generates 3-dimensional rotations and with the basis: serve as the angular momentum operators! and...
TRANSCRIPT
00
00
000
1
i
i
00
000
00
2
i
i
000
00
00
3 i
i
generates 3-dimensional rotations
and with the basis:
0
2/
2/1
,
1
0
0
,
0
2/
2/1
ii serve as the angularmomentum operators!
0210
21021
0210
xJ
020
202
020
i
ii
i
J y
100
010
001
zJ
and with the simpler basis:
1
0
0
,
0
1
0
,
0
0
1
And of course we already had
010
0
0
000
00
00
0
100
0
22
21
21
221
221
ii
i
i
i
i
U 3U =
†
100
000
001
000
0
0
0
100
0
22
21
21
221
221
ii
i
i
U 1U =†
U 2U =†
020
202
020
i
ii
i
0210
21021
0210
[ 12 21 ] = i3
U†[ 12 21 ]U =U†i3U
U†12UU†21U = iU
†3U
U†1UU†2UU
†2UU†1U = iU
†3U
Since U†U = UU
† = IJx
Jx JyJy JxiJz
0210
21021
0210
00
00
000
i
i
000
000
000
000
000
000 00
000
00
i
i
000
000
000
000
000
000
020
202
020
i
ii
i
a
bcxyz,
This 6×6 matrix also satisfies the same algebra:
000
000
000
000
000
000
000
000
000
000
000
000
3-dimensional transformations (like rotations) are not limited to 3-dimensional “representations”
100
010
0011 z
JJ
2/10
02/12/1 zJJ
2/3000
02/100
002/10
0002/3
2/3 zJJ
20000
01000
00000
00010
00002
2 zJJ
Besides the infinite number of similarity transformations that couldproduce other 3×3 matrix representations of this algebra
The 3-dimensional representation
in the orthonormal basis
that diagonalizes z is the
“DEFINING” representation
of vector rotations
R ( ) = eiJ·/ħ^
can take many forms
ℓ = 2mℓ = 2, 1, 0, 1, 2
L2 = 2(3) = 6|L| = 6 = 2.4495
ℓ = 1mℓ = 1, 0, 1
L2 = 1(2) = 2|L| = 2 = 1.4142
2
1
0
1
0
Note the always odd number of possible orientations:
A “degeneracy” in otherwise identical states!
Spectra of the alkali metals
(here Sodium)all show
lots of doublets
1924: Pauli suggested electrons posses some new, previously un-recognized & non-classical 2-valued property
Perhaps our working definition of angular momentum was too literal…too classical
perhaps the operator relations
yzxxz
xyzzy
zxyyx
LiLLLL
LiLLLL
LiLLLL
may be the more fundamental definition
Such “Commutation Rules”are recognized by mathematicians as
the “defining algebra” of a non-abelian
(non-commuting) group[ Group Theory; Matrix Theory ]
Reserving L to represent orbital angular momentum, introducing the more generic operator J to represent any or all angular momentum
yzxxz
xyzzy
zxyyx
JiJJJJ
JiJJJJ
JiJJJJ
study this as an algebraic group
Uhlenbeck & Goudsmit find actually J=0, ½, 1, 3/2, 2, … are all allowed!
ms = ± 12
spin “up”spin “down”
s = ħ = 0.866 ħ 3 2
sz = ħ 12
| n l m > | > = nlm12
12
10( )
“spinor”
the most general state is a linear expansion in this 2-dimensional basis set
1 0 0 1( ) = + ( ) ( )
with 2 + 2 = 1
spin : 12p, n, e, , , e , , , u, d, c, s, t, b
leptons quarks
the fundamental constituents of all matter!
obviously:
1
1
2
2
0
0
20
0
zs
How about the operators sx, sy ?eigenvalues of each are also ħ/2 but
their matrices are not diagonal in this basis
s- 2
1
2
1
2
1
2
1 = 1ħ
s- 2
1
2
1
2
1
2
1= 1ħ
s-
01
00ħ
s+
00
10ħ
obviously work on the basis we’ve defined
You already know these as the Pauli matrices
01
10
2
xs
0
0
2 i
isy
10
01
2
zs
obeying the same commutation rule: zyx i ],[but 2-dimensionally!
What if we used THESE as generators?
01
10x
0
0
i
iy
10
01z
01
10x
0
0
i
iy
10
01z
Should still describe “rotations”. Its 3-components will still require
3 continuously variable independent parameters:
x , y , z
But this is not the defining representationand can not act on 3-dimensional space vectors.
These are operators that obviously act on the SPINORS(the SPIN space, not the 3-dimensional wave functions.
Spinors are 2-component objects
intermediate between scalars (1-component) and vectors (3-component)
When we rotate the “coordinate system” scalars are unchanged.Vector components are mixed by the prescriptions we’ve outlined.
What happens to SPINORS?
/2
1 ie
actually can only act on the spinor part
// 2
121
)()( imnmn
i erre
/21ie
/2
1 ie
The rotations on 3-dim vector space involved ORTHOGONAL operators
These carry complex elements, cannot be unitary!
Rt = R-1 i.e. RtR = RRt = 1
As we will see later this is a UNITARY MATRIX
of determinate = 1
Let’s stay with the simplified case of rotation = z (about the z-axis)
^
!2
)2/(
!5
)2/(
!4
)2/(
!3
)2/(
!2
)2/(
21
6655443322/2
1 zzzzzzi iiiiiie
!6
)2/(
!5
)2/(
!4
)2/(
!3
)2/(
!2
)2/(
21
6655443322/2
1 zzzzzzi iiiiiie
Notice: zz =
10
01
10
01
110
01
and obviously: zzz = z
!6
)2/(
!4
)2/(
!2
)2/(1
664422 zzz iii( )
!7
)2/(
!5
)2/(
!3
)2/()2/(
753 ( )+ iz
/2
1ie
)()(22
/ sincos21 z
i ie
)()(22
/ sincos21 z
i ie
an operator analog to: ei = = coscos + i + isinsin
)()(22
/ sincos21 iei
1 00 1( ) 1 0
0 -1( )
Let’s look at a rotation of 2 (360o)
)()(2
22
2/)2( sincos21 z
i ie z
= 1 + 0This means
A 360o rotation does not bring a spinor “full circle”.Its phase is changed by the rotation.
Limitations of Schrödinger’s Equation
1-particle equation
),()(),(2
),( 21
22
txxVtxxm
txt
i
),,(),()()(
),,(2
),,(2
),,(
212121
2122
2
2
2
2121
2
2
2
21
txxxxVxVxV
txxxm
txxxm
txxt
i
2-particle equation:
mutual interaction
But in many high energy reactionsthe number of particles is not conserved!
np+e++e
n+p n+p+3
e+ p e+ p + 6 + 3
Let’s expand the DEL operator from 3- to 4-dimensions
x i.e.
tc 1
0
x1 y
2z
3
lowered
since operates on x
then ipand as we’ve argued before,
the starting point for a relativistic QM equation:
022 cmpp
0222 cm
0)( 22 cm The Klein-Gordon
Equation
or if you prefer: 0)(1 22
2
2
2
mc
tc
But this equation has a drawback:Look at Schrödinger’s Equation for a free particle
22
2
mti
02
22
m
i
22
2
mi
*22
*
2
mi
*( ) ( )
0)(2
)( *22*2
** m
i
0)(2
)( **2
*
mti
probabilitydensity
probabilitycurrent density
The ContinuityEquation
starting from the Klein-Gordon Equation 0)(1 22
2
2
2
mc
tc
*( ) ( )* = 022
222
2
mc
ct
2222
2
2
mc
ct
0*22
*22
*222*2*2
2
2
2*
mcmc
cctt
)()( **2**
ct
not positive definite
The Klein-Gordon Equation is 2nd order in t!
Need to know initial (t=0) state as well as (0)
much morecomplicatedtime evolution