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Quantum Mechanics
p2
E
0
2m
 h2 2
ih 
0
t 2m
Classical – non relativistic
Quantum Mechanical : Schrodinger eq
Quantum Mechanics
p2
E
0
2m
2

i

 2  0
t 2m
E 2  p 2  m2
2
 t 2  2  m2
Classical – non relativistic
Quantum Mechanical : Schrodinger eq
Classical – relativistic
(Natural units h  c  1)
Quantum Mechanical - relativistic :
Klein-Gordon (Schrodinger) equation

Relativistic QM - The Klein Gordon equation (1926)
 (x)
Scalar particle (field) (J=0) :
E p m
2
2
2
Energy eigenvalues
2
  t 2  2  m2
(natural units)
E  (p2  m2 )1/ 2 ???
1927 Dirac tried to eliminate negative solutions by writing a relativistic
equation linear in E (a theory of fermions)
1934 Pauli and Weisskopf revived KG equation with E<0 solutions as E>0
solutions for particles of opposite charge (antiparticles). Unlike Dirac’s hole
theory this interpretation is applicable to bosons (integer spin) as well as to
fermions (half integer spin).
As we shall see the antiparticle states make the field theory causal
Physical interpretation of Quantum Mechanics
i t  21m  2  0
Schrödinger equation (S.E.)

t
i ( S .E.)  i (S .E.)
*
*
= 
 .j  0 continuity eq.
2
“probability density”
QuickTime™ and a
decompressor
are needed to see this picture.
j   2im ( *   * )
“probability current”
Divergence Theorem
 .M dV=  M .n dS


Physical interpretation of Quantum Mechanics
i t  21m  2  0
Schrödinger equation (S.E.)

t
i ( S .E.)  i (S .E.)
*
*
= 
 .j  0 continuity eq.
j   2im ( *   * )
2
“probability density”
“probability current”
2
 t 2  2  m2
Klein Gordon equation
  2E N
2
Negative probability?
  Neip. x ,   2 E N
  dV    d
V
  i( * t   t )
*
3
x  2E
j  i( *   * )
j   (  , j)
2
f p  e
ip. x
1
2 p0V
Normalised free particle solutions
Physical interpretation of Quantum Mechanics
i t  21m  2  0
Schrödinger equation (S.E.)

t
i ( S .E.)  i (S .E.)
*
*
= 
 .j  0 continuity eq.
j   2im ( *   * )
2
“probability density”
“probability current”
2
 t 2  2  m2
Klein Gordon equation
  i(
Pauli and Weisskopf
* 
t


*
t
)

j   e (  , j)  jEM

t
 .j  0 =   j
j  i( *   * )
j   (  , j)
Physical interpretation of Quantum Mechanics
i t  21m  2  0
Schrödinger equation (S.E.)

t
i ( S .E.)  i (S .E.)
*
*
= 
 .j  0 continuity eq.
j   2im ( *   * )
2
“probability density”
“probability current”
2
 t 2  2  m2
Klein Gordon equation
EM  i(
Pauli and Weisskopf
* 
t


*
t
)

j   e (  , j)  jEM

t
 .j  0 =   j
j  i( *   * )
j   (  , j)

Relativistic QM - The Klein Gordon equation (1926)
 (x)
Scalar particle (field) (J=0) :
2
  t 2  2  m2
E p m
2
2
2
Relativistic notation :
(     m2 )  0
(natural units)
4 vector notation
A  ( A0 , A), B   ( B0 , B)
contravariant
A  ( A0 ,  A), B  ( B0 , B)
A  g A
covariant
A  g  A
A.B  A B  A B  A0 B0  A.B
4 vectors
E
( , p)  p 
c

  (
, )
ct
(ct , x)  x 
  (


, )
ct
p i 

p p   E 2  p
2
  2  
Field theory of


Scalar particle – satisfies KG equation
(     m2 )  0

Classical electrodynamics, motion of charge –e in EM potential
is obtained by the substitution :

Quantum mechanics :
A   (A0 , A)
p   p   eA
i   i   eA
The Klein Gordon equation becomes:
(     m2 )  V 
where V  ie(  A  A   )  e2 A2
em  4e
2
The smallness of the EM coupling,
1
137 ,
Make a “perturbation” expansion of V in powers of
V
means that it is sensible to
 em
ie(  A  A   )
The pion propagator
Want to solve :
Solution :
where
and
(     m2 )  V
 ( x)   ( x)   d 4 x '  F ( x ' x)V ( x ') ( x ')
(     m2 )  0
(     m2 ) F ( x ' x)   4 ( x ' x)
Feynman propagator
d
4
4
Dirac Delta function
x '  ( x ' x) f ( x ')  f ( x)
The pion propagator
(     m2 )  V
Want to solve :
 ( x)   ( x)   d 4 x '  F ( x ' x)V ( x ') ( x ')
Solution :
(     m2 )  0
where
(     m2 ) F ( x ' x)   4 ( x ' x)
and
Simplest to solve for propagator in momentum space by taking Fourier transform
1
(2
2 2) 2
 ip .( x '  x )
e

(     m2 ) F ( x ' x)d 4 ( x ' x)  (221 )2  e ip.( x ' x ) 4 ( x ' x)d 4 ( x ' x)
2
( p 2  m 2 ) F ( p) 
•F ( p) 

1
(2 )2
1
,
 p 2  m2  i
1
(2 )2
 F (x)   (2 )4  d p e
1
4
 ip.x
1
p 2  m2  i
The Born series
 ( x)   ( x)   d x '  F ( x ' x)V ( x ') ( x ')
4
Since V(x) is small can solve this equation iteratively :
 ( x)   ( x)   d 4 x '  F ( x ' x)V ( x ') ( x ')
  d 4 x ''  d 4 x '''  F ( x '' x)V ( x '') F ( x ''' x '')V ( x ''') ( x ''')  ...
x
Interpretation :
V ( x ')
x'
x '''
x ''
 F ( x ' x)
But energy eigenvalues
E  (p2  m2 )1/ 2
???
Feynman – Stuckelberg interpretation

  ( E  0)
t
e
  ( E  0)
ei (  E )( t )
 iEt
Two different time orderings giving same observable event :
t1
t2
  
   
 
time
t2
t1
space
t1
t2
  
     
 
t1
t2
  
   
 
t1
t2
  
     
 
time
space

 p  p2  m2
 F ( x ' x)   (2 )4  d p e
1
4
ip.( x '  x )
1
p2  m2 i
 i  (2 )3 2 e
d3 p

1/ 2
 i p t ' t ip.(x'-x)
p
(p0 integral most conveniently evaluated using contour integration via Cauchy’s theorem )
 F ( x)   (2 )4  d p e
4
1
p  m  i
2
2

p  p  m  i
2
2
o
 F ( x ' x)   (21 )4  d p e
3
2
 i p.( x '  x )
 ip. x
1
p  m2 i
2


p0   p  m
 dp  p  
2

1/ 2
eip0 (t 't )
0
0
2
p

i   p i
 i  p0    p  i 

}
I
 If t '- t  0, choose contour such that p0  ipI
I 
 i i
e
p
  t ' t 
p ( t ' t )
 If t '- t  0, choose contour such that p0  ipI
I 
 i i
e
p
( pI +ve)  e ip0 (t 't )  e pI (t 't )
  t  t '
p ( t ' t )
( pI +ve)  e  ip0 (t 't )  e  pI (t 't )
t1
t2
  
   
 
t1
t2
  
     
 
time
space
 F ( x ' x)   (2 )4  d p e
4
1

 ip.( x '  x )
1
p 2  m2 i
 i  (2 )3 2 e
d3 p
 i p t ' t ip .( x '  x )
p

 F ( x  x ')  i  d p f ( x ') f ( x) (t ' t )  i  d p f ( x ') f p* ( x) (t  t ')
3
p
where
f p  e
ip. x
1
2 p0V
*
p
3
p
are positive and negative energy solutions to free KG equation