View
1
Download
0
Category
Preview:
Citation preview
P. C. Deshmukh Department of PhysicsIndian Institute of Technology MadrasChennai 600036
pcd@physics.iitm.ac.in 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
August‐September 2012 PCD STiAP Unit 3 2
( )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
Relativistic Quantum Mechanics of the
Hydrogen Atom
Unit 3(i)
First: Transition from Non-relativistic
to Relativistic Dynamical Variables
Then: quantization!
August‐September 2012 PCD STiAP Unit 3 3
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 )
August‐September 2012 PCD STiAP Unit 3 5
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
− =
August‐September 2012 PCD STiAP Unit 3 6
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 ?
August‐September 2012 PCD STiAP Unit 3
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.
August‐September 2012 PCD STiAP Unit 3 8
2
2
1
' 1-
ΔΔ = ≥ Δ
−
= ≤
t
L l l
τ τβ
β
v/c 1= ⟨β
August‐September 2012 PCD STiAP Unit 3 9
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
August‐September 2012 PCD STiAP Unit 3 10
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’
August‐September 2012 PCD STiAP Unit 3 11
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
( )( )2ds dx dxμμ=
( ) ( ) ( ) ( )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= + + +
August‐September 2012 PCD STiAP Unit 3 12
( )( )2ds dx dxμμ=
( )( ) ( )( ) ( )( ) ( )( )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
August‐September 2012 PCD STiAP Unit 3 13
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
August‐September 2012 PCD STiAP Unit 3
( ) ( ) ( ) ( )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⎨ ⎬=
−
⎩ ⎭
−
August‐September 2012 PCD STiAP Unit 3 15
( ) ( )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 − − − =
August‐September 2012 PCD STiAP Unit 3 16
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”
August‐September 2012 PCD STiAP Unit 3 17
( )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
August‐September 2012 PCD STiAP Unit 3 18
( ) 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
August‐September 2012 PCD STiAP Unit 3 19
( ) 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 =
August‐September 2012 PCD STiAP Unit 3 20
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 = −
August‐September 2012 PCD STiAP Unit 3 21
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
= − =
August‐September 2012 PCD STiAP Unit 3 22
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 = + + +⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎝ ⎠
August‐September 2012 PCD STiAP Unit 3 23
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.
August‐September 2012 PCD STiAP Unit 3 24
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
?
August‐September 2012 PCD STiAP Unit 3 25
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
August‐September 2012 PCD STiAP Unit 3
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
August‐September 2012 PCD STiAP Unit 3 27
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
August‐September 2012 PCD STiAP Unit 3 28
; 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
∂ ∂⎛ ⎞= ∂ = ≡ ∇⎜ ⎟∂ ∂⎝ ⎠
August‐September 2012 PCD STiAP Unit 3 29
( )
( )
1 2 3
0
;
1
op opq q p ,p ,p p p i
Ep mc i ic c t ct
→ ≡ → = − ∇
∂ ∂= γ = → =
∂ ∂
( )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
∂ ∂⎛ ⎞= ∂ = ≡ ∇⎜ ⎟∂ ∂⎝ ⎠
August‐September 2012 PCD STiAP Unit 3 30
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
= − =
August‐September 2012 PCD STiAP Unit 3 31
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 = =
∂ ∂
≡ = − ∇
→
→
August‐September 2012 PCD STiAP Unit 3 32
Klein-Gordon equation
Quantization!
Equation of Motion ….
Time evolution of state vector
Questions? pcd@physics.iitm.ac.in
2 2 2 2 2 2
22 2
2
mp p c m v
E p.p m cc
μμ = γ − γ
= − =
August‐September 2012 PCD STiAP Unit 3 33
P. C. Deshmukh Department of PhysicsIndian Institute of Technology MadrasChennai 600036
pcd@physics.iitm.ac.in
Unit 3(ii) Lecture 14
Relativistic Quantum Mechanics of the
Hydrogen Atom - 2
Select/Special Topics in Atomic Physics
( )QUANTIZATION of p m m c,vμ μ= η = γ
August‐September 2012 PCD STiAP Unit 3 34
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 = =
∂ ∂
≡ = − ∇
→
→
August‐September 2012 PCD STiAP Unit 3 35
( )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
August‐September 2012 PCD STiAP Unit 3 36
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
August‐September 2012 PCD STiAP Unit 3
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
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
August‐September 2012 PCD STiAP Unit 3 38
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 ? ?
August‐September 2012 PCD STiAP Unit 3 39
( )( )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 −
August‐September 2012 PCD STiAP Unit 3 40
( )( )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
≡
August‐September 2012 PCD STiAP Unit 3 41
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 =
August‐September 2012 PCD STiAP Unit 3 42
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
August‐September 2012 PCD STiAP Unit 3 43
( ) ( )
( ) ( )
( ) ( )
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 = − − −
August‐September 2012 PCD STiAP Unit 3 44
( ) ( )
( ) ( )
( ) ( )
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 = − − −
August‐September 2012 PCD STiAP Unit 3 45
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
August‐September 2012 PCD STiAP Unit 3 46
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 , ,=
August‐September 2012 PCD STiAP Unit 3 47
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 =
August‐September 2012 PCD STiAP Unit 3 48
( ) 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
−⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥+ + +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥− − −⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥− − −⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦
⎡ ⎤⎢ ⎥⎢ ⎥− =⎢ ⎥⎢ ⎥⎣ ⎦
August‐September 2012 PCD STiAP Unit 3 49
( ) 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
⎤⎥⎥ =
⎢ ⎥⎢ ⎥
⎦
August‐September 2012 PCD STiAP Unit 3 50
( )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.
August‐September 2012 PCD STiAP Unit 3 51
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
August‐September 2012 PCD STiAP Unit 3 53
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
∂ ∂− =
∂ ∂
August‐September 2012 PCD STiAP Unit 3 54
( ) ( )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= − + + +
∂ ∂ ∂ ∂ ∂
August‐September 2012 PCD STiAP Unit 3 55
x
d L Ldt v x
∂ ∂=
∂ ∂
August‐September 2012 PCD STiAP Unit 3 56
( ) ( )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 + + +
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 ˆe e e
B Ax y z
A A A
A AA A A Aˆ ˆ ˆe e ey z x z x y
⎡ ⎤⎢ ⎥∂ ∂ ∂⎢ ⎥= ∇× = ⎢ ⎥∂ ∂ ∂
⎢ ⎥⎢ ⎥⎣ ⎦∂ ∂⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂⎛ ⎞= − + − + −⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠ ⎝ ⎠
x y z
x y z
y yz z x x
ˆ ˆ ˆe 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
∂⎛ ⎞∂ ∂ ∂⎛ ⎞× = − − −⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠
August‐September 2012 PCD STiAP Unit 3 57
( ) ( ) 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!
August‐September 2012 PCD STiAP Unit 3 58
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
qL m v v v q (r,t) v A (r,t) v A (r,t) v A (r,t)c
= + + − 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 = −⎨ ⎬⎜ ⎟⎝ ⎠⎩ ⎭
August‐September 2012 PCD STiAP Unit 3 59
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 =⎜ ⎟⎝ ⎠
August‐September 2012 PCD STiAP Unit 3 60
( )( )
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
⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥−⎣ ⎦⎣ ⎦
August‐September 2012 PCD STiAP Unit 3 61
( )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’
August‐September 2012 PCD STiAP Unit 3 62
( )4 1 4 1
2
4 4i c mc e
t × ××
∂ψ = α π + β + φ ψ
∂i
Dirac equation in ‘standard representation’
( )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 = ⎢ ⎥−⎣ ⎦
August‐September 2012 PCD STiAP Unit 3 63
( )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 = ⎢ ⎥−⎣ ⎦
August‐September 2012 PCD STiAP Unit 3 64
( )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
= +
= ± +
August‐September 2012 PCD STiAP Unit 3 65
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?
Electron spin requires two components,- but Dirac equation admits
4-component wavefunctionDirac equation – admits ‘negative energy’ solutions
4-component wavefunctionMulticomponent wavefunction : non-zero spin
2 2 4 2
2 4 2
E m c p.pc
E m c p.pc
= +
= ± +
August‐September 2012 PCD STiAP Unit 3 66
[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
Questions? pcd@physics.iitm.ac.in
August‐September 2012 PCD STiAP Unit 3 67
2 2 4 2
2 4 2
E m c p.pc
E m c p.pc
= +
= ± +
Electron spin requires two components,- but Dirac equation admits
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
P. C. Deshmukh Department of PhysicsIndian Institute of Technology MadrasChennai 600036
pcd@physics.iitm.ac.in
Unit 3(iii) Lecture 15
Relativistic Quantum Mechanics of the
Hydrogen Atom
Select/Special Topics in Atomic Physics
Reduction of Dirac Eq. to two-component formPAULI Equation
4 1
2 1
2 1×
×
×
ϕ⎛ ⎞ψ = ⎜ ⎟χ⎝ ⎠
Foldy-Wouthuysen Transformations - 1
August‐September 2012 PCD STiAP Unit 3 68
Electron spin requires two components,- but Dirac equation admits
4-component wavefunctionDirac equation – admits ‘negative energy’ solutions
4-component wavefunctionMulticomponent wavefunction : non-zero spin
2 2 4 2
2 4 2
E m c p.pc
E m c p.pc
= +
= ± +
August‐September 2012 PCD STiAP Unit 3 69
[1] Reduction to 2-component ‘Pauli’ relation.
[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
August‐September 2012 PCD STiAP Unit 3 70
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
August‐September 2012 PCD STiAP Unit 3 71
[1] Reduction to 2-component ‘Pauli’ relation.
( )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.
August‐September 2012 PCD STiAP Unit 3 72
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
August‐September 2012 PCD STiAP Unit 3 73
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.
August‐September 2012 PCD STiAP Unit 3 74
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 = − × −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
August‐September 2012 PCD STiAP Unit 3 75
( )e p A A pc
π× π = − × + ×
August‐September 2012 PCD STiAP Unit 3 76
( )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
August‐September 2012 PCD STiAP Unit 3 77
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
August‐September 2012 PCD STiAP Unit 3 78
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
August‐September 2012 PCD STiAP Unit 3 79
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
⇒
August‐September 2012 PCD STiAP Unit 3 80
22e s Bmc
i
August‐September 2012 PCD STiAP Unit 3 81
ei
r p= ×
1 ˆiA Ac
μ =
r
p
2e vi eT r
= =π ( )21 1
2 21 1
2 2 2
v evrˆ ˆe 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
August‐September 2012 PCD STiAP Unit 3 82
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
Dirac equation in ‘standard representation’
( )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
August‐September 2012 PCD STiAP Unit 3 83
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.
August‐September 2012 PCD STiAP Unit 3 84
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.
August‐September 2012 PCD STiAP Unit 3 85
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.
August‐September 2012 PCD STiAP Unit 3 86
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
August‐September 2012 PCD STiAP Unit 3 87
"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.
August‐September 2012 PCD STiAP Unit 3 88
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).
August‐September 2012 PCD STiAP Unit 3 89
August‐September 2012 PCD STiAP Unit 3 90
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
August‐September 2012 PCD STiAP Unit 3 91
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 →∞
August‐September 2012 PCD STiAP Unit 3 92
( ) 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
August‐September 2012 PCD STiAP Unit 3 93
August‐September 2012 PCD STiAP Unit 3 94
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
Questions? pcd@physics.iitm.ac.in
August‐September 2012 PCD STiAP Unit 3 95
2 2 4 2
2 4 2
E m c p.pc
E m c p.pc
= +
= ± +
Electron spin requires two components,- but Dirac equation admits
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
P. C. Deshmukh Department of PhysicsIndian Institute of Technology MadrasChennai 600036
pcd@physics.iitm.ac.in
Unit 3(iv) Lecture 16
Relativistic Quantum Mechanics of the
Hydrogen Atom
Select/Special Topics in Atomic Physics
Reduction of Dirac Eq. to two-component formPAULI Equation
4 1
2 1
2 1×
×
×
ϕ⎛ ⎞ψ = ⎜ ⎟χ⎝ ⎠
Foldy-Wouthuysen Transformations - 2
August‐September 2012 PCD STiAP Unit 3 96
H it
∂ψψ =
∂
iS' eψ → ψ = ψ
August‐September 2012 PCD STiAP Unit 3
( ) 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⎜ ⎟⎢ ⎥∂ ∂⎝ ⎠⎣ ⎦
iS iS iS iSH' e He i e et
+ − + −∂⇒ = −
∂
'i H' 't
∂ψ= ψ
∂
OBJECTIVE: odd operators play an ignorable role.
August‐September 2012 PCD STiAP Unit 3 98
( )4 1
4 1
2
4 4Free electron: i c p mc
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
August‐September 2012 PCD STiAP Unit 3 99
( )4 1
4 1
2
4 4Free electron: i c p mc
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
August‐September 2012 PCD STiAP Unit 3 100
( ) ( ) ( )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
August‐September 2012 PCD STiAP Unit 3 101
( )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
August‐September 2012 PCD STiAP Unit 3 102
( )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
August‐September 2012 PCD STiAP Unit 3 103
( )
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 ⋅⎣ ⎦ ⎣ ⎦
August‐September 2012 PCD STiAP Unit 3 104
( )
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
August‐September 2012 PCD STiAP Unit 3 105
( )
( ) ( )
( )
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⎜ ⎟⎝ ⎠
August‐September 2012 PCD STiAP Unit 3 106
( )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 ⋅⎢ ⎥
⎣ ⎦
August‐September 2012 PCD STiAP Unit 3 107
( ) 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 βα ⋅ β = −α ⋅p p
( ) 2
22
ω ωα + +
β
⎛ ⎞ ⎛ ⎞= β⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⎛ ⎞ ⎛ ⎞+ ⎜ ⎟ ⎜ ⎟⎝
ω α ⋅ ω−
⎠ ⎝ ⎠
iH'
sin sinp p
p pcos c p cos mcmc mc
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 +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦⎢ ⎥⎣ ⎦
August‐September 2012 PCD STiAP Unit 3 108
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 ⎜ ⎟ ⎜+⎢ ⎥⎣
⎟⎝ ⎠ ⎠⎦⎝
August‐September 2012 PCD STiAP Unit 3 109
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
August‐September 2012 PCD STiAP Unit 3 110
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?
iS iS iS iSH' e He e i et
+ − + −∂⎛ ⎞= + −⎜ ⎟∂⎝ ⎠
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⎝ ⎠
August‐September 2012 PCD STiAP Unit 3 111
iS iS iS iSH' e He i e et
+ − + −∂⎛ ⎞= − ⎜ ⎟∂⎝ ⎠
1 where i S i S' lim F( ) F( ) e e+ 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
August‐September 2012 PCD STiAP Unit 3 112
1 where i S i S' lim F( ) F( ) e e+ 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⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
[ ] 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⎣ ⎦
August‐September 2012 PCD STiAP Unit 3 113
[ ]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⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
August‐September 2012 PCD STiAP Unit 3 114
iS iS iS iSH' e He e i et
+ − + −∂⎛ ⎞= + −⎜ ⎟∂⎝ ⎠
August‐September 2012 PCD STiAP Unit 3 115
iS iS iS iSH' e He e i et
+ − + −∂⎛ ⎞= + −⎜ ⎟∂⎝ ⎠
[ ] [ ] [ ]
[ ]
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
+ −
⎡ ⎤⎡ ⎤ ⎡ ⎤= + ⎣ ⎦ ⎣ ⎦⎣ ⎦
∂⎛ ⎞⎡ ⎤⎡ ⎤⎡ ⎤ + −⎜ ⎟⎣ ⎦⎣ ⎦⎣ ⎦ ∂⎝ ⎠
August‐September 2012 PCD STiAP Unit 3 116
iS iS iS iSH' e He e i et
+ − + −∂⎛ ⎞= + −⎜ ⎟∂⎝ ⎠
[ ] [ ] [ ]
[ ]
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 .....
⎡ ⎤⎡ ⎤ ⎡ ⎤= + ⎣ ⎦ ⎣ ⎦⎣ ⎦
⎡ ⎤⎡ ⎤⎡ ⎤⎣ ⎦⎣ ⎦⎣ ⎦
⎡ ⎤⎡ ⎤ ⎡ ⎤− − + +⎣ ⎦ ⎣ ⎦⎣ ⎦
August‐September 2012 PCD STiAP Unit 3 117
[ ] [ ] [ ]
[ ]
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⎛ ⎞⎜ ⎟⎝ ⎠
○
August‐September 2012 PCD STiAP Unit 3 118
[ ] [ ] [ ]
[ ]
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⎛ ⎞⎜ ⎟⎝ ⎠
○
August‐September 2012 PCD STiAP Unit 3 119
[ ] [ ] [ ]
[ ]
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⎣ ⎦⎣ ⎦⎣ ⎦
⎡ ⎤⎡ ⎤ ⎡ ⎤− − +⎣ ⎦ ⎣ ⎦⎣ ⎦
August‐September 2012 PCD STiAP Unit 3 120
[ ] [ ] [ ]
[ ]
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 ……
August‐September 2012 PCD STiAP Unit 3 121
[ ] [ ] [ ]
[ ]
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 − − +⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦
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
August‐September 2012 PCD STiAP Unit 3 122
[ ] [ ] [ ]
[ ]
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 − − +⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦
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⎣ ⎦⎣ ⎦
August‐September 2012 PCD STiAP Unit 3 123
[ ] [ ]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,! !
i iS, S, S S,S S, S,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⎜ ⎟⎝ ⎠
∼
August‐September 2012 PCD STiAP Unit 3 124
[ ]
[ ]
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
+ − + −∂⎛ ⎞= + −⎜ ⎟∂⎝ ⎠
August‐September 2012 PCD STiAP Unit 3 125
Questions? pcd@physics.iitm.ac.in
August‐September 2012 PCD STiAP Unit 3 126
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⎣ ⎦ ⎣ ⎦
β θ βθθ = θ ε − +
i' , , ,mc m c m c m c
i' ,mc m c mc
The leading ‘odd’ term is now of
2= β + ε + θH' mc ' '
1 m⎛ ⎞⎜ ⎟⎝ ⎠
○
P. C. Deshmukh Department of PhysicsIndian Institute of Technology MadrasChennai 600036
pcd@physics.iitm.ac.in
Unit 3(v) Lecture 17
Relativistic Quantum Mechanics of the
Hydrogen Atom
Select/Special Topics in Atomic Physics
Reduction of Dirac Eq. to two-component formPAULI Equation
4 1
2 1
2 1×
×
×
ϕ⎛ ⎞ψ = ⎜ ⎟χ⎝ ⎠
Foldy-Wouthuysen Transformations - 3
August‐September 2012 PCD STiAP Unit 3 127
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
August‐September 2012 PCD STiAP Unit 3 128
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 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
+ − + −∂⎛ ⎞= + −⎜ ⎟∂⎝ ⎠
August‐September 2012 PCD STiAP Unit 3 129
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⎣ ⎦ ⎣ ⎦
August‐September 2012 PCD STiAP Unit 3 130
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
August‐September 2012 PCD STiAP Unit 3 131
[ ]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∇× + ×∇ =
August‐September 2012 PCD STiAP Unit 3 132
{ }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!
August‐September 2012 PCD STiAP Unit 3 133
( )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
August‐September 2012 PCD STiAP Unit 3 134
[ ]{ }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 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 +⎨ ⎬ ⎨ ⎬∂ ∂⎩ ⎭ ⎩ ⎭
August‐September 2012 PCD STiAP Unit 3 135
[ ]{ }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⎜ ⎟ ⎣ ⎦⎝ ⎠
[ ], 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 ⋅⎣ ⎦ ⎣ ⎦
August‐September 2012 PCD STiAP Unit 3 136
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⋅ ×⎣ ⎦ ×
August‐September 2012 PCD STiAP Unit 3 137
[ ]{ }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⎜ ⎟ ⎣ ⎦⎝ ⎠
[ ]{ } ( )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
August‐September 2012 PCD STiAP Unit 3 138
[ ]{ }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⎜ ⎟ ⎣ ⎦⎝ ⎠
[ ]{ }( )
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
August‐September 2012 PCD STiAP Unit 3 139
[ ]{ }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⎜ ⎟ ⎣ ⎦⎝ ⎠
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 VˆE e rr r r
∂ ∂= − = −
∂ ∂
2 2
14
Vrm
r pr
ec
∂×
∂σ ⋅
August‐September 2012 PCD STiAP Unit 3 140
[ ]{ }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⎜ ⎟ ⎣ ⎦⎝ ⎠
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
⎡ ⎤⎛ ⎞⎢ ⎥= + −⎜ ⎟⎢ ⎥⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞ ⎛ ⎞= + − + −⎢ ⎥⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦
= − +
August‐September 2012 PCD STiAP Unit 3 141
August‐September 2012 PCD STiAP Unit 3 142
( )
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:
August‐September 2012 PCD STiAP Unit 3 143
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
August‐September 2012 PCD STiAP Unit 3 144
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
August‐September 2012 PCD STiAP Unit 3 145
2s = σ
2
2
2
ec
ame
⎛ ⎞α = ⎜ ⎟
⎝ ⎠
=
( )
2 4
2 2
22
2
12
2
nmZ eE
nZ
mcn
= −
α= −
Ref.: Bransden & Joachain – Physics of Atoms & Molecules / Ch 5
( )( )
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
August‐September 2012 PCD STiAP Unit 3 146
( )22
22n
ZE mc
nα
= −
( )( )
2
31 14
12 12
spin orbit n
j( j ) ( )H E Z
n−
⎧ ⎫+ − + −⎨ ⎬⎩ ⎭= − α
⎛ ⎞+ +⎜ ⎟⎝ ⎠
Ref.: Bransden & Joachain – Physics of Atoms & Molecules / Ch 5
August‐September 2012 PCD STiAP Unit 3 147
( )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= −
Ref.: Bransden & Joachain – Physics of Atoms & Molecules / Ch 5
( )( )
( )
( )
( )
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≠
August‐September 2012 PCD STiAP Unit 3 148
( )( )
( )
( )
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= −
Ref.: Bransden & Joachain – Physics of Atoms & Molecules / Ch 5
August‐September 2012 PCD STiAP Unit 3 149
( )
( )
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= −
Ref.: Bransden & Joachain – Physics of Atoms & Molecules / Ch 5
August‐September 2012 PCD STiAP Unit 3 150
( )
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≠
August‐September 2012 PCD STiAP Unit 3 151
( )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
= + −⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟+⎢ ⎥⎝ ⎠⎣ ⎦
August‐September 2012 PCD STiAP Unit 3 152
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
August‐September 2012 PCD STiAP Unit 3 153
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 '
August‐September 2012 PCD STiAP Unit 3 154
2Spherical symmetryDirac HamiltonianH c p mc V(r)= α⋅ + β +
Strategy:
separation into
radial and angular parts
How shall we handle this term?
August‐September 2012 PCD STiAP Unit 3 155
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 ⋅
August‐September 2012 PCD STiAP Unit 3 156
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⋅⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠
August‐September 2012 PCD STiAP Unit 3 157
( )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
August‐September 2012 PCD STiAP Unit 3 158
2⎛ ⎞= α + α β + β +⎜ ⎟⎝ ⎠
SphDirac r r r
iH c p K mc V(r)r
Questions? pcd@physics.iitm.ac.in
Next: solution to the H atom
2Spherical symmetryDirac HamiltonianH c p mc V(r)= α⋅ + β +
( )( )
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 +⎜ ⎟∂⎝ ⎠
P. C. Deshmukh Department of PhysicsIndian Institute of Technology MadrasChennai 600036
pcd@physics.iitm.ac.in
Unit 3(vi) Lecture 18
Relativistic Quantum Mechanics of the
Hydrogen AtomSpherical symmetry of the Coulomb potential
Select/Special Topics in Atomic Physics
August‐September 2012 PCD STiAP Unit 3 159
August‐September 2012 PCD STiAP Unit 3 160
2Spherical symmetryDirac HamiltonianH c p mc V(r)= α⋅ + β +
Strategy:
separation into
radial and angular parts
How shall we handle this term?
Hydrogen atom – solution to Dirac Hamiltonian
August‐September 2012 PCD STiAP Unit 3 161
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 ⋅
August‐September 2012 PCD STiAP Unit 3 162
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⋅⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠
August‐September 2012 PCD STiAP Unit 3 163
( )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
August‐September 2012 PCD STiAP Unit 3 164
2⎛ ⎞= α + α β + β +⎜ ⎟⎝ ⎠
SphDirac r r r
iH c p K mc V(r)r
2Spherical symmetryDirac HamiltonianH c p mc V(r)= α⋅ + β +
( )( )
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 +⎜ ⎟∂⎝ ⎠
August‐September 2012 PCD STiAP Unit 3 165
2⎛ ⎞= α + α β + β +⎜ ⎟⎝ ⎠
SphDirac r r r
iH c p K mc V(r)r
2Spherical symmetryDirac HamiltonianH c p mc V(r)= α⋅ + β +
( )( )
4 4 4 4 4 4 4 4
4 4 4 4 4 4
1
i.e. 1
× × × ×
× × ×
= +Ξ ⋅
= +Ξ ⋅
K
K
β
β
21SphDirac rH - ic K mc V(r)
r r r∂⎛ ⎞= α + − β + β +⎜ ⎟∂⎝ ⎠
August‐September 2012 PCD STiAP Unit 3 166
( ) ( ) ( )( )( )
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
August‐September 2012 PCD STiAP Unit 3 167
( ) 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
August‐September 2012 PCD STiAP Unit 3 168
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 = ± +⎜ ⎟⎝ ⎠
= ∓
August‐September 2012 PCD STiAP Unit 3 169
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 = ± +⎜ ⎟⎝ ⎠
= ∓
August‐September 2012 PCD STiAP Unit 3 170
2SphDirac r r r
cH c p i K mc V(r)r
= α + α β + β +
21SphDirac rH - ic K mc V(r)
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
cc p i mc V(r) u E ur κ κ
κ⎡ ⎤β α + α β + β + = β⎢ ⎥⎣ ⎦
August‐September 2012 PCD STiAP Unit 3 171
2r r r n m nj n m
cc p i mc V(r) u E ur κ κ
κ⎡ ⎤− α β − α + + β = β⎢ ⎥⎣ ⎦
2 1r r ;βα = −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
cc p i mc V(r) u E ur κ κ
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⎢ ⎥⎣ ⎦
August‐September 2012 PCD STiAP Unit 3 172
( ) ( )
( ) ( )( )
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
August‐September 2012 PCD STiAP Unit 3 173
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 + + =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠
August‐September 2012 PCD STiAP Unit 3 174
( )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 ˆe 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
u uccp i mc V(r) Eur
+ +
−
⎛ ⎞σ σ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞κ− + + =⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟σ σ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠
August‐September 2012 PCD STiAP Unit 3 175
( )20 0 00 0 0 0
r rr nj
r r
u uccp i mc V(r) Eur
+ +
−
⎛ ⎞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! × → ×
August‐September 2012 PCD STiAP Unit 3 176
2r r r n m nj n m
cc p i mc V(r) u E ur κ κ
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
cc p i mc V(r) u E ur κ κ
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
ccp i mc V(r) u E ur κ κ
κ⎡ ⎤β + β + α β + α = β α⎢ ⎥⎣ ⎦
2r r r n m nj r n m
ccp i mc V(r) u E ur κ κ
κ⎡ ⎤β + − α + βα = βα⎢ ⎥⎣ ⎦
(Eq.X)
(Eq.Y)
August‐September 2012 PCD STiAP Unit 3 177
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
ccp i mc V(r) u E ur κ κ
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⎢ ⎥⎣ ⎦
+
August‐September 2012 PCD STiAP Unit 3 178
( ) ( )
( ) ( )( )
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
ˆ uiQ (r) (r)r+κ κ
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! × → ×
August‐September 2012 PCD STiAP Unit 3 179
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 = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
August‐September 2012 PCD STiAP Unit 3 180
( )
( )
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 = ⎜ ⎟⎛ ⎞ ⎛ ⎞⎜ ⎟= + −⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠
August‐September 2012 PCD STiAP Unit 3 181
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 = − →
August‐September 2012 PCD STiAP Unit 3 182
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
August‐September 2012 PCD STiAP Unit 3 183
( )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
ˆ uiQ (r) (r)r+κ κ
κ−κ −κ
Ω⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟Ω ⎝ ⎠⎝ ⎠
August‐September 2012 PCD STiAP Unit 3 184
( )2
n mr r
n m n mnj
ˆiQ (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
ˆ uiQ (r) (r)r+κ κ
κ−κ −κ
Ω⎛ ⎞ ⎛ ⎞= = ⇒⎜ ⎟ ⎜ ⎟Ω ⎝ ⎠⎝ ⎠
( )2
n mr
n m n mr nj r
ˆP (r) (r)ccp ir r
ˆ ˆiQ (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
ˆiQ (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 ± =
August‐September 2012 PCD STiAP Unit 3 185
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− + = − −
August‐September 2012 PCD STiAP Unit 3 186
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
− − − −⎡ ⎤ = × × =⎢ ⎥⎣ ⎦
Questions? pcd@physics.iitm.ac.in
Recommended