤− ξ ξ ξ ξ⎢ ⎥ = ξ − + − +

−⎢ ⎥⎣

=

⎦

ξ

ξ =

∑

∑

( ) ( )

( )( )

2 42 422 2

233

3

1 1 1 1 12 4

1 13

.....

H'p

p m

p pp pmc ! m mc ! m

c p mcp p pp .....

m mcc ! m

⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ − ω − ω⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦α +

⎧ ⎫⎡ ⎤βα ⋅ ⎪ ⎪⎛ ⎞ ⎛ ⎞ ⎛ ⎞ω − ω +⎨ ⎬⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦⎪ ⎪⎩ ⎭

⎧ ⎫+⎪ ⎪

⎪ ⎪= β⎨ ⎬⎪ ⎪+ +⎪ ⎪⎩ ⎭

i

( )2p p p p pcos c p mcmc m mc

H' sim

np

βα ⋅⎛ ⎞ ⎛ ⎞⎡ ⎤⎛ ⎞ ⎛ ⎞= + β⎢ ⎥⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ω ω α +⎜ ⎟ ⎜ ⎟

⎣ ⎠ ⎝ ⎠ ⎦⎝i

( )2p p pcos c p mcmc mc

H' sinp

⎡ ⎤⎛ ⎞ ⎛ ⎞ω β= + β⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

α ⋅ ωα +i

pm

⎛ ⎞ω = ω⎜ ⎟⎝ ⎠


( )2p p pcos c p mcmc mc

H' sinp

⎡ ⎤⎛ ⎞ ⎛ ⎞ω β= + β⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

α ⋅ ωα +i

pm

⎛ ⎞ω = ω⎜ ⎟⎝ ⎠

( )

( )( )

2

2

p pcos c p cos mcmc mcp p p p pc mc

m

H'

sin sinp pc mc

ω ωα + +

β α ⋅ α ω βα

⎛ ⎞ ⎛ ⎞= β⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞+ β⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⋅ ω+

i

i

( ) 2

22

H'

sin sinp p

p pcos c p cos mcmc mc

p p p pc mcmc mc

⎛ ⎞ ⎛ ⎞=ω ω

α + +

β ω βα ⋅

β⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞+ β⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝

ω+

⎠

i

2 2 2 2 2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2 2 2 2 2

2 2 2 2

2 2 2 2

2 2 2 22 2 2 2 2 2 2 2

2 2 2 22 2 2 2 2 2

1 0 0 00 1 0 0

00

1 00 00 10

pp

p

p pp

p p

× × × × × ×

× × × × × ×

× ×

× ×

× ×× × × ×

× ×× × ×

σ σ⎡ ⎤ ⎡ ⎤ ⎡ ⎤βα = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥− σ −σ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

σ ⋅⎡ ⎤βα ⋅ = ⎢ ⎥−σ ⋅⎣ ⎦

σ ⋅ −σ ⋅⎡ ⎤ ⎡ ⎤βα ⋅ β = =⎢ ⎥ ⎢ ⎥−−σ ⋅ −σ ⋅⎣ ⎦⎣ ⎦ 2 20

p×

⎡ ⎤= −α ⋅⎢ ⎥

⎣ ⎦


( ) 2

22

H'

sin sinp p


p p p pc mcmc mc

⎛ ⎞ ⎛ ⎞=ω ω

α + +

β ω βα ⋅

β⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞+ β⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝

ω+

⎠

i βα ⋅ β = −α ⋅p p

( ) 2

22

ω ωα + +

β

⎛ ⎞ ⎛ ⎞= β⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞+ ⎜ ⎟ ⎜ ⎟⎝

ω α ⋅ ω−

⎠ ⎝ ⎠

iH'

sin sinp p


p p p pc mcmc mc

2

2

⎡ ⎤ω⎛ ⎞ ⎛ ⎞= β +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎡ ⎤⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝

ω+⎢ ⎥

⎣ ⎦α ⋅ ω

−⎠⎣ ⎦

ω

H' sin

si

p pmc cos pcmc mc

p p ppc cos mcmcp mc

n

2 2H p p p p pmc cos pc pc cos mcmc mc mc mc

' sin sinp

⎡ ⎤ω ω α ⋅ ω ω+ −

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= β +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦⎢ ⎥⎣ ⎦


2 1p p p p mc pmc cos pc sin pc cosmc mc

H' tam mc

np c p

⎡ ⎤ω ω α ⋅ ω ω+

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= β +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦−⎢ ⎥

⎣ ⎦

1choose: p mc ptanm p mc

−⎛ ⎞ω = ω =⎜ ⎟⎝ ⎠

11 1 1 1 0tan tanmc p mc p mc ptanp mc p mc p mc

−⎛ ⎞⎛ ⎞ = − = − =⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

ω−

only ‘even’ operatorno ‘odd’ operator

p

mc

2 2 2m c p+

FW triangle

2 p pmc cos pc sinmc mc

H' ⎡ ⎤ω ω⎛ ⎞ ⎛ ⎞= β ⎜ ⎟ ⎜+⎢ ⎥⎣

⎟⎝ ⎠ ⎠⎦⎝


1choose: p mc ptanm p mc

−⎛ ⎞ω = ω =⎜ ⎟⎝ ⎠ p

mc

2 2 2m c p+

2 p pmc cos pc sinmc mc

H' ⎡ ⎤ω ω⎛ ⎞ ⎛ ⎞= β ⎜ ⎟ ⎜+⎢ ⎥⎣

⎟⎝ ⎠ ⎠⎦⎝

1 1 1

2 2USING:

1 1− − −δ δδ = =

+ δ + δtan sin cos

2 2 2 2

2 2

2 2 2 2

1

1 1

pmcmc pH' cc m c p

p pm c m c

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥+ = +⎢ ⎥

+ +

β

⎢ ⎥⎣ ⎦

= β


4 1

4 1

2

4 4

ei c p A mc et c×

×

×

∂ψ ⎛ ⎞⎛ ⎞= α − + β + φ ψ⎜ ⎟⎜ ⎟∂ ⎝ ⎠⎝ ⎠i

iS iS iS iSH' e He e i et

+ − + −∂⎛ ⎞= + −⎜ ⎟∂⎝ ⎠

( )A,φElectron in an EM field

CHOICE of the operator S?


+ − + −∂⎛ ⎞= + −⎜ ⎟∂⎝ ⎠

1 iS iS i S i S' e e lim e e+ − + ξ − ξ

ξ→Ω = Ω = Ω

1 where i S i S' lim F( ) F( ) e e+ ξ − ξ

ξ→Ω = ξ ξ = Ω

1 0 0

n n

nn

F( )' limn!

∞

ξ→= ξ=

⎧ ⎫⎡ ⎤ξ ∂ ξ⎪ ⎪Ω = ⎨ ⎬⎢ ⎥∂ξ⎣ ⎦⎪ ⎪⎩ ⎭∑

0 2 2 3 3 4 4

2 3 410 0 0 0

00 1 2 3 4

F F F F' lim F( ) ...! ! ! ! !ξ→

ξ= ξ= ξ= ξ=

⎧ ⎫⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞ξ ξ ∂ ξ ∂ ξ ∂ ξ ∂⎪ ⎪Ω = ξ = + + + +⎨ ⎬⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟∂ξ ∂ξ ∂ξ ∂ξ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭

2 3 4

2 3 40 0 0 0

1 1 102 6 24

F F F F' F( ) ..ξ= ξ= ξ= ξ=

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂Ω = ξ = + + + + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟∂ξ ∂ξ ∂ξ ∂ξ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

0

n

n

Fn, ?ξ=

⎛ ⎞∂∀ =⎜ ⎟∂ξ⎝ ⎠



+ − + −∂⎛ ⎞= − ⎜ ⎟∂⎝ ⎠


ξ→Ω = ξ ξ = Ω

2 3 4

2 3 40 0 0 0

1 1 102 6 24

F F F F' F( ) ..ξ= ξ= ξ= ξ=

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂Ω = ξ = + + + + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟∂ξ ∂ξ ∂ξ ∂ξ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

0

n

n

Fn, ?ξ=

⎛ ⎞∂∀ =⎜ ⎟∂ξ⎝ ⎠

( ) ( )= iS i S i S i S i SF( ) e e e iSe+ ξ − ξ + ξ − ξ∂ ξΩ + Ω −

∂ξ

=i i S i S i S i SF( ) e S e i e Se+ ξ − ξ + ξ − ξ∂ ξΩ − Ω

∂ξ [ ]i S i Sie S, e+ ξ − ξ−

= Ω



ξ→Ω = ξ ξ = Ω

2 3 4

2 3 40 0 0 0

1 1 102 6 24

F F F F' F( ) ..ξ= ξ= ξ= ξ=

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂Ω = ξ = + + + + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟∂ξ ∂ξ ∂ξ ∂ξ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

[ ] i S i SF( ) ie S, e+ ξ − ξ−

∂ ξ= Ω

∂ξ[ ]

22

2 i S i SF( ) i e S, S, e+ ξ − ξ∂ ξ ⎡ ⎤= Ω⎣ ⎦∂ξ

[ ]3

33 i S i SF( ) i e S, S, S, e+ ξ − ξ∂ ξ ⎡ ⎤⎡ ⎤= Ω⎣ ⎦⎣ ⎦∂ξ

[ ] n

n i S i Sn

F( ) i e S, S, S,... S, e+ ξ − ξ∂ ξ ⎡ ⎤⎡ ⎤⎡ ⎤= Ω⎣ ⎦⎣ ⎦⎣ ⎦∂ξ

[ ]0

n

nn

F( ) i S, S, S,... S,ξ=

⎡ ⎤∂ ξ ⎡ ⎤⎡ ⎤⎡ ⎤= Ω⎢ ⎥ ⎣ ⎦⎣ ⎦⎣ ⎦∂ξ⎣ ⎦


[ ]0

n

nn

F( ) i S, S, S,... S,ξ=

⎡ ⎤∂ ξ ⎡ ⎤⎡ ⎤⎡ ⎤= Ω⎢ ⎥ ⎣ ⎦⎣ ⎦⎣ ⎦∂ξ⎣ ⎦

0

For 0 0 n

n

F( )n , F( )ξ=

⎡ ⎤∂ ξ= = ξ = = Ω⎢ ⎥∂ξ⎣ ⎦

[ ]0

For 1 F( )n , i S,ξ=

⎡ ⎤∂ ξ= = Ω⎢ ⎥∂ξ⎣ ⎦

[ ]2

22

0

For 2 F( )n , i S, S,ξ=

⎡ ⎤∂ ξ ⎡ ⎤= = Ω⎢ ⎥ ⎣ ⎦∂ξ⎣ ⎦

[ ]3

33

0

For 3 F( )n , i S, S, S,ξ=

⎡ ⎤∂ ξ ⎡ ⎤⎡ ⎤= = Ω⎢ ⎥ ⎣ ⎦⎣ ⎦∂ξ⎣ ⎦

2 3 4

2 3 40 0 0 0

1 1 102 6 24

F F F F' F( ) ..ξ= ξ= ξ= ξ=

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂Ω = ξ = + + + + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟∂ξ ∂ξ ∂ξ ∂ξ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠



+ − + −∂⎛ ⎞= + −⎜ ⎟∂⎝ ⎠



+ − + −∂⎛ ⎞= + −⎜ ⎟∂⎝ ⎠

[ ] [ ] [ ]

[ ]

2 3

4

+ S, S,H + S, S,H +2 3

+ S, S,H +.... 4

iS iS

i iH' H i S,H S,! !

i S, S, e i e! t

+ −

⎡ ⎤⎡ ⎤ ⎡ ⎤= + ⎣ ⎦ ⎣ ⎦⎣ ⎦

∂⎛ ⎞⎡ ⎤⎡ ⎤⎡ ⎤ + −⎜ ⎟⎣ ⎦⎣ ⎦⎣ ⎦ ∂⎝ ⎠



+ − + −∂⎛ ⎞= + −⎜ ⎟∂⎝ ⎠

[ ] [ ] [ ]

[ ]

2 3

4

+ S, S,H + S, S,H +2 3

+ S, S,H +.... 4

iS iS

i iH' H i S,H S,! !

i S, S, e i e! t

+ −

⎡ ⎤⎡ ⎤ ⎡ ⎤= + ⎣ ⎦ ⎣ ⎦⎣ ⎦

∂⎛ ⎞⎡ ⎤⎡ ⎤⎡ ⎤ + −⎜ ⎟⎣ ⎦⎣ ⎦⎣ ⎦ ∂⎝ ⎠

[ ] [ ] [ ]

[ ]

2 3

4

+ S, S,H + S, S,H +2 3

+ S, S,H +.... 4

1 2 6

i iH' H i S,H S,! !

i S, S,!

iS S,S S, S,S .....

⎡ ⎤⎡ ⎤ ⎡ ⎤= + ⎣ ⎦ ⎣ ⎦⎣ ⎦

⎡ ⎤⎡ ⎤⎡ ⎤⎣ ⎦⎣ ⎦⎣ ⎦

⎡ ⎤⎡ ⎤ ⎡ ⎤− − + +⎣ ⎦ ⎣ ⎦⎣ ⎦


[ ] [ ] [ ]

[ ]

2 3

4

+ S, S,H + S, S,H +2 3

+ S, S , H +.... 4

1 2 6

i iH' H i S,H S,! !

i S, S,!

iS S,S S, S,S .....

⎡ ⎤⎡ ⎤ ⎡ ⎤= + ⎣ ⎦ ⎣ ⎦⎣ ⎦

⎡ ⎤⎡ ⎤⎡ ⎤⎣ ⎦⎣ ⎦⎣ ⎦

⎡ ⎤⎡ ⎤ ⎡ ⎤− − + +⎣ ⎦ ⎣ ⎦⎣ ⎦

( )2

2

H mc c p eA e

mc

=β + α⋅ − + φ

=β + θ+ ε S such that the role of odd operators

1diminishes by

Choose

m⎛ ⎞⎜ ⎟⎝ ⎠

○


[ ] [ ] [ ]

[ ]

2 3

4

+ S, S,H + S, S,H +2 3

+ S, S , H +.... 4

1 2 6

i iH' H i S,H S,! !

i S, S,!

iS S,S S, S,S .....

⎡ ⎤⎡ ⎤ ⎡ ⎤= + ⎣ ⎦ ⎣ ⎦⎣ ⎦

⎡ ⎤⎡ ⎤⎡ ⎤⎣ ⎦⎣ ⎦⎣ ⎦

⎡ ⎤⎡ ⎤ ⎡ ⎤− − + +⎣ ⎦ ⎣ ⎦⎣ ⎦

( )2

2

H mc c p eA e

mc

=β + α⋅ − + φ

=β + θ+ ε

2H mc≈β

S such that the role of odd operators 1diminishes by

Choose

m⎛ ⎞⎜ ⎟⎝ ⎠

○


[ ] [ ] [ ]

[ ]

2 3

4

+ S, S,H + S, S,H +2 3

+ S, S , m + 4

1 2 6

i iH' H i S,H S,! !

i S, S,!

iS S,S S, S,S .

⎡ ⎤⎡ ⎤ ⎡ ⎤= + ⎣ ⎦ ⎣ ⎦⎣ ⎦

⎡ ⎤⎡ ⎤⎡ ⎤β⎣ ⎦⎣ ⎦⎣ ⎦

⎡ ⎤⎡ ⎤ ⎡ ⎤− − +⎣ ⎦ ⎣ ⎦⎣ ⎦

[ ] [ ] [ ]

[ ]

1 S, S,H S, S,H +2 6

1 + S, S , m + 24

1 2 6

iH' H i S,H S,

S, S,

iS S,S S, S,S .

⎡ ⎤⎡ ⎤ ⎡ ⎤= + − −⎣ ⎦ ⎣ ⎦⎣ ⎦

⎡ ⎤⎡ ⎤⎡ ⎤β⎣ ⎦⎣ ⎦⎣ ⎦

⎡ ⎤⎡ ⎤ ⎡ ⎤− − +⎣ ⎦ ⎣ ⎦⎣ ⎦


[ ] [ ] [ ]

[ ]

2 3

4

+ S, S,H + S, S,H +2 3

1 + S, S , m 4 2 6

i iH' H i S,H S,! !

i iS, S, S S,S S, S,S .!

⎡ ⎤⎡ ⎤ ⎡ ⎤≡ + ⎣ ⎦ ⎣ ⎦⎣ ⎦

⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤⎡ ⎤β − − +⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦

( )2

2

H mc c p eA e

mc

= β + α⋅ − + φ

= β + θ+ ε - in the first four terms

1 22iSmc−βθ

=

2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 22

1 0 00 1 0

2

× × × ×

× × × ×

σ⎡ ⎤ ⎡ ⎤ ⎛ ⎞− ⋅ −⎜ ⎟⎢ ⎥ ⎢ ⎥− σ ⎝ ⎠⎣ ⎦ ⎣ ⎦=

ei c p Ac

mc

choose

First Foldy-Wouthuysen transformation ……


[ ] [ ] [ ]

[ ]

2 3

4

+ S, S,H + S, S,H +2 3

1 + S, S , m 4 2 6

i iH' H i S,H S,! !


⎡ ⎤⎡ ⎤ ⎡ ⎤≡ + ⎣ ⎦ ⎣ ⎦⎣ ⎦

⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤⎡ ⎤β − − +⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦

1 22iSmc−βθ

=I FWT:[ ] 222

ii S,H i , mcmc−

−

− βθ⎡ ⎤= β + θ + ε⎢ ⎥⎣ ⎦

22 −

βθ⎡ ⎤θ −⎢ ⎥⎣ ⎦,

mc2

22 −

βθ⎡ ⎤= β +⎢ ⎥⎣ ⎦, mc

mc 22 −

βθ⎡ ⎤ε⎢ ⎥⎣ ⎦,

mc

( ) ( ) ( )2 22 2

1 1 12 2 2mc mc

= βθβ −β θ + βθ − θβθ + βθε − εβθ

( ) ( ) ( )2 22 2

1 1 12 2 2mc mc

= −ββθ −β θ + βθ + βθθ + βθε −βεθ

βθ = −θββε = εβ

[ ] [ ]2

2 2

12

i S,H ,mc mc− −

βθ= −θ + + β θ ε


[ ] [ ] [ ]

[ ]

2 3

4

+ S, S,H + S, S,H +2 3

1 + S, S , m 4 2 6

i iH' H i S,H S,! !


⎡ ⎤⎡ ⎤ ⎡ ⎤≡ + ⎣ ⎦ ⎣ ⎦⎣ ⎦

⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤⎡ ⎤β − − +⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦

1 22iSmc−βθ

=I FWT:

βθ = −θββε = εβ

[ ] [ ]{ }2

2 2i iS, S,H S, i S,H

− −− −⎡ ⎤⎡ ⎤ =⎣ ⎦ ⎣ ⎦

[ ] [ ]

[ ]

2 2

2 2

2

2 2 2

1 2 2 2

1 2 2 2

i iS, S,H S, ,mc mc

i i , ,mc mc mc

− −−−

−−

⎡ ⎤βθ⎡ ⎤ = −θ + + β θ ε⎢ ⎥⎣ ⎦⎣ ⎦

⎡ ⎤− βθ βθ= −θ + + β θ ε⎢ ⎥

⎣ ⎦

[ ] [ ]2 2 3

2 2 4 2 4

12 2 2 8i S, S,H , ,

mc m c m c− −

βθ θ⎡ ⎤ ⎡ ⎤= − − − θ θ ε⎣ ⎦⎣ ⎦


[ ] [ ]2

2 2

12

i S,H ,mc mc− −

βθ= −θ + + β θ ε

[ ] [ ] [ ]

[ ]

2 3

4

+ S, S,H + S, S,H +2 3

1 + S, S , m 4 2 6

i iH' H i S,H S,! !


⎡ ⎤⎡ ⎤ ⎡ ⎤≡ + ⎣ ⎦ ⎣ ⎦⎣ ⎦

⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤⎡ ⎤β − − +⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦

1 22−βθ

=iSmc

I FWT:

2 2 4 2 2 2 8i i i iS,S S, ,

mc m c −−−

⎡ ⎤⎛ ⎞− − − βθ −⎡ ⎤ ⎡ ⎤= = θ θ⎢ ⎥⎜ ⎟ ⎣ ⎦⎣ ⎦ ⎝ ⎠⎣ ⎦

1 22iSmc−βθ

=

2

1'odd' terms

H' mc ' '

m

⇒ ≡ β + ε + θ

⎛ ⎞Ο⎜ ⎟⎝ ⎠

∼


[ ]

[ ]

2

2 4

2 3 6 2 4 2 4

3

2 2 4 2

12 8 8 8

2 3 2

H' mc ' 'where

i' , , ,mc m c m c m c

i' ,mc m c mc

= β + ε + θ

βθ βθ ⎡ ⎤⎡ ⎤ε = ε + − − θ θ ε − θ θ⎣ ⎦ ⎣ ⎦

β θ βθθ = θ ε − +

The leading ‘odd’ term is now of 1 m⎛ ⎞⎜ ⎟⎝ ⎠

○

2

2

eH mc c p A ec

mc

⎛ ⎞= β + α⋅ − + φ⎜ ⎟⎝ ⎠

= β + θ+ ε

1 1 1 1 iS iS iS iSH' e He e i et

+ − + −∂⎛ ⎞= + −⎜ ⎟∂⎝ ⎠





+ − + −∂⎛ ⎞= + −⎜ ⎟∂⎝ ⎠

[ ]

[ ]

2 4

2 3 6 2 4 2 4

3

2 2 4 2

12 8 8 8

2 3 2

βθ βθ ⎡ ⎤⎡ ⎤ε = ε + − − θ θ ε − θ θ⎣ ⎦ ⎣ ⎦



i' ,mc m c mc

The leading ‘odd’ term is now of

2= β + ε + θH' mc ' '

1 m⎛ ⎞⎜ ⎟⎝ ⎠

○


[email protected]

Unit 3(v) Lecture 17


Hydrogen Atom



4 1

2 1

2 1×

×

×

ϕ⎛ ⎞ψ = ⎜ ⎟χ⎝ ⎠



The Foldy-Wouthuysen

transformations also enable us cast

the Dirac theory in a form which

displays the different interaction terms

between an electron and an applied

field in easily interpretable form.Ref.: Bjorken & Drell, RQM, Ch.4


Bransden & Joachain: Physics of Atoms & Molecules

[ ]

[ ]

2

2 4

2 3 6 2 4 2 4

3

2 2 4 2

12 8 8 8

2 3 2

H' mc ' 'where


i' ,mc m c mc

= β + ε + θ

βθ βθ ⎡ ⎤⎡ ⎤ε = ε + − − θ θ ε − θ θ⎣ ⎦ ⎣ ⎦


The leading ‘odd’ term is now of 1 m⎛ ⎞⎜ ⎟⎝ ⎠

○

2

2

eH mc c p A ec

mc

⎛ ⎞= β + α⋅ − + φ⎜ ⎟⎝ ⎠

= β + θ+ ε


+ − + −∂⎛ ⎞= + −⎜ ⎟∂⎝ ⎠


To reduce the importance of the ‘odd’ terms further:

2 22i 'Smc−βθ

=II FWT:

2 2 2 2 iS iS iS iSH'' e H'e e i et

+ − + −∂⎛ ⎞= + −⎜ ⎟∂⎝ ⎠

2H'' mc ' "= β + ε + θ III FWT: 3 22i "Smc−βθ

=

3 3 2iS iSH''' e H" i e mc 't

+ −∂⎛ ⎞= − = β + ε⎜ ⎟∂⎝ ⎠

[ ]2 4

22 3 6 2 4 2 4

12 8 8 8

iH''' mc , , ,mc m c m c m cβθ βθ ⎡ ⎤⎡ ⎤= β + ε + − − θ θ ε − θ θ⎣ ⎦ ⎣ ⎦


2

2

2 2

12 2 2

ec p Ac e ep A p A

mc mc m c c

⎛ ⎞⎛ ⎞α ⋅ −⎜ ⎟⎜ ⎟ ⎧ ⎫⎧ ⎫θ ⎛ ⎞ ⎛ ⎞⎝ ⎠⎝ ⎠= = α ⋅ − α ⋅ −⎨ ⎬⎨ ⎬⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎩ ⎭⎩ ⎭

2

2

0 012 2 0 0

0 012 0 0

mc m

m

⎧ ⎫⎧ ⎫⎛ ⎞ ⎛ ⎞σ σθ ⎪ ⎪⎪ ⎪= ⋅ π ⋅ π⎨ ⎬⎨ ⎬⎜ ⎟ ⎜ ⎟σ σ⎪ ⎪⎪ ⎪⎝ ⎠ ⎝ ⎠⎩ ⎭⎩ ⎭

⎛ ⎞⎛ ⎞σ ⋅ π σ ⋅ π= ⎜ ⎟⎜ ⎟

σ ⋅ π σ ⋅ π⎝ ⎠⎝ ⎠

{ }2

24 42

1 1 2 2

imc m ×

θ= π + σ ⋅ π× π


[ ]2 4

22 3 6 2 4 2 4

12 8 8 8

iH''' mc , , ,mc m c m c m cβθ βθ ⎡ ⎤⎡ ⎤= β + ε + − − θ θ ε − θ θ⎣ ⎦ ⎣ ⎦

2

2

4 4

012 2 0

1 12

mc m

m ×

σ ⋅ π σ ⋅ π⎛ ⎞θ= ⎜ ⎟σ ⋅ π σ ⋅ π⎝ ⎠

= σ ⋅ π σ ⋅ π

{ }2

24 42

1 1 2 2

imc m ×

θ= π + σ ⋅ π× π

( ) ( )

( )

2

2

e ep A p Ac c

e ep p p A A p A Ac c

ie A Ac

⎛ ⎞ ⎛ ⎞π× π = − × −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= × − × + × + ×

= ∇× + ×∇

( ) ( ) ( )( ) ( ) ( )

A A f Af A f

A f A f A f

∇× + ×∇ = ∇× + ×∇

= ∇× − ×∇ + ×∇

( ) A A f Bf∇× + ×∇ =


{ }2

24 42

1 1 2 2

imc m ×

θ= π + σ ⋅ π× π

( ) A A f Bf∇× + ×∇ ≡

22

4 42

1 1 2 2

e Bmc m c ×

θ ⎧ ⎫= π − σ ⋅⎨ ⎬⎩ ⎭ 2

2

contributes to

in 2

H'''mcβθ

( )A A A B A∇× ≡ ∇× − ×∇ ≡ − ×∇

Note the difference!


( )ie A Ac

π× π = ∇× + ×∇

[ ]2 4

22 3 6 2 4 2 4

12 8 8 8

iH''' mc , , ,mc m c m c m c

⎛ ⎞θ θ ⎡ ⎤⎡ ⎤= β + − + ε − θ θ ε − θ θ⎜ ⎟ ⎣ ⎦ ⎣ ⎦⎝ ⎠

[ ]{ }2 4

22 3 6 2 4

1 2 8 8

H''' mc , , imc m c m c

⎛ ⎞θ θ ⎡ ⎤= β + − + ε − θ θ ε + θ⎜ ⎟ ⎣ ⎦⎝ ⎠

[ ] ( ) ( )d, i c p eA ,e icdt

d dc p,e ce A,e icdt dt

⎡ ⎤θ ε + θ = α ⋅ − φ + α ⋅ π⎣ ⎦

⎧ ⎫α π⎛ ⎞ ⎛ ⎞⎡ ⎤= α ⋅ φ − α ⋅ φ + ⋅ π + α ⋅⎡ ⎤ ⎨ ⎬⎜ ⎟ ⎜ ⎟⎣ ⎦ ⎣ ⎦ ⎝ ⎠ ⎝ ⎠⎩ ⎭

[ ] d d, i i ec , icdt dt

⎧ ⎫α π⎛ ⎞ ⎛ ⎞⎡ ⎤θ ε + θ = − α ⋅∇ φ + ⋅ π + α ⋅⎨ ⎬⎜ ⎟ ⎜ ⎟⎣ ⎦ ⎝ ⎠ ⎝ ⎠⎩ ⎭( )

( ) ( ) ( ), f f

f f f f f f

⎡ ⎤α ⋅∇ φ = α ⋅∇φ − φα ⋅∇⎣ ⎦

= α ⋅∇φ − φα ⋅∇ = α ⋅∇φ + α ⋅∇ φ − φα ⋅∇ = α ⋅∇φ


[ ]{ }2 4

22 3 6 2 4

1 2 8 8


⎛ ⎞θ θ ⎡ ⎤= β + − + ε − θ θ ε + θ⎜ ⎟ ⎣ ⎦⎝ ⎠

[ ] d d, i i ce , icdt dt

⎧ ⎫α π⎛ ⎞ ⎛ ⎞⎡ ⎤θ ε + θ = − α ⋅∇ φ + ⋅ π + α ⋅⎨ ⎬⎜ ⎟ ⎜ ⎟⎣ ⎦ ⎝ ⎠ ⎝ ⎠⎩ ⎭

( ) 1 A, Ec t

⎛ ⎞∂⎡ ⎤α ⋅∇ φ = α ⋅∇φ = α ⋅ − −⎜ ⎟⎣ ⎦ ∂⎝ ⎠

[ ] 1 A d d, i i ce E icc t dt dt

⎛ ⎞ ⎧ ⎫∂ α π⎛ ⎞ ⎛ ⎞θ ε + θ = α ⋅ + + ⋅ π + α ⋅⎜ ⎟ ⎨ ⎬⎜ ⎟ ⎜ ⎟∂ ⎝ ⎠ ⎝ ⎠⎩ ⎭⎝ ⎠

[ ] A d d, i i ce E i e ict dt dt

⎧ ⎫∂ α π⎛ ⎞ ⎛ ⎞θ ε + θ = α ⋅ + α ⋅ + ⋅ π + α ⋅⎨ ⎬⎜ ⎟ ⎜ ⎟∂ ⎝ ⎠ ⎝ ⎠⎩ ⎭

[ ] [ ]1 1 d d,H ; ,Hdt i t dt i tα ∂α π ∂π⎧ ⎫ ⎧ ⎫= α + = π +⎨ ⎬ ⎨ ⎬∂ ∂⎩ ⎭ ⎩ ⎭


[ ]{ }2 4

22 3 6 2 4

1 2 8 8


⎛ ⎞θ θ ⎡ ⎤= β + − + ε − θ θ ε + θ⎜ ⎟ ⎣ ⎦⎝ ⎠

[ ], iθ ε + θ

[ ]

[ ]

1

1

,Hi tAi ce E i e ic

t A,H ei t

⎧ ⎫∂α⎧ ⎫α + ⋅ π⎨ ⎬⎪ ⎪∂⎩ ⎭∂ ⎪ ⎪= α ⋅ + α ⋅ + ⎨ ⎬⎛ ⎞⎧ ⎫∂ ∂⎪ ⎪⎪ ⎪+ α ⋅ π −⎜ ⎟⎨ ⎬⎜ ⎟⎪ ⎪∂⎪ ⎪⎩ ⎭⎝ ⎠⎩ ⎭

[ ] [ ]1 1 A Ai ce E i e c ,H ,H iet t

⎧ ⎫∂ ∂⎪ ⎪= α ⋅ + α ⋅ + α ⋅ π + α ⋅ π − α ⋅⎨ ⎬∂ ∂⎪ ⎪⎩ ⎭

[ ], i i ce Eθ ε + θ = α ⋅

[ ]{ }2 4 2 4

1 8 8

i ce, , i , Em c m c

⎡ ⎤⎡ ⎤θ θ ε + θ = θ α ⋅⎣ ⎦ ⎣ ⎦


2 48ie c c , Em c

ep Ac

⎛ ⎞−⎜ ⎟⎝ ⎠

⎡ ⎤= α ⋅ α ⋅⎢ ⎥

⎣ ⎦

2 28ie , Em c

p⎡ ⎤= α ⋅ α ⋅⎣ ⎦ ( )2 2 8

ie Ec

pEpm

= α ⋅ α ⋅ − α ⋅ α ⋅

[ ]{ } ( )2 4 2 2 1 8 8

ie, , i E Em c m

pc

p⎡ ⎤θ θ ε + θ = α ⋅ α ⋅ − α ⋅ α ⋅⎣ ⎦

[ ]{ }2 4 2 2

0 00 01 8 8 0 00

0

p pE Eie, ,p p

im c m c E E

⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞σ ⋅ σ ⋅σ ⋅ σ ⋅⎡ ⎤θ θ ε + θ = −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎣ ⎦ ⎜ ⎟ ⎜ ⎟σ ⋅ σ ⋅σ ⋅ σ ⋅⎝ ⎠ ⎝ ⎠⎝ ⎠

⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭⎝ ⎠

[ ]{ }2 4 2 2

0 01

8 8 0 0

E Eie, , im c m

pc E

p

p E p

⎛ ⎞ ⎛ ⎞σ ⋅ σ ⋅ σ ⋅ σ ⋅⎡ ⎤

⎧ ⎫⎪ ⎪⎨ ⎬θ θ ε + θ = −⎜ ⎟ ⎜ ⎟⎣ ⎦ ⎜ ⎟ ⎜ ⎟σ ⋅ σ ⋅ σ ⋅ σ ⋅⎪ ⎪⎩ ⎠ ⎭⎝ ⎠ ⎝

[ ]{ } ( )2 4 2 2 4 41

8 81ie, , i E E

mp

cp

c m ×⎡ ⎤θ θ ε + θ = σ ⋅ σ ⋅ − σ ⋅ σ ⋅⎣ ⎦

[ ]{ } ( )2 2 42 44

1 1 8 8

ie, , i E i Ep E i Em c m

p pc

p ×⎡ ⎤θ θ ε + θ = ⋅ + σ⋅ − ⋅ − σ⋅ ×⎣ ⎦ ×


[ ]{ }2 4

22 3 6 2 4

1 2 8 8


⎛ ⎞θ θ ⎡ ⎤= β + − + ε − θ θ ε + θ⎜ ⎟ ⎣ ⎦⎝ ⎠

[ ]{ } ( )2 2 42 44

1 1 8 8

ie, , i E i Ep E i Em c m

p pc

p ×⎡ ⎤θ θ ε + θ = ⋅ + σ⋅ − ⋅ − σ⋅ ×⎣ ⎦ ×

( ) ( )( ) ( )( ) ( ) ( )

Ef i Ef i E f i E f

i E E i

p

p pf E E f

⋅ = − ⋅ = − ⋅ − ⋅

⎡ ⎤= − ⋅ +

∇ ∇ ∇

⎡ ⎤∇ ∇⎣ ⎦⋅ − = ⋅ + ⋅⎣ ⎦

[ ]{ }( ) ( )

2 4

2 2 2 2 2 2 2 2 2 2

1 8

8 8 8 8 8

, , im c

ie ie e ie eE E E E Em c m c

pm c m c m c

p p p p

⎡ ⎤θ θ ε + θ =⎣ ⎦

= ⋅ + ⋅ − σ ⋅ − ⋅ + σ ⋅ ××

( ) ( )p E p E E p⋅ = ⋅ + ⋅The range of applicability of the gradient does not go beyond the bracket in the first term on the rhs


[ ]{ }2 4

22 3 6 2 4

1 2 8 8


⎛ ⎞θ θ ⎡ ⎤= β + − + ε − θ θ ε + θ⎜ ⎟ ⎣ ⎦⎝ ⎠

[ ]{ }( )

2 4

2 2 2 2 2 2

1 8

8 8 8

, , im c

ie e eE E Em c

p pm

pc m c

×

⎡ ⎤θ θ ε + θ =⎣ ⎦

= ⋅ − σ ⋅ + σ ⋅ ×

( )p E p E E p× = × − ×

( )2 2 2 2 2 2 2 28 8 8 8ie e e eE E E Em c m

p p p pc m c m c

⋅ − σ ⋅ + σ= × ×⋅ + σ ⋅ ×

( )2 2

2 2 2 2 2 2 8 8

cur 4

le ie ediv E E Em c m c m c

p+ σ + ×⋅ ⋅= σ


[ ]{ }2 4

22 3 6 2 4

1 2 8 8


⎛ ⎞θ θ ⎡ ⎤= β + − + ε − θ θ ε + θ⎜ ⎟ ⎣ ⎦⎝ ⎠

ec c p Ac

⎛ ⎞θ = α π = α −⎜ ⎟⎝ ⎠

i i

( )

2

42

3 2

2 2

2 2 2 2 2 2

12 8 2

curl 8 8

4

H''' mc e Bm m c mc

e ie ee div E

ep Apc

pE Em c m c m c

⎛ ⎞⎜ ⎟⎜ ⎟= β + − −β σ ⋅ +⎜ ⎟⎜ ⎟⎝ ⎠

+ φ

⎛ ⎞−⎜

− − σ ⋅ − σ

⎟⎝ ⎠

×⋅

1r

V VÊ e rr r r

∂ ∂= − = −

∂ ∂

2 2

14

Vrm

r pr

ec

∂×

∂σ ⋅


[ ]{ }2 4

22 3 6 2 4

1 2 8 8


⎛ ⎞θ θ ⎡ ⎤= β + − + ε − θ θ ε + θ⎜ ⎟ ⎣ ⎦⎝ ⎠

22

4 42

1 1 2 2

e Bmc m c ×

θ ⎧ ⎫= π − σ ⋅⎨ ⎬⎩ ⎭

[ ]{ } ( )2 2

2 4 2 2 2 2 2 2

1 8 8 8

curl 4

e ie e, , i div E E Em c m c m c m c

p⎡ ⎤θ θ ε + θ = + σ ⋅ + σ ⋅⎣ ⎦ ×

Relativistic K.E. correction

( )

2

12 2 2 4 22

12 2 2

2 4 22 4 1

RelK.E.T E mc

p c m c mc

p cm c mcm c

= −

= + −

⎛ ⎞⎛ ⎞= + −⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠1

2 22

2 2

2 42

2 4

3 2

1 1

1 11 12 8

2 8

RelK.E.

pT mcm c

p pmc ..mc mc

p p ....m m c

⎡ ⎤⎛ ⎞⎢ ⎥= + −⎜ ⎟⎢ ⎥⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞ ⎛ ⎞= + − + −⎢ ⎥⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

= − +



( )

2

42

3 2

2 2

2 2 2 2 2 2

12 8 2

1 cu8 8 4

rl

epH''' mc e B e

m m c mc

e ie ediv E Em c m

Ap

c m c

c

V r pr r

⎛ ⎞⎜ ⎟⎜ ⎟= β + − −β σ ⋅ + φ⎜ ⎟⎜ ⎟⎝ ⎠

⎛ ⎞−⎜ ⎟⎝ ⎠

∂×

∂− − σ ⋅ + σ ⋅

( )

2

42

3 2

2 2

2 2 2 2 2 2

12 8 2

1c l8 4

r 8

u

H''' mc e Bm m c mc

e ie ee div E Em c m

ep Apc

Vrc rc m

⎛ ⎞⎜ ⎟⎜ ⎟= β + − −β σ ⋅ +⎜ ⎟⎜ ⎟⎝ ⎠

+

⎛ ⎞−⎜ ⎟⎝ ⎠

∂∂

φ − − σ ⋅ + σ ⋅

Magnetic dipole term

Relativistic K.E. correction

Darwin correctionzitterbewegung

Spin-orbit interaction (Thomas)

( )

2 24 2 2

3 2 2 2

2 2 22 2

1 18 2 2 2 2

1 1 22 2n n n

p p pm c m m c m c m

E V E E V Vm c m c

⎛ ⎞ ⎛ ⎞−− = − =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

− − ⎡ ⎤= − = − +⎣ ⎦

2ZeVr

= − 2 23 2

1 1 1 112

r n a r n a= =

⎛ ⎞+⎜ ⎟⎝ ⎠

&

( )

2 4 2 2 2

2 2 2 2

2 22 22 2

2 2

12 2

2 2

nmZ e me Z eE

n nZe Zmc mc

c n n

= − = −

α⎛ ⎞= − = −⎜ ⎟

⎝ ⎠

H.W.Show that:


2ec

⎛ ⎞α = ⎜ ⎟

⎝ ⎠

2 3142

Rel. nK.E.

Z nE En

⎡ ⎤⎢ ⎥α⎛ ⎞ ⎢ ⎥Δ = − −⎜ ⎟ ⎛ ⎞⎢ ⎥⎝ ⎠ +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

Ref.: Bransden & Joachain – Physics of Atoms & Molecules / Ch 5

( )

2 2 2 2

2 2 3

1 14 4

1 24

spin orbitradial

e V e VHm c r r m c r re sZe

m c r

−

∂ ∂= σ ⋅ = σ ⋅

∂ ∂

−= − ⋅

( ){ }

2

2 2 3

2 3 2

2 23 3

12

1 1 112 212

spin orbitZeH sm c r

Ze Z j( j ) ( ) s(s )m c n a

− = ⋅

⎡ ⎤= + − + − +⎢ ⎥⎛ ⎞ ⎣ ⎦+ +⎜ ⎟

⎝ ⎠

Spin-orbit correction- same order as ‘relativistic mass’ correction


2s = σ

2ec

⎛ ⎞α = ⎜ ⎟

⎝ ⎠

2 2

14

Vrm

r pr

ec

∂×

∂σ ⋅

( )

2 4

2 2

22

2

12

2

nmZ eE

nZ

mcn

= −

α= −

( )

( )

( )

2 32

2 23 3

332 2 2

2 2 23

424 2

3

31 14

14 12

31 14

14 12

31 14

14 12

spin orbit

Z j( j ) ( )ZeHm c n a

Z j( j ) ( )Ze mem c n

j( j ) ( )eZ mcc n

−

⎧ ⎫+ − + −⎨ ⎬⎩ ⎭=

⎛ ⎞+ +⎜ ⎟⎝ ⎠⎧ ⎫+ − + −⎨ ⎬⎛ ⎞ ⎩ ⎭= ⎜ ⎟ ⎛ ⎞⎝ ⎠ + +⎜ ⎟

⎝ ⎠⎧ ⎫+ − + −⎨ ⎬⎛ ⎞ ⎩ ⎭= ⎜ ⎟ ⎛ ⎞⎝ ⎠ + +⎜ ⎟

⎝ ⎠Spin-orbit correction- same order as ‘relativistic mass’ correction


2s = σ

2

2

2

ec

ame

⎛ ⎞α = ⎜ ⎟

⎝ ⎠

=

( )

2 4

2 2

22

2

12

2

nmZ eE

nZ

mcn

= −

α= −


( )( )

4 2

3

31 14

14 12

1 3 1 3 3for 1 1 12 4 2 2 4

1 3 1 1 3for 1 1 12 4 2 2 4

spin orbit

j( j ) ( )H Z mc

n

j , j( j ) ( ) ( )

j , j( j ) ( ) ( )

−

⎧ ⎫+ − + −⎨ ⎬⎩ ⎭= α

⎛ ⎞+ +⎜ ⎟⎝ ⎠

⎧ ⎫⎧ ⎫ ⎛ ⎞⎛ ⎞= + + − + − = + + − + − =⎨ ⎬ ⎨ ⎬⎜ ⎟⎜ ⎟⎩ ⎭ ⎝ ⎠⎝ ⎠⎩ ⎭

⎧ ⎫⎧ ⎫ ⎛ ⎞⎛ ⎞= − + − + − = − + − + −⎨ ⎬ ⎨ ⎬⎜ ⎟⎜ ⎟⎩ ⎭ ⎝ ⎠⎝ ⎠⎩

1= − −⎭

Spin-orbit correction- same order as ‘relativistic mass’ correction


( )22

22n

ZE mc

nα

= −

( )( )

2

31 14

12 12

spin orbit n

j( j ) ( )H E Z

n−

⎧ ⎫+ − + −⎨ ⎬⎩ ⎭= − α

⎛ ⎞+ +⎜ ⎟⎝ ⎠



( )2

2

1 3142

Rel. nK.E.

nE E Zn

⎡ ⎤⎢ ⎥⎢ ⎥Δ = − α −

⎛ ⎞⎢ ⎥+⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

( )( )

( )

( )

( )

4 2

3

4 2

3

4 2

3

31 14

14 12

1when 12 4 12

1when 12 42

spin orbit

spin orbit

spin orbit

j( j ) ( )H Z mc

n

Z mcj , H

n

Z mcj , H

n

−

−

−

⎧ ⎫+ − + −⎨ ⎬⎩ ⎭= α

⎛ ⎞+ +⎜ ⎟⎝ ⎠

α= + =

⎛ ⎞+ +⎜ ⎟⎝ ⎠

− α= − =

⎛ ⎞+⎜ ⎟⎝ ⎠

( )

2 4

2 2

22

2

12

2

nmZ eE

nZ

mcn

= −

α= −


( )( )

( )

( )

( )

4 2

3

4 42 2

1 13 32 2

31 14

14 12

1 14 1 42 2

spin orbit

spin orbit spin orbitj j

j( j ) ( )H Z mc

n

Z mc Z mcH ; H

n n

−

− −= + = −

⎧ ⎫+ − + −⎨ ⎬⎩ ⎭= α

⎛ ⎞+ +⎜ ⎟⎝ ⎠

α − α= =

⎛ ⎞ ⎛ ⎞+ + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

0≠


( )( )

( )

( )

4 2

3

4 2

3

1 1 114 12

114 12

spin orbitsplitting

Z mcn

Z mcn

− ⎡ ⎤αΔ = −⎢ ⎥

+ ⎛ ⎞⎢ ⎥⎣ ⎦ +⎜ ⎟⎝ ⎠

⎡ ⎤⎢ ⎥α −⎢ ⎥=⎛ ⎞⎢ ⎥+ +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

( )

2 4

2 2

22

2

12

2

nmZ eE

nZ

mcn

= −

α= −



( )

( )

2 2 2

2 2 2 2 2

2 2 2 23

2 2 2 2 2

2 23

2 2

8 8

48 8

2

'''Darwin r

r

e e Ze ˆh div E em c m c r

eZe Ze (r )m c r m c

Ze (r )m c

⎛ ⎞−= − = − ∇ ⋅⎜ ⎟

⎝ ⎠⎛ ⎞

= ∇ ⋅ = πδ⎜ ⎟⎝ ⎠

π= δ

( )

2 23

0 0 0 02 2

22 22

0 02 2

2

02

'''Darwin n, ,m n, ,m

n, ,m n

Zeh (r )m c

ZZe (r ) Em c n

= = = =

= =

π= ψ δ ψ

απ= ψ = = −0=

( )

2 4

2 2

22

2

12

2

nmZ eE

nZ

mcn

= −

α= −



( )

2 23

0 0 0 02 2

2 22 2

0 02 2

2

0 2

'''Darwin n, ,m n, ,m

nn, ,m

Zeh (r )m c

EZe (r ) Zm c n

= = = =

= =

π= ψ δ ψ

π= ψ = = − α

( )2

2

1 3142

Rel. nK.E.

nE E Zn

⎡ ⎤⎢ ⎥⎢ ⎥Δ = − α −

⎛ ⎞⎢ ⎥+⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

( )( )

2

31 14

12 12

spin orbit n

j( j ) ( )H E Z

n−

⎧ ⎫+ − + −⎨ ⎬⎩ ⎭= − α

⎛ ⎞+ +⎜ ⎟⎝ ⎠

0=

0≠


( )2

2

1 3142

Rel. nK.E.

nH E Zn

⎡ ⎤⎢ ⎥⎢ ⎥= − α −

⎛ ⎞⎢ ⎥+⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

( )( )

2

31 14

12 12

spin orbit n

j( j ) ( )H E Z

n−

⎧ ⎫+ − + −⎨ ⎬⎩ ⎭= − α

⎛ ⎞+ +⎜ ⎟⎝ ⎠

( )2

2

31 42

Relativistic Relativisticall nj n

Z nEn j

⎛ ⎞⎜ ⎟α

Δ = Δ = −⎜ ⎟⎜ ⎟+⎝ ⎠

( )2 ''' nDarwin

Eh Zn

= − α

( )2

2

31 1 42

Relativisticnj n

Z nE En j

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟α

= + −⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟+⎢ ⎥⎝ ⎠⎣ ⎦


Dirac Equation for spherically symmetric potential

References:

Bjorken and Drell: Relativistic Quantum MechanicsGriener: Relativistic Quantum MechanicsTrigg: ‘Quantum Mechanics’Messiah: Quantum Mechanics

FOR THE COULOMB POTENTIAL,

THE DIRAC EQUATION HAS EXACT ANALYTICAL

SOLUTIONS


Strategy for the Coulomb potential:

Eliminate all features except the radial features using constants of motion.

2DiracHamiltonian

eH mc c p A ec

⎛ ⎞= β + α⋅ − + φ⎜ ⎟⎝ ⎠

2

2

DiracElectrostaticfield

H mc c p e

mc c ei

= β + α⋅ + φ

= β + α⋅ ∇ + φ

2DiracFreeparticle

H mc e= β + φ

0DiracFreeparticle

H ,−

⎡ ⎤≠⎢ ⎥

⎣ ⎦0Dirac

Freeparticle

H ,s−

⎡ ⎤≠⎢ ⎥

⎣ ⎦

0DiracCoulombH ,

−⎡ ⎤ ≠⎣ ⎦ 0Dirac

CoulombH ,s−

⎡ ⎤ ≠⎣ ⎦

2zH, j , j

j s= + & 'parity '


2Spherical symmetryDirac HamiltonianH c p mc V(r)= α⋅ + β +

Strategy:

separation into

radial and angular parts

How shall we handle this term?


1

0 1 001 0 0

0 D Diracσ⎛ ⎞ σ⎛ ⎞ ⎛ ⎞ρ = Ξ = = σ = σ⎜ ⎟ ⎜ ⎟σ⎝

α = ⎜ ⎟σ⎝⎠ ⎠⎠ ⎝

1

1

0 1 0 01 0 0 0

0 1 0 01 0 0 0

σ σ⎛ ⎞⎛ ⎞ ⎛ ⎞ρ Ξ = = = α⎜ ⎟⎜ ⎟ ⎜ ⎟σ σ⎝ ⎠⎝ ⎠ ⎝ ⎠

σ σ⎛ ⎞⎛ ⎞ ⎛ ⎞ρ α = = = Ξ⎜ ⎟⎜ ⎟ ⎜ ⎟σ σ⎝ ⎠⎝ ⎠ ⎝ ⎠

0 1 02 0 2 0 1 2

sσ⎛ ⎞ ⎛ ⎞

= = σ = Ξ⎜ ⎟ ⎜ ⎟σ⎝ ⎠ ⎝ ⎠

0 1 00 0 1

p p pσ⎛ ⎞ ⎛ ⎞

Ξ ⋅ = ⋅ = σ ⋅⎜ ⎟ ⎜ ⎟σ⎝ ⎠ ⎝ ⎠

2

1p r r pr

Ξ ⋅ = Ξ ⋅ Ξ ⋅ Ξ ⋅


2

1p r r pr

Ξ ⋅ = Ξ ⋅ Ξ ⋅ Ξ ⋅

( ) ( )2

1 rep r r p i r p r p ir r

Ξ ⋅Ξ ⋅ = Ξ ⋅ ⋅ + Ξ ⋅ × = ⋅ + Ξ ⋅

( ) 1 1θ ϕ

⎛ ⎞∂ ∂ ∂⋅ = ⋅ − ∇ = − ⋅∇ = − ⋅ + +⎜ ⎟∂ ∂θ θ ∂ϕ⎝ ⎠

r r r rˆ ˆ ˆr p re i i re i re ˆ ˆ ˆr r r sin

e e e

∂⋅ = − = +

∂ rr p i r rp ir

1∂⎛ ⎞= − +⎜ ⎟∂⎝ ⎠∂

= − −∂

rp ir r

iir r

since =

∂− −

∂

rrp

i r ir

( )r riˆp e pr

⎛ ⎞Ξ ⋅ = Ξ ⋅ + + Ξ ⋅⎜ ⎟⎝ ⎠

( )1

0 11 0 r r

iˆp p e pr

⎛ ⎞ ⎛ ⎞α ⋅ = ρ Ξ⋅ = Ξ⋅ + +Ξ⋅⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠


( )r rip pr

⎛ ⎞α ⋅ = α + +Ξ⋅⎜ ⎟⎝ ⎠

( ) ( )1

0 11 0 r r r r

i iˆp p e p pr r

⎛ ⎞ ⎛ ⎞ ⎛ ⎞α ⋅ = ρ Ξ⋅ = Ξ⋅ + +Ξ⋅ = α + +Ξ⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

( ) ( )r ri i ip i i

r r r r r∂ ∂⎛ ⎞ ⎛ ⎞α ⋅ = α − − + +Ξ⋅ = α − + Ξ⋅⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

2= α⋅ +β + SphDiracH c p mc V(r)

( ) ( )4 4 4 4 4 4 4 4 4 4 4 4 4 41 i.e. 1K Kβ β× × × × × × ×= + Ξ ⋅ = +Ξ ⋅

2⎛ ⎞= α + α β + β +⎜ ⎟⎝ ⎠

SphDirac r r r

iH c p K mc V(r)r

r rip p Kr

⎛ ⎞α ⋅ = α + β⎜ ⎟⎝ ⎠

New operator:Ref: Greiner page 174


2⎛ ⎞= α + α β + β +⎜ ⎟⎝ ⎠

SphDirac r r r

iH c p K mc V(r)r


Next: solution to the H atom


( )( )

4 4 4 4 4 4 4 4

4 4 4 4 4 4

1

i.e. 1

× × × ×

× × ×

= +Ξ ⋅

= +Ξ ⋅

K

K

β

β

Reference: Greiner page 174

21SphDirac rH - ic K mc V(r)

r r r∂⎛ ⎞= α + − β + β +⎜ ⎟∂⎝ ⎠


[email protected]

Unit 3(vi) Lecture 18


Hydrogen AtomSpherical symmetry of the Coulomb potential





Strategy:

separation into

radial and angular parts

How shall we handle this term?

Hydrogen atom – solution to Dirac Hamiltonian


1

0 1 001 0 0

0 D Diracσ⎛ ⎞ σ⎛ ⎞ ⎛ ⎞ρ = Ξ = = σ = σ⎜ ⎟ ⎜ ⎟σ⎝

α = ⎜ ⎟σ⎝⎠ ⎠⎠ ⎝

1

1

0 1 0 01 0 0 0

0 1 0 01 0 0 0

σ σ⎛ ⎞⎛ ⎞ ⎛ ⎞ρ Ξ = = = α⎜ ⎟⎜ ⎟ ⎜ ⎟σ σ⎝ ⎠⎝ ⎠ ⎝ ⎠

σ σ⎛ ⎞⎛ ⎞ ⎛ ⎞ρ α = = = Ξ⎜ ⎟⎜ ⎟ ⎜ ⎟σ σ⎝ ⎠⎝ ⎠ ⎝ ⎠

0 1 02 0 2 0 1 2

sσ⎛ ⎞ ⎛ ⎞

= = σ = Ξ⎜ ⎟ ⎜ ⎟σ⎝ ⎠ ⎝ ⎠

0 1 00 0 1

p p pσ⎛ ⎞ ⎛ ⎞

Ξ ⋅ = ⋅ = σ ⋅⎜ ⎟ ⎜ ⎟σ⎝ ⎠ ⎝ ⎠

2

1p r r pr

Ξ ⋅ = Ξ ⋅ Ξ ⋅ Ξ ⋅


2

1p r r pr

Ξ ⋅ = Ξ ⋅ Ξ ⋅ Ξ ⋅

( ) ( )2

1 rep r r p i r p r p ir r

Ξ ⋅Ξ ⋅ = Ξ ⋅ ⋅ + Ξ ⋅ × = ⋅ + Ξ ⋅

( ) 1 1θ ϕ

⎛ ⎞∂ ∂ ∂⋅ = ⋅ − ∇ = − ⋅∇ = − ⋅ + +⎜ ⎟∂ ∂θ θ ∂ϕ⎝ ⎠

r r r rˆ ˆ ˆr p re i i re i re ˆ ˆ ˆr r r sin

e e e

∂⋅ = − = +

∂ rr p i r rp ir

1∂⎛ ⎞= − +⎜ ⎟∂⎝ ⎠∂

= − −∂

rp ir r

iir r

since =

∂− −

∂

rrp

i r ir

( )r riˆp e pr

⎛ ⎞Ξ ⋅ = Ξ ⋅ + + Ξ ⋅⎜ ⎟⎝ ⎠

( )1

0 11 0 r r

iˆp p e pr

⎛ ⎞ ⎛ ⎞α ⋅ = ρ Ξ⋅ = Ξ⋅ + +Ξ⋅⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠


( )r rip pr

⎛ ⎞α ⋅ = α + +Ξ⋅⎜ ⎟⎝ ⎠

( ) ( )1

0 11 0 r r r r

i iˆp p e p pr r

⎛ ⎞ ⎛ ⎞ ⎛ ⎞α ⋅ = ρ Ξ⋅ = Ξ⋅ + +Ξ⋅ = α + +Ξ⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

( ) ( )r ri i ip i i

r r r r r∂ ∂⎛ ⎞ ⎛ ⎞α ⋅ = α − − + +Ξ⋅ = α − + Ξ⋅⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

2= α⋅ +β + SphDiracH c p mc V(r)

( ) ( )4 4 4 4 4 4 4 4 4 4 4 4 4 41 i.e. 1K Kβ β× × × × × × ×= + Ξ ⋅ = +Ξ ⋅

2⎛ ⎞= α + α β + β +⎜ ⎟⎝ ⎠

SphDirac r r r

iH c p K mc V(r)r

r rip p Kr

⎛ ⎞α ⋅ = α + β⎜ ⎟⎝ ⎠

New operator:Ref: Greiner page 174


2⎛ ⎞= α + α β + β +⎜ ⎟⎝ ⎠

SphDirac r r r

iH c p K mc V(r)r


( )( )

4 4 4 4 4 4 4 4

4 4 4 4 4 4

1

i.e. 1

× × × ×

× × ×

= +Ξ ⋅

= +Ξ ⋅

K

K

β

β

Reference: Greiner page 174


r r r∂⎛ ⎞= α + − β + β +⎜ ⎟∂⎝ ⎠


2⎛ ⎞= α + α β + β +⎜ ⎟⎝ ⎠

SphDirac r r r

iH c p K mc V(r)r


( )( )

4 4 4 4 4 4 4 4

4 4 4 4 4 4

1

i.e. 1

× × × ×

× × ×

= +Ξ ⋅

= +Ξ ⋅

K

K

β

β


r r r∂⎛ ⎞= α + − β + β +⎜ ⎟∂⎝ ⎠


( ) ( ) ( )( )( )

2 2 2

2 2 2

2 2 2

2

2

since 2

j j j s s s s

s j s

j s s

= ⋅ = + ⋅ + = + ⋅ +

⋅ = − −

⋅σ = − − → = σ

( )( ) ( )2 2

2

i i iσ ⋅ σ ⋅ = + σ ⋅ × = + σ ⋅

= − σ ⋅

( ) ( )( )2 2j s⋅ σ = − σ ⋅ σ ⋅ − σ ⋅ −

( )( )2 = σ ⋅ σ ⋅ + σ ⋅

( )

( )

22 2

22

324

14

j = σ ⋅ + σ ⋅ +

⎡ ⎤= σ ⋅ + −⎣ ⎦

quantum numberκ


( ) 22 2 1

4j ⎡ ⎤+ = σ ⋅ +⎣ ⎦

( )( )

4 4 4 4 4 4 4 4

4 4 4 4 4 4

1

i.e. 1

× × × ×

× × ×

= +Ξ ⋅

= +Ξ ⋅

K

K

β

β

( )( )2 2

2

1 0 1 014 0 1 0 1

0 0 1 00 0 0 1

j K K K K

K KK

K K

⎛ ⎞ ⎛ ⎞+ = β β = ⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ⎝ ⎠

⎛ ⎞⎛ ⎞ ⎛ ⎞= =⎜ ⎟⎜ ⎟ ⎜ ⎟− −⎝ ⎠⎝ ⎠ ⎝ ⎠

2 2 2 2 21 11 14 4

j( j ) j( j )⎡ ⎤κ = + + = + +⎢ ⎥⎣ ⎦2

2 1 114 2

j( j ) j⎡ ⎤ ⎛ ⎞κ = + + = +⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠

121for 2

j

j

⎛ ⎞κ = ± +⎜ ⎟⎝ ⎠

= ∓

2The operator would give a good quantum number

K


Dirac ‘Good Quantum Numbers’:

1 0 0

0 1 0D

PP P P

P⎛ ⎞ ⎛ ⎞

= β = =⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ⎝ ⎠

0DH , j−

⎡ ⎤ =⎣ ⎦

[ ] 0DH ,K−=

[ ] 0D DH ,P−=

( )4 4 4 4 4 4 4 41 × × × ×= +Ξ ⋅K β

0 1 02 0 2 0 1 2

sσ⎛ ⎞ ⎛ ⎞

= = σ = Ξ⎜ ⎟ ⎜ ⎟σ⎝ ⎠ ⎝ ⎠

j s= +

n, ,m n, j, ,mκ ≡ ω12

j⎛ ⎞κ = + ω⎜ ⎟⎝ ⎠

11 for 211 for 2

j

j

ω = + = −

ω = − = +

121for 2

j

j

⎛ ⎞κ = ± +⎜ ⎟⎝ ⎠

= ∓


2

12

12

32

32

52

52

12

10 1 1 1 0 1211 1 1 1 1 1231 1 2 1 1 1232 1 2 1 2 1252 1 3 1 2 1253 1 3 1 3 12

jOrbital Parity j j ( )

s

p

p

d

d

f

ω+ω

κ ω + −

+ − − +

− + + −

− − − −

+ + + +

+ − − +

− + + −

11 for 211 for 2

j

j

ω = + = −

ω = − = +

12

j⎛ ⎞κ = + ω⎜ ⎟⎝ ⎠

121for 2

j

j

⎛ ⎞κ = ± +⎜ ⎟⎝ ⎠

= ∓


2SphDirac r r r

cH c p i K mc V(r)r

= α + α β + β +


r r r∂⎛ ⎞= α + − β + β +⎜ ⎟∂⎝ ⎠

2r r r n m nj n m

cc p i K mc V(r) u E ur κ κ

⎡ ⎤α + α β + β + =⎢ ⎥⎣ ⎦

2r r r n m nj n m

cc p i mc V(r) u E ur κ κ

κ⎡ ⎤α + α β + β + =⎢ ⎥⎣ ⎦

2r r r n m nj n m


κ⎡ ⎤β α + α β + β + = β⎢ ⎥⎣ ⎦


2r r r n m nj n m


κ⎡ ⎤− α β − α + + β = β⎢ ⎥⎣ ⎦

2 1r r ;βα = −α β β =

2r r r n m nj n m


κ⎡ ⎤α + α β +β + =⎢ ⎥⎣ ⎦

2r r r n m nj n m


κ⎡ ⎤β α + α β + β + = β⎢ ⎥⎣ ⎦

(Eq.A)

(Eq.B)

( )12

Eq.A Eq.B+ →

( ) ( )

( ) ( )( )

2

1 1 1 12 2 11 1 21 12 2

r r r

n m nj n m

i cc pr u E u

mc V(r)κ κ

κ⎡ ⎤α −β − α −β +⎢ ⎥= + β⎢ ⎥

⎢ ⎥+ + β + + β⎢ ⎥⎣ ⎦


( ) ( )

( ) ( )( )

2

1 1 1 12 2 11 1 21 12 2

r r r

n m nj n m

i cc pr u E u

mc V(r)κ κ

κ⎡ ⎤α −β − α −β +⎢ ⎥= + β⎢ ⎥

⎢ ⎥+ + β + + β⎢ ⎥⎣ ⎦

1 n mn m

n m

ˆ uP (r) (r)u

ˆ uiQ (r) (r)r+κ κ

κ−κ −κ

Ω⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟Ω ⎝ ⎠⎝ ⎠

( ) 01 0 1 0 0 01 2

0 1 0 1 0 2u u uu u u u+ + +

− − − −

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞− β = − = =⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

( ) 1 0 1 0 2 01 2

0 1 0 1 0 0 0u u u uu u u+ + + +

− − −

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ β = + = =⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

2

0 01 2 22 2 1 2

2 01 12 22 0 2 0

r r r

nj

i cc pu ur u

Eu u

mc V(r)

− − +

+ +

⎡ ⎤⎛ ⎞ ⎛ ⎞κα − α +⎢ ⎥⎜ ⎟ ⎜ ⎟

⎛ ⎞⎝ ⎠ ⎝ ⎠⎢ ⎥ = ⎜ ⎟⎢ ⎥⎛ ⎞ ⎛ ⎞ ⎝ ⎠⎢ ⎥+ +⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

Spherical Harmonic Spinors j mΩ


2

0 01 2 22 2 1 2

2 01 12 22 0 2 0

r r r

nj

i cc pu ur u

Eu u

mc V(r)

− − +

+ +

⎡ ⎤⎛ ⎞ ⎛ ⎞κα − α +⎢ ⎥⎜ ⎟ ⎜ ⎟

⎛ ⎞⎝ ⎠ ⎝ ⎠⎢ ⎥ = ⎜ ⎟⎢ ⎥⎛ ⎞ ⎛ ⎞ ⎝ ⎠⎢ ⎥+ +⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

( )200 0r r r nj

u ucc p i mc V(r) Eur

+ +

−

⎛ ⎞ ⎛ ⎞ ⎛ ⎞κ⎛ ⎞α − α + + =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠


( )200 0r r r nj

u ucc p i mc V(r) Eur

+ +

−

⎛ ⎞ ⎛ ⎞ ⎛ ⎞κ⎛ ⎞α − α + + =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠

00 1 0 1 0 001 0 1 0 0 0

rr r r r

r

ˆ ˆ ê e eσσ σ ⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞

α = Ξ⋅ = ⋅ = ⋅ = ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ σσ σ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠

'odd' r :α

( )200 0r r r nj

u uccp i mc V(r) Eur

+ +

−

⎛ ⎞ ⎛ ⎞ ⎛ ⎞κ⎛ ⎞α − α + + =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠

[ ] 0 r r,p−

α = ⇒

( )20 0 00 0 0 0

r rr nj

r r


+ +

−

⎛ ⎞σ σ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞κ− + + =⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟σ σ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠


( )20 0 00 0 0 0

r rr nj

r r


+ +

−

⎛ ⎞σ σ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞κ− + + =⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟σ σ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠

( )2

00

0 00

r r

nj

r r

ccp iu ur

mc V(r) Euccp i

r

+ +

−

⎛ ⎞κ⎛ ⎞− σ⎜ ⎟⎜ ⎟ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎝ ⎠⎜ ⎟ + + =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟κ⎛ ⎞ ⎝ ⎠ ⎝ ⎠⎝ ⎠− σ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

( )2r r nj

ccp i u mc V(r) u E ur − + +

κ⎛ ⎞− σ + + =⎜ ⎟⎝ ⎠This result has neither the ‘odd’ operators, nor angle-dependent operators. Essentially, we have separated the radial part.

4 1 2 1 reductionNOTE! × → ×


2r r r n m nj n m


κ⎡ ⎤α + α β +β + =⎢ ⎥⎣ ⎦ (Eq.A)

( )We had operated upon Eq.A by .Now, we do so by r .

β

α

2r r r r n m nj r n m


κ⎡ ⎤α α + α β + β + = α⎢ ⎥⎣ ⎦

2r r r n m nj r n m

ccp i mc V(r) u E ur κ κ

κ⎡ ⎤+ β + α β + α = α⎢ ⎥⎣ ⎦

2r r r n m nj r n m


κ⎡ ⎤β + β + α β + α = β α⎢ ⎥⎣ ⎦

2r r r n m nj r n m


κ⎡ ⎤β + − α + βα = βα⎢ ⎥⎣ ⎦

(Eq.X)

(Eq.Y)


2r r r n m nj r n m


κ⎡ ⎤+ β + α β + α = α⎢ ⎥⎣ ⎦

2r r r n m nj r n m


κ⎡ ⎤β + − α + βα = βα⎢ ⎥⎣ ⎦

(Eq.X)

(Eq.Y)

( )12

Eq.X Eq.Y+ →

( ) ( )

( ) ( )( )

2

1 11 1 12 2 11 1 21 12 2

r

n m nj r n m

r r

ccp ir u E u

mc V(r)κ κ

κ⎡ ⎤+ β + β +⎢ ⎥= α −β⎢ ⎥

⎢ ⎥+ α β − + α −β⎢ ⎥⎣ ⎦

+

( ) ( )

( ) ( )( )

2

1 11

1 1

rn m nj r n m

r r

ccp ir u E u

mc V(r)κ κ

κ⎡ ⎤+ β + + β⎢ ⎥ = α −β⎢ ⎥−α −β + α −β⎢ ⎥⎣ ⎦

+


( ) ( )

( ) ( )( )

2

1 11

1 1

rn m nj r n m

r r

ccp ir u E u

mc V(r)κ κ

κ⎡ ⎤+ β + + β⎢ ⎥ = α −β⎢ ⎥−α −β + α −β⎢ ⎥⎣ ⎦

+

( ) ( )01 2 & 1 2

0u u uu u u+ + +

− − −

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞− β = + β =⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠

1 n mn m

n m

ˆ uP (r) (r)u


κ−κ −κ

Ω⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟Ω ⎝ ⎠⎝ ⎠

( )2 0 00r r nj r

uccp i V(r) mc Eu ur

+

− −

⎛ ⎞ ⎛ ⎞⎛ ⎞κ⎛ ⎞+ + − α = α⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠

( )2r r nj r

: 'odd'ccp i u V(r) mc u E u

r + − −

α ⇒

κ⎛ ⎞+ + − σ = σ⎜ ⎟⎝ ⎠

4 1 2 1 reductionNOTE! × → ×


Spherical Harmonic Spinors j mΩ

( )12

11 22

1 12 2'

s'

s

definition

j m m ' s,mm m

ˆY (r) m m jm=−

→=−

⎛ ⎞Ω = χ ζ ⎜ ⎟⎝ ⎠

∑ ∑

( ) ( ) ( )12

11 22

1 12 2' s

ss

j m ' s sm m m ,mm

ˆY (r) , m m m , m jm= −

=−

⎛ ⎞Ω = χ ζ = − ⎜ ⎟⎝ ⎠

∑

( )

( )

1 112 22

1 112 22

1 1 1 12 2 2 2

1 1 1 1 2 2 2 2

s'

'

j m ',mm m

',m m

ˆY (r) , m m , , jm

ˆY (r) , m m , , jm

⎛ ⎞ =−= +⎜ ⎟⎝ ⎠

⎛ ⎞= −⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞Ω = χ ζ = + −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

+

⎛ ⎞ ⎛ ⎞χ ζ = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠


( )

( )

1 112 22

1 112 22

1 1 1 12 2 2 2

1 1 1 1 2 2 2 2

s'

'

j m ',mm m

',m m

ˆY (r) , m m , , jm

ˆY (r) , m m , , jm

⎛ ⎞ =−= +⎜ ⎟⎝ ⎠

⎛ ⎞= −⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞Ω = χ ζ = + −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

+

⎛ ⎞ ⎛ ⎞χ ζ = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

12

12

0 1 1 1 11 2 2 2 2

1 1 1 1 1 0 2 2 2 2

'

'

j m 'm m

'm m

ˆY (r) , m m , , jm

ˆY (r) , m m , , jm

⎛ ⎞= +⎜ ⎟⎝ ⎠

⎛ ⎞= −⎜ ⎟⎝ ⎠

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

+

⎛ ⎞ ⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

12

12 2 rows 1 column

1 1 1 12 2 2 2

1 1 1 12 2 2 2

'

'

'm m

j m

'm m

ˆY (r) , m m , , jm

ˆY (r) , m m , , jm

⎛ ⎞= −⎜ ⎟⎝ ⎠

⎛ ⎞= +⎜ ⎟⎝ ⎠ ×

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


12

12

1 1 1 12 2 2 2

1 1 1 12 2 2 2

'

'

'm m

j m

'm m

ˆY (r) , m m , , jm

ˆY (r) , m m , , jm

⎛ ⎞= −⎜ ⎟⎝ ⎠

⎛ ⎞= +⎜ ⎟⎝ ⎠

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

1 1 1 12 2 2 2

1 1 1 12 2 2 2

'

'

, m m , , jm ?

, m m , , jm ?

⎛ ⎞ ⎛ ⎞= − =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞= + − =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Depends on12

j = ±

1 1 1 1 12 2 2 2 2 2

1 1 1 1 12 2 2 2 2 2

'

'

j m, m m , , j mj

j m, m m , , j mj

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

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

1 1 1 1 1 12 2 2 2 2 2 2

1 1 1 1 1 12 2 2 2 2 2 2

'

'

j m, m m , , j mj

j m, m m , , j mj

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

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

1 2

j← = +

1 2

j = − →


1 12 2

1 12 2

1 2

2

2

'

'

j , m m

j m

j , m m

for j

j m ˆY (r)j

j m ˆY (r)j

⎛ ⎞ ⎛ ⎞= − = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞= − = +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= +

⎛ ⎞+⎜ ⎟⎜ ⎟

Ω = ⎜ ⎟−⎜ ⎟

⎜ ⎟⎝ ⎠

1 12 2

1 12 2

1 2

12 2

12 2

'

'

j , m m

j m

j , m m

for j

j m ˆY (r)j

j m ˆY (r)j

⎛ ⎞ ⎛ ⎞= + = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞= + = +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= −

⎛ ⎞− +−⎜ ⎟

+⎜ ⎟Ω = ⎜ ⎟

+ +⎜ ⎟⎜ ⎟+⎝ ⎠

Spherical HarmonicSpinors j mΩ


( )2r r nj

ccp i u mc V(r) u E ur − + +

κ⎛ ⎞− σ + + =⎜ ⎟⎝ ⎠

( )2r r nj r

ccp i u V(r) mc u E ur + − −

κ⎛ ⎞+ + − σ = σ⎜ ⎟⎝ ⎠

1 12 2

1 12 2

1 2

2

2

'

'

j , m m

j m

j , m m

for j

j m ˆY (r)j

j m ˆY (r)j

⎛ ⎞ ⎛ ⎞= − = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞= − = +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= +

⎛ ⎞+⎜ ⎟⎜ ⎟

Ω = ⎜ ⎟−⎜ ⎟

⎜ ⎟⎝ ⎠

1 12 2

1 12 2

1 2

12 2

12 2

'

'

j , m m

j m

j , m m

for j

j m ˆY (r)j

j m ˆY (r)j

⎛ ⎞ ⎛ ⎞= + = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞= + = +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= −

⎛ ⎞− +−⎜ ⎟

+⎜ ⎟Ω = ⎜ ⎟

+ +⎜ ⎟⎜ ⎟+⎝ ⎠

1 n mn m

n m

ˆ uP (r) (r)u


κ−κ −κ

Ω⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟Ω ⎝ ⎠⎝ ⎠


( )2

n mr r

n m n mnj

îQ (r) (r)ccp ir r

ˆ ˆP (r) (r) P (r) (r)mc V(r) Er r

κ −κ

κ κ κ κ

Ωκ ⎛ ⎞⎛ ⎞− σ +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠Ω Ω⎛ ⎞ ⎛ ⎞+ + =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

1 n mn m

n m

ˆ uP (r) (r)u


κ−κ −κ

Ω⎛ ⎞ ⎛ ⎞= = ⇒⎜ ⎟ ⎜ ⎟Ω ⎝ ⎠⎝ ⎠

( )2

n mr

n m n mr nj r

ˆP (r) (r)ccp ir r

ˆ îQ (r) (r) iQ (r) (r)V(r) mc Er r

κ κ

κ −κ κ −κ

Ωκ ⎛ ⎞⎛ ⎞+ +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠Ω Ω⎛ ⎞ ⎛ ⎞+ − σ = σ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

{ }

{ }

n mr r

nr m

nm

ˆP (r) (r)ur

P (r) ˆ(r)r

P (r) ˆ(r)r

κ κ+

κκ

κ−κ

Ωσ = σ

= σ Ω

= −Ω

{ }

{ }

n mr r

nr m

nm

îQ (r) (r)ur

iQ (r) ˆ(r)r

iQ (r) ˆ(r)r

κ −κ−

κ−κ

κκ

Ωσ = σ

= σ Ω

= −Ω

ru ?σ ± =


2

2

n

n

dP(r)c c P(r) (E mc V(r))Q(r)dr r

dQ(r)c c Q(r) (E mc V(r))P(r)dr r

κ

κ

κ− + = − + −

κ+ = − − −

and G(r) iP(r) F(r) iQ(r)= − = −

2

2

n

n

dP(r)c c P(r) (E mc V(r))Q(r)dr r

dQ(r)c c Q(r) (E mc V(r))P(r)dr r

κ

κ

κ− − = − + −

κ− = − − −

writing for instead of

− κ+ κ

2

2

n

n

dP(r)c c P(r) (E mc V(r))Q(r)dr rdQ(r)c c Q(r) (E mc V(r))P(r)

dr r

κ

κ

κ+ = + + −

κ− + = − −


and G(r) iP(r) F(r) iQ(r)= − = −

2

2

n

n

dP(r)c c P(r) (E mc V(r))Q(r)dr rdQ(r)c c Q(r) (E mc V(r))P(r)

dr r

κ

κ

κ+ = + + −

κ− + = − −

References:

Eq.4.13/p.55/Bjorken and Drell

Eq.XX.170/p928/Messiah, Vol.II

Eq.2.14/Ian P. Grant, Adv. in Phys. 19 (1970)

Eq.4.2.20/Pratt, Ron, Tseng, Rev.Mod.Phys. (1973)

Eq.2/Burke & Grant/Proc.Phys.Soc. 90 (1967)

2 1 1 1 2 2 dimensions

c ML T LT L ML Tr

− − − −⎡ ⎤ = × × =⎢ ⎥⎣ ⎦


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