hamiltonian symmetry: unitary transformations€¦ · translations and rotations • taylor...
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FermiGasy
Hamiltonian Symmetry: Unitary Transformations
Translations and Rotations• Taylor expansion and unitary transformations• Translation invariance
1D periodic lattice• Rotational symmetries
Angular momentum operators, algebraKets for states w/good angular momentumLadder operatorsSpherical harmonicsRotational matrix elements
• Rotationally symmetric energy eigen functionsSquare well, Bessel functions
• Intrinsic spinPauli matrices, spinors
• Coupling of angular momenta, Clebsch-GordanWigner Eckart TheoremSpherical tensors
• Exchange symmetry
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Translations
: ( ) ( )Translation r r r a and r r → = + =(r) ’(r’)
r
a
Active view: move particle and wf, not coordinate system
Express ’ as function of (Taylor expansion)
( )
( )
2
0
1( ) ( ) ( ) ( ) ( ) ...
2!
1( ) ( )
!
n a
n
r r a r a r a r
a r e rn
−
=
= − = − + − −
= − =
ˆ
ˆ
ˆ : ( ) ( ) ( )
ˆ: ( ) ( ) 1 ( )
ia p
a
ia p
Momentum operator p r e r e ri
iInfinitesimal a r e r a p r
− −
−
= → = =
= −
( )
( )
ˆˆ:
ˆˆ: 1
ia p
Operator of translations U a e
iinfinitesimal U a a p
−
=
= −
r
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Unitary Translation Operators
1
1 1 1
ˆ ˆ( ) ( ) ( ) ( ) ( ) ( )
ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( )
r U a r and therefore r U a r
r U a U a r U a U a U a U a
−
− − −
= =
→ = → = = ( )
ˆ
ˆ1
1 †
†
ˆ
ˆ ( )
ˆ ˆ( ) ( )
ˆ ˆ( ) ( )
ia p
ia p
U a e
U a e
U a U a
U a U a
−
+ −
−
=
=
=
= −
11
ˆ( )
ˆ ˆ ˆ( ) ( ) (
( )
( )
ˆ ˆ ˆ( ) ( ) ( ) ( ) ( )
ˆ ˆ ˆ( ) ( ) ( ) ( ) ( ))
Translate position dependent operator A r
A r a r a U
U a
a A r r
U a
A r
A r U a rr rA U a
r
− −
−
= − − = =
= =
1ˆ ˆ ˆ ˆ( ) ( ) ( ) ( )A r U a A r U a− =
†
1
ˆ ˆ ˆ ˆ( ) ( ) ( ) ( ) ( )| ( ) ( )| ( )
ˆ ˆ( )| ( ) ( )| (
( )
) ( )
(
(
)
)
U a r U a r r U a U a r
r U a U a r
r
r r
r
−
= = =
=
Unitary Operator
Unitary operator preserves norm/probability/ matrix elements
ˆ ˆ( ) ( ) ( ) ( )r A r r A r =
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Translation Invariance
ˆ ˆ( ) ( )H r H r a= − →If Hamiltonian translation invariant2
0
1
ˆˆ ˆ ˆ( ) ( ) ( )
2
ˆ ˆ ˆ ˆ ˆ( ) ( ) ( ) ( ) ( )
pH r V H r a H r
m
H r U a H r U a H r−
= − + = − =
= =
x
V(x)
mV0
ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) ( ) ( ) , 0U a H r H r U a H U → = → =
ˆ
ˆ ˆˆ ˆ ˆ ˆ0 , ( ) , 1 ,
ˆˆ ˆ, 0 ( ) ( ) ( )
ip r
p p p
i iH U a H p a H p a a
H p H r E r with r e Plane waves
Plane waves are linear momentum EF
−
= = − = −
→ = → =
−
pIf the Hamiltonian of a system is translation invariant, it commutes with the operator of translations=momentum operator ( ).
Then and have simultaneous eigen functions
Then solutions of Schrödinger Equ. (V=const) are linear-momentum eigen functions.
pH
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Periodic Boundary Conditions
Electron moving in linear Coulomb lattice: periodic ion sites x, x+d, x+2d,…x+Nd →discrete translational symmetry (Dx=d lattice constant). Operator Approximate potential VC → V
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ˆdT
( ) ( )( )
ˆ ˆ, 0
i p x i k xx
d
V x piecewise constant x e e
H T common eigen functions
= → =
→ = →
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
ˆ ˆ:
ˆ2
1
d d
niN
N N Nd
u x d u x Periodic boundaryT eigen functions T u x t u x
T u x u x Nd t u x u x
condition
t t e
− = =
= + = = →
+
= → =
=
Bloch Theorem
e-
A+
approximate
x
x
VC(x)
V(x)
d
For the specific system (SEq.)!
( ) ( ) ( ) ( ) ( )ˆˆ ˆ2 2
22
dH x x V x
m dxH x H x d == − + → +
( ), , , ,
1 20 1 2
2
odd NNn
even NN
−=
( ) ( ) ( ); ;2i k xnkn n
nk periodic function
du
Nx e xx
=
=
Bloch Function
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( ) ( )
( )
( ) :
,
: ; :b b
Region I Valence Ban
Electron mass m binding energy E
k m V E m E V
periodic function x
d
−
= − = −
→
0 02 2
( )( )
e e
e e
ik x ik xb b
x xb b
nd x nd bA Ax
nd b x n dB B
−+ −
−+ −
+ + =
+ + + 1
( ) ( ) i kxku x x e = →
( )( ) ( )
( ) ( ) ( )( )
e e
e e
i k k x k k xb b
k kik x ik xb b
nd x nd bA Au x u x d
nd b x n dB B
− − ++ −
− − ++ −
+ + = =
+ + + 1
0
x
nd (n+1)d
-E
0
Determine constants A± and B± from matching conditions @ x=0, x=d.
( ) ( ) ( ) ( )
!
,
( ) sinh sin cosh
( ) cos( )
cosb bb b b b
b b
with
kf E d b k b d b k b
f E d
k
k
→
− = − + −
=
2 2
2 ( )
Allowed energy b
E wit
and
h f E 1
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( )
( ) ( )
,
( ) :
: ;
;
c
Electron mass m binding energy E
Region II Conduction Band
k m E V
periodic function x for nd b x
For V E
n d
−
→
= −
→ +
+
−
−1
12
1
( ) e e ;ik x ik xc cx A A −
+ −= +
( ) ( ) i kxku x x e = →
-E
Determine constants A± from matching conditions @ x=d.
( ) ( ) ( ) ( )
!
( ) co ,
( ) sin sin cos cos
s( )
c bc b c b
b c
similar function with
k kf E k d b k b k d b k b
k
f
k
E k d→
+ = − + −
=
2 2
2
0
x
nd (n+1)d
0
-E
( )
Allowed energy b
E wit
and
h f E 1
Crystal Band Structure
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( )f kb
→k b
( ) ( )) 1
Allowed energy ba
E kb wit
d
h f E
n
FermiGasy
Hamiltonian Symmetry: Unitary Transformations
Translations and Rotations• Taylor expansion and unitary transformations• Translation invariance
1D periodic lattice• Rotational symmetries
Angular momentum operators, algebraKets for states w/good angular momentumLadder operatorsSpherical harmonicsRotational matrix elements
• Rotationally symmetric energy eigen functionsSquare well, Bessel functions
• Intrinsic spinPauli matrices, spinors
• Coupling of angular momenta, Clebsch-GordanWigner Eckart TheoremSpherical tensors
• Exchange symmetry
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Rotations about one (z) Axis
Example: rotation about z axis ˆ( ): ( ) ( ) ( ) ( )z wr r ith r rR = =
2
0
:
1( ) ( ) ( ) ( ) ( )
2!
1( ) ( )
!
n
n
In polar coordinates
e rn
−
=
= − = − + −
= − =
ˆ ˆ: :z zRemember J L x yi i y x
= = = −
ˆˆ ( )
iJz
zR e
−
=Operator of finite rotations by about z axis
ˆˆ: ( ) ( ) 1 ( )
iJz
z
iInfinitesimal rotations by about z r e r J r
− = −
y
x
q
( , , )r r q =
( , , )r r q = +
z
ˆ
( ) ( ) ( )
.( )
iJz
Orbital angular momentum r e r e r
op z component
− −
→ = =
( ) ( )ˆ2
ˆ ˆ(2 ) ˆ1
iJz
z zJr R r e n integer n
−
= = → = →
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Quantum States with Rotational Symmetry
ˆˆ ( ) ; , ,
iJi i
iR e i x y z
−
= =Operators of rotations about x,y,z axes
ˆˆ: ( ) ( ) 1 ( ) ; , ,
iJi i
i i
iInfinitesimal r e r J r i x y z
− = − =
Attention: Order of successive rotations is important.
If H rotationally invariant, the H eigen functions are also angular momentum J eigen functions conserved in each state. How well can components be measured? J
z
M
,
ˆ ˆ ˆ,
x y z
x y z
From classical Poisson bracket relations
L L L and cyclic one expects commutators
L L i L different components incompatible
=
= →
Calculate commutators →
Angular Momentum Commutators
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Lz
Lz=M
ˆ ˆ
:
Definition of orbital angular momentum L r p i r
In Cartesian coordinates
= = −
ˆ ˆ ˆ; ;x y zL i y z L i z x L i x yz y x z y x
= − − = − − = − −
2
2
2
2
1 ˆ ˆ
1 ˆ ˆ
x y
y x
L L y z z xz y x z
yx
L L z x y zx z z y
xy
x y and cyclic p
zxy z
yzz x
zy x
z
ermuta
xx
zy
t
z
yxz z
xx y
yzx z y
x
z
iy
− = − − =
= + +
−
= − − =
= + +
= −
−
−
−
−
ons
ˆ ˆ ˆ,x y zL L i L and cyclic =
Angular Momentum Commutators
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Lz
Lz=M
ˆ ˆ
:
Definition of orbital angular momentum L r p i r
In Cartesian coordinates
= = −
ˆ ˆ ˆ; ;x y zL i y z L i z x L i x yz y x z y x
= − − = − − = − −
( ) ( ) ( ) ( )
2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ, , , , ,
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 0
z x x z x z x y y z y z y
x y y x y x x y
L L L L L L L L L L L L L L
L i L i L L L i L i L L
= + + +
= − + − + + + + =
2 2 2 2 2 2 2 2
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ, ; , ; ,
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ, , , ,
x y z y z x z x y
x y z z x z y z z z
L L i L L L i L L L i L
L L L L L L L L L L L L
= = =
= + + → = + +
2 (" ") z
Simultaneous measurements are possible
for L and only one z component L
z-axis:= quantization axis, arbitrary axis, but simple in spherical coordinates.Adopt consistent quantization relations for all angular momenta, include spins.
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Operator Algebra
2, ,
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ, 0, : , , ,
ˆ ˆ ˆ,
x y z x y z y z x z x y
i ijkj k
J J J J i J J J i J J J i JBut
J J i J
= = = =
→ =
Representations of operator Lie algebra →
ijk totally antisymmetric for non-cyclic permutations
ˆ ˆ ˆ
3
ˆ ; ; ˆ ˆ ˆ:: xz xy yJ J i
component non Hermitian spherical tensor
J
in rotations transform amo
J J
ng
iJ
themselve
J
s
−+ = −
− −
=+
2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ, 0 , , , 2z z zJ J J J J J J J J J J + + − − + − = = + = − =
ˆ ˆ( ) expi i i
iR J
= −
Assume J2, Jz angular-momentum basis states,since only one projection can be ‘sharp’ [Ji,Jk]≠0
J
M
What are J2 operator eigen values L?
2 2ˆ ˆz
J J J JJ J M
M M M M= L =
M integer or half integer
Jz
M
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Ladder Operator Matrix Elements
ˆˆ: ; ˆ ˆ ˆˆ :ˆ ;: x x yyzNon Hermitian spherical tensor J J J J J iJiJ −+ = = −+−
M integer or half integer
Jz
M
( )
( ) ( ) ( )
( ) ( )
2
1 2
, 1
, :
ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ( 1) ( 1) ( 1)
: ( 1) ( 1) 1
0
ˆ
!
1
z z
JJ M M
In J M basis
J J J JJ J J J
M M M M
J JJ J J J J M M
M M
Also J J M M J M J M
J JJ J M J
Norm must be
M J
MM M
+ + − +
=
= − + = + − +
+ − + = − + +
= +
→
=
( )2 2 2 2
2
1ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ( 1)2
ˆ ˆ ˆ ˆ ˆ ˆ( 1)
z z z z z
z z
From J J J J J J J J J J J J J J J
J J J J J
+ − − + − + − +
− +
= + + → = + + = + +
→ = − +
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Scanning Angular Momentum States
( )
ˆ ?
ˆ ˆ ˆ ˆ ( 1) ( 1)
ˆ ˆ ˆ,
ˆ ˆ1
z
z
z
Acti Apply J J Jon of operators J on states
JJ J J J M M
MJ J
J
M
J J
M M
=
= + = =
( ) ( )ˆ 11
J JJ J M J M
M M = +
Jz
M
( )2
2 2 2
2
ˆ ˆ ˆ ˆ1ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ2
z zzz
z zz
J J J JUse J J J J J J J
J J J J
− ++ − − +
+ −
+ += + + → =
+ −
Largest projection Mmax =J: 2
0 2(
2
)
2 ˆ ˆ ˆ (ˆˆ 1)z z
JJ
JJ J J J
J JJ J J
J JJ− +
= ==
= + + =
+
2 2 2
2 2 2 22 ( 1) ( 1)
ˆ ˆ ˆ ˆ ˆ ( 1)1 1 1
z z
J J J
J J JJ J J J J J J
J J J− +
= = − = −
= + + = + − − −
etc, for all M
→ EV: L =J(J+1)ħ2
“Ladder” operators. Normalization:
(2 1)steps through J
Mvalues M J
+
( )2
:
ˆ 1
Length of J
J J J= +
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Matrix Elements
( ) ( )1 1ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ: ;2 2
x y x yUse J J iJ J J J J J J to calculatei
+ − + −= → = + = −
1 1ˆ ˆ( )( 1) ( )( 1)2 21 1
ˆ ˆ( )( 1) ( )( 1)2 21 1
x x
y y
J J J JJ J M J M J J M J M
M M M M
J J J Ji iJ J M J M J J M J M
M M M M
= − + + = + − ++ −
−= − + + = + − +
+ −
1: : (Group SU(2))2
Special treatment spin J S= =
0 1 0 1 0ˆ ˆ ˆ( ) ( ) ( )2 2 21 0 0 0 1
x y z
x y z
iS S S
i
− = = =
−
ˆ :2
S Pauli matrices→ =
Half-integer angular momenta do not correspond to rotations of classical objects: Rotation about 3600 does not return wave function to original!
( )2 2
2 2
1ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
ˆ ˆ( 1) ; ; 1 2 1
ˆ:2
2
:
z s ss
x
s
y
s s
zDefine equivalent spin ops S S iS S S S S
S S S SS S S S m S m
m
S S
m m m
+ − − +
= + = = →
→ = + +
=
=
!M J
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Quantum States with Rotational Symmetry
ˆˆ ( ) ; , ,
iJi i
iR e i x y z
−
= =Operators of rotations about x,y,z axes
ˆˆ: ( ) ( ) 1 ( ) ; , ,
iJi i
i i
iInfinitesimal r e r J r i x y z
− = − =
Attention: Order of successive rotations is important.
If H rotationally invariant, the H eigen functions are also angular momentum eigen functions (J2, Jz) conserved in each state.
Jz
M
( ) ( )
( )
, , ( ) ,
,
JJ M
definesJM
r u r
J Jr
M M
q q
q
=
= ⎯⎯⎯⎯⎯→
q : polar
: azimuth
2 2ˆ ˆ; z
J J J JJ J M
M M M M= L =
Angular momentum eigen functions and eigen values
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J=L Eigen Functions in r Representation
ˆ sin cot
ˆ cos cot sin ; ˆz
x
y
L i cos
L i L i
q q
q q
= +
= −
= −+
1 2
1( , ) ( ˆ)( 1) ( , )L Lm mY l m l m YLq q
−
+ += − + +
Orbital angular momentum has relation to rotation in 3D space
22 2
2 2
1 1ˆ sinsin sin
L qq q q q
= − +
ˆ cot. iL eLadder op i qq
= +
Angular momentum involves only angular dependence of wave function. Differential equation:
ˆ( , ): ( , ) ( , ) ( , )L L L Lm z m m mY L Y i Y mYq q q q
= − =
( , ) ( ) eL L i m
m mY Y q q =
Construct entire series by applying ladder operators L:
Lz
m
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Recursive Derivation of L2, Lz Wave Functions
ˆ0 ( )
:
( )
cot ( )
( ) sin
LLL
L LL
L
Mini quantum number m L
maximum anti alignment with z axis
L Y
Y a Starting fcn for con
mum
L Y
struction
q qq
q
q
q
−−
−
−
= −
− −
→ = −
→ = =
=
( 1)ˆ ˆ( , ) ( ) cot ( )L L im L i mm m mL Y L Y e m Y e qq q q q
q
++ +
= = −
( ). ˆ cotiLLadder op es i qq
= +
1 2
1( ) ( )(
(
1) c
)
o
si
t (
n
)L Lm m
L LLStarting fcn recursive construction
Y L m L m m
a
Y
Y
q q qq
q q
−
−
+
→
= − + + −
=
1 22( )!
( ) ( 1) sin sin(2 )!( )! cos
L mL L m m Lm
L mY a
L L mq q q
q
+
+ − = −
+
Lz
m
ˆzL
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Recursive Derivation of L2, Lz Wave Functions
2*, ,
0 0
1 (2 1)!1 sin ( ) ( )
42 !L L L L L
Determine constant from normalization
Ld
L
a
d a
q q q q
− −
+ = → =
Spherical Harmonics
2 1 ( )!( , ) ( 1) (cos )
4 ( )!
L m im mm L
L L mY e P
L m
q q
+ −= −
+
( ) ( )( )21(cos ) sin cos 1
cos2 !
L m Lmm
L LP
Lq q q
q
+
= −
Legendre Polynomials PL
0
2 1( , )
4(cos )L
L
LY Pq
q
+=
( ) ( )1 2
2( 1) (2 1)! ( )!( , ) sin sin
4 (2 )!( )! cos2 !
L mL mm LL im
m L
L L mY e
L L mL
q q q q
++ − + − =
+
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Summary: Spherical Harmonics
21 2
1 2 1 21 20 0
sin ( , ) ( , )L L
L L m mm md d Y Y
q q q q =
Stationary wave functions for “good” L,: (e.g. wf of rigid rotor)
Orthonormality/Closure
0
( , ) ( , )L
LLm m
L m L
f c Yq q +
= =−
=
2
0 0
sin ( , ) ( , )L LLm m mc Y f d d Y f
q q q q = =
Arbitrary function
( , ) ( 1) ( , ˆ: ( , ) ( 1) ( ,) )L m Lm m
L L Lm mParitY YY y Yq q q q
−= − = −
1 1 2 2
4(cos ) ( , ) ( , )
2 1
LL L
L m mm L
P Y YL
q q q
=−
= +
q1
2
r
rAngular correlation
( )( )
!2 1( , ): ( 1)
4 !( )L m
mm imL
L mLY P
Le
m
−+ = −
+
2L+1 functions = dimension of H
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Separation of Variables: HF Interaction
( )
( )
*1 1 2 2 1 2
*1 1 2 2
:
4cos ( , ) ( , )
2 1
4 41 ( , ) ( , )
2
,
2 1 1
m mm
m
m mm
Addition Theorem of spherical harmonics
P Y Y
Y Y
r r
q q q
q
q
q −
= =+
= −+ +
=
( )
int 1, ,
,
0
0
1* 4( ,
4( , ) )
2
( )
ˆ ( )
112 1
i
p p p pp
p
i i i ii
p i pp
i p i pi
m
m mm
i
Electron nucleus sum over protons hyperfine interactions
e e r rH r P
r rr
scalar product o
e r
f
e rY
T
Y
T sepa
+
−− −
−
= =
= −
=
+ +
= ,i prate tensors T T
protons electronsonly only
q
1
2
r
r
Representation in Cartesian Coordinates
After Wolfram Mathworld
http://mathworld.wolfram.com/SphericalHarmonic.html
W. Udo Schröder, 2019
Sym
metr
ies
Tra
ns&
Rot
25
q
q
From Euler’s equation
H-3px R(r)·Y1(q,)
W. Udo Schröder, 2019
Sym
metr
ies
Tra
ns&
Rot
26
Example: Spherical Harmonics (Dipole)
( )
( )
11
10
11
1 3 1 3 1( , ) sin
2 2 2 2
1 3 1 3( , ) cos
2 2
1 3 1 3 1( , ) sin
2 2 2 2
i
i
rY r e x iy
rY r z
rY r e x iy
q q
q q
q q
−−
= − = − +
= + =
= + = + −
Spherical harmonics , irreducible tensor degree k=1 (Vector)
→ Structure of generic irreducible tensor of degree k=1 (Vector) in Cartesian
coordinates:
( )
( )
11
10
11
1
2
1
2
x y
z
x y
T T iT
T T
T T iT−
= − +
=
= + −
Irreducible representation T of 3D vector constructed from Cartesian coordinates Tx, Ty, Tz, like spherical harmonics.
T will transform like a spherical harmonic, here like Y1
W. Udo Schröder, 2019
Sym
m T
rans
& R
ot
III-1
neo
27
Legendre Polynomials
2 2( 1) ; 0,1,2,....L L L L= + =
Quantization of angular momentum
2 2(cos ) ( 1) (cos )ˆL LP L LL Pq q= +
Even L → even functions PL
Odd L → odd functions PL
21( ) ( 1)
2 !
LL
L L LP x x
L x
= −
( )
0
1
22
33
4 24
5 35
( ) 1
( )
1( ) 1
2
1( ) (5 3 )
2
1( ) (35 30 3)
8
1( ) (63 70 15 )
8
:
P x
P x x
P x x
P x x x
P x x x
P x x x x
=
=
= −
= −
= − +
= − +
ort
hogonal basis
set
= Polynomial of finite order L
x:=cosq
:
ˆ ( 1) ( 1)L LL L L
Note
P P = − → = −
W. Udo Schröder, 2019
Sym
m T
rans
& R
ot
III-1
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28
Orthogonality of Legendre Polynomials
Different L: out of phase. Overlap integral = 0
Larger L for PL → smaller wavelength l → larger momenta p → larger a.m. L.
( )( ) ( )
1
1 2 1 20 1 1
2 !(cos ) (cos )
2 1 !sin m m
L L L L
L md P P
L L m
q qq q+
=+ −
cos( )d q−cosd d dq = −
q integral in polar coordinate system
W. Udo Schröder, 2019
Sym
m T
rans
& R
ot
III-1
neo
29
Legendre Polynomials: Polar Plots (only PL(cosq)>0)
PL
q
Different L: out of phase except 00. Polar plot sign sensitive → |PL|
W. Udo Schröder, 2019
Sym
m T
rans
& R
ot
III-1
neo
30
Legendre Polynomials P1|m|
2 20 1
1 1( ) ( ) 2 ( )f P Pq q q= +
( , , ) 1, ( ),cos(0 )
sin( )cos( )
sin( )sin( )
cos( )
m
Lr r P
x r
y r
z r
q q
q
q
q
=
=
=
=
z
x
yq
r
Plot PLm=0
01P
11P
W. Udo Schröder, 2019
Sym
m T
rans
& R
ot
III-1
neo
31
Spherical Harmonics P1|m|cos (m)
( , , ) 1, ( ),cos( )
sin( )cos( )
sin( )sin( )
cos( )
m
Lr r P m
x r
y r
z r
q q
q
q
q
=
=
=
=
z
x
yq
r
Plot Re YLm
10y
11y
2 21 10 1( ) ( ) 2 ( )f y yq q q= +
W. Udo Schröder, 2019
Sym
m T
rans
& R
ot
III-1
neo
32
Legendre Polynomials P2|m|
12P
22P
02P
2 2 20 1 22 2 2( ) ( ) 2 ( ) 2 ( )f P P Pq q q q= + +
Completeness
W. Udo Schröder, 2019
Sym
m T
rans
& R
ot
III-1
neo
33
Spherical Harmonics Y2m=P2
|m|cos(m)
2 2 22 2 20 1 2( ) ( ) 2 ( ) 2 ( )f Y Y Yq q q q= + +
20y
21y
22y
Completeness
W. Udo Schröder, 2019
Sym
m T
rans
& R
ot
III-1
neo
34
Decomposition of Plane Waves
Plane wave can be decomposed into concentric spherical elementary waves (like Huygens)k
r r k=
z
q
cos
0
"
( , )
,
"
m
ikz ik
m
rm m
m
e
Linear momen
e c Y
tum p k
moving in z quantization direction
symmetric about z axis c
q q
=
→
→
=
=
cos0 0
0
( ) 2 sin ( ) 4 (2 1) ( )ikrc r d Y e i j kr
q q q q = = + Spherical Besselfunctions ( )j kr
cos0 0 0( , ) ( ) (4 2 ,( 1) )ikz ikre e c Y i j kr Yq qq = = = +
*4 (2 1) ( ) ( , ) ( , )m
ik rm r r m k k
m
e i j kr Y Y q q =−
=+
= +
:mFor arbitrary direction k use Addition Theorem for Y
W. Udo Schröder, 2019
Sym
m T
rans
& R
ot
III-1
neo
35
Angular Wave Packets
WPt q t,( )
0
100
L
cL P L cos q( ),( ) sin L( ) t( )( )=
=
cL 1−( )L
2 L 1+( )=
L( ) L L 1+( )=
cL 1−( )L
2 L 1+( )1−
=
Stationary WF (sharp L, m) : extended in q,.
Wave packets (LC over L,m): localized in q,.
FermiGasy
Hamiltonian Symmetry: Unitary Transformations
Translations and Rotations• Taylor expansion and unitary transformations• Translation invariance
1D periodic lattice• Rotational symmetries
Angular momentum operators, algebraKets for states w/good angular momentumLadder operatorsSpherical harmonicsRotational matrix elements
• Rotationally symmetric energy eigen functionsSquare well, Bessel functions
• Intrinsic spinPauli matrices, spinors
• Coupling of angular momenta, Clebsch-GordanWigner Eckart TheoremSpherical tensors
• Exchange symmetry
W. Udo Schröder, 2019
Sym
m T
rans
& R
ot
III-1
neo
37
Cartesian Representation of Rotations & Operators
: 3' decompose into counter clock wise rotat nsrr io→ −
" "'
" "' '
" "' '
" "' '
z y z
x x x x
y y y y
z z z z q
⎯⎯⎯→ ⎯⎯⎯→ ⎯⎯⎯→
"
"
)" (zR
x
y
x
y
z z
=
'
"
"
"'
"' ( )
"' "y
x
y
zz
x
yR q
=
'"
( ) '"
'
'
'' "z
x
y
x
R y
zz
=
Order of rotations matter ! Rots do not “commute”!
2
q1
−q1
2−2
1
3
( )( ) ( )( ) ( )( ) ( )( )( )
2 2 1 1 2 2 1 1 3 1 2
2 2 1 1 2 2 1 1
2 1 3
inf . 1 2 :
( ) ( ) ( ) ( ) ( )
ˆ ˆ ˆ ˆ1 1 1 1
ˆ
correspond to rotation about (3 )axis
Successive rotations about axes and
R R R R R
J J J J
J
third
q q q
q q
q
− −
+ + − −
→ −
−q12
2q1
( ) ˆJ Matrix J=
ˆ ˆ ˆ( ) exp exp expz y z
i i iR J J J q
= − − −
= Rots about original axes Invert sequence!
W. Udo Schröder, 2019
Sym
m T
rans
& R
ot
III-1
neo
38
Cartesian Representation of Rotations & Operators
" "'
" "' '
" "' '
" "' '
z y z
x x x x
y y y y
z z z z q
⎯⎯⎯→ ⎯⎯⎯→ ⎯⎯⎯→
': 3decompose into rotatr onsr i→
"
"
)" (zR
x
y
x
y
z z
=
'
"
"
"'
"' ( )
"' "y
x
y
zz
x
yR q
=
'"
( ) '"
'
'
'' "z
x
y
x
R y
zz
=
cos 0
( ) cos 0
0 0 1z
sin
R sin
−
=
cos 0
( ) 0 1 0
0 cosy
sin
R
sin
q q
q
q q
= −
1 0 0
( ) 0 cos
0 cosxR sin
sin
= −
3x3 Rotational Matrices=“Representation” of special orthogonal group SO(3)
0
0 0 0( )ˆ 0 0 1
0 1 0
xx
RJ
i i
=
= − = −
0 0 1
ˆ 0 0 0
1 0 0yJ
i
−
=
0 1 0
ˆ 1 0 0
0 0 0zJ
i
= −
Set of angular momentum operators and their relations =“Lie Algebra”
W. Udo Schröder, 2019
Sym
m T
rans
& R
ot
III-1
neo
39
Representation of Rotational Operators
ˆ ˆ ˆˆ ˆ ˆ ˆ( , , ): ( ) ( ) ( )
i i iJ J Jz y z
z y zD R R R e e e
− − −
= =
( ), , .Euler angles successive rotations
Effect on wave functions/kets → rotation matrix
( ) ( )ˆ, , : , ,JM M
J JD D
M M =
ˆz
J JJ M
M M= ( ) ( )( ), , :J i M M J
M M M MD e d − +
→ =reduced rotation matrix(y axis)
operates in the space of fixed J
Effect of rotation: (q,) → (q’,’) ( )ˆ( , ): , , ( , )J JM MY D Yq q =
Wigner’s D function
( ) *0( , , ) 4 2 1 ( , )J J
M MD L Y q q = +
Unitary transformation
( ) ( ) ( )
( ) ( )
1 †
*
, , , , , ,
, , , ,J JMK KM
D D D
D D
− = − − − =
= − − −
order of angles!
Explicit RotOp Representation
W. Udo Schröder, 2019
Sym
m T
rans
& R
ot
III-1
neo
40
cos 0
: ( ) cos 0
0 0 1z
sin
Finite rotation about z axis R sin
q q
q q q
−
− =
( )( )
( )( )
2 2 12 2 1
0 0
0 1 0 1? :
2 ! 2 1 !1 0ˆexp
1 0z
n nn n
n n
iJRotation Check
n nq
q q+ +
= =
− − = −
+− −
−
0
:
1 0 1 ˆ:1 1 0
1ˆ1 0
. . . z
z
Infinitely small
x x iJ
y y
Matrix rep of ang mom op Ji
q
qq q
q
− = − = −
=
−
− →
1 1
cos2 :
cos
x sin xeffective D
y sin y
q q
q q
− → =
( )( )
( )( )
2 2 1
0 0
( 1) ( 1)
2 ! 2 1 !
cos si
1 0 0 1
0 1 1 0
1 0 0 1
0 1 1
nsin
si0 n coc
sos
n nn n
n nn n
q q
q
q
+
= =
−
− −=
=
− = −
+
−
-
FermiGasy
Hamiltonian Symmetry: Unitary Transformations
Translations and Rotations• Taylor expansion and unitary transformations• Translation invariance
1D periodic lattice• Rotational symmetries
Angular momentum operators, algebraKets for states w/good angular momentumLadder operatorsSpherical harmonicsRotational matrix elements
• Rotationally symmetric energy eigen functionsSquare well, Bessel functions
• Intrinsic spinPauli matrices, spinors
• Coupling of angular momenta, Clebsch-GordanWigner Eckart TheoremSpherical tensors
• Exchange symmetry
W. Udo Schröder, 2019
Sym
mT
rans
& R
ot
III-1
neo
42
Energy/Angular Momentum Eigen Functions
ˆ2ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) , ( ) 0 , ,
iLz z
z z z z zR e H R H L H L
−
= → = = =
2 2ˆ: ( , ) ( 1) ( , );
ˆ ( , ) ( , )
m m
z m m
EF L Y YRequ
Y
ire
L mY
q q
q q
= +
=
→Rotational invariance: Orbital L is conserved
L, mL “are good quantum numbers” → quantize
: ( ) ( )central poParticl te rntiin al V Ve r=
Radial and angular d.o.f. are independent → compatible observables L2,Lz, pr
22 ( ) ( ) ( ) : ( ) ( ) ( , )
2
LL mV r r E r separated variables r j r Y
m q
− + = → =
( ) ( , ).
:. .
LL
mLL
LL LSpatial rej r
pr
m mof am statY
eq =
W. Udo Schröder, 2019
Sym
mT
rans
& R
ot
III-1
neo
43
Energy/Angular Momentum Eigen Functions
2 22 2 2
2
( )
2 2
1 ˆ2
r
centrifugaradial tangential
m m
l
Lmr
+
→ −
1ˆ
:
r
Gradient For
r r
r r
m
pi
a
r
u
r
l
= +
→ =
2 ˆ: ( ) ( ) ( , ) ( , ) ,L LL m m zSeparate variables r j r Y Y EF of L L q q = → =
( )22
2
(1( ) )
1)(
22r L Lj r E j
L L
rmV r
mr
++
− + =
( )2
2ˆ ( ) ( ) ( ) ( ) ( )2
rH r V r r E r E E rm
q
= − + = = +
( ) ( )2
2Effective ( ) potential :
( 1),
2eff
L LV r Lcentrifugal V r
mr
+= +
Radial equation:
W. Udo Schröder, 2019
Sym
m T
rans
& R
ot
III-1
neo
44
TISE Solutions for Central Potentials
( )
Not Hermitian
with :
ˆ
1 ˆ
ˆ. . ?
then
:
: , ( '
ˆˆ.2
ˆ
!
)
u u u
r u
ur u
r
U
r r rGradient formula r r
qu op p r p
rse symmetrized op p p r
r i as req d
p
p
=
= + =
+
=
=
22 2
2
1 1ˆ ˆˆ2
1ˆ
r r
r rp p p i r
r rp
rr rr
r
= + = −
→ =
2
2
2
2 2
2 2 2 2
2 2
2
2
2
2
2
( ) ( ) ( , ) ( ) ( , )
,
ˆ
( 1)
2 2
2 ( ) ( ) ( , ) 0 |: ( ) ( )
( ) 2 ( ) ( 1) 0 |
1
)(
() 2
L
L rL
V r r Y E r Ym mr
mr V r E r Y Y
r mr V r E L L mr
rr r
r rr
r
rrr rr
q q
q q
−
+
+ + =
− + − + =
− + − +
+ =
22 ( ) ( ) ( ) ( ) ( , )
2
LL mL
V r E r r j r Ym
q
− + = → =
W. Udo Schröder, 2019
Sym
m T
rans
& R
ot
III-1
neo
45
TISE Solutions for Central Potentials
2
2
2
ˆ( , ) ( , )
2
( 1)2
( . . )
L Lm L m
L
LY E Y
E E L L
Moment of inertia s p mr
q q =
= = +
=
22 2
,2
,
, ,
2
,
:
( 1)( ) ( ) 0
2 2
( )( ) : ( )
1 :
( )
n Lr
n Lrn L n L n Lr r r
a
Therefore
L LV r E r
m mr
u rr r
r
D RadiaUnivers l for central potenti
r
al
e u
r
sl
r
E
S r
S
t
− + ++ + − =
= → =
General Conclusion:
For all rot. symmetric H one already knows a) angular wave functions:
b) the energies associated with angular motion
Ex: Particle mass m distance r : Sphere of radius R, mass M:
( , )LmY q
2 ( 1) 2LE E L Lq = = +
2mr =
( ) 22 5 MR =
, for
specific L, radia
)
l
(
q#
nr
r
LRadial wf r
n
u Total energy Radial qu#
+− + + =
2
2
2 2
, , ,2
( 1)( ) ( ) ( )
2 2n L n L n Lr r r
L LV r u r E u r
m r mr
Mean-Field Symmetries
W. Udo Schröder, 2019
Sym
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rans
& R
ot
III-1
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46
( ) ( ) ( )
( )
2
2
1,2...
n n nV r r E rm
Mean field V r confines sy
a
discrete energystem
main qu ntumnumb
u
e
spectr m
rs n
−
D + =
→
=
3D → 1D Schrödinger problem
Oscillator
Woods Saxon
Square Well
( ) ( ):
, ,
→ →=If central potential
Spherical symmetry rotational invariance decoupled radial and angular motion
noangle dependent torqr uV es L conservedV r
( ) ( ) ( )( )
( )
( ) ( )
( ) ( )2
0
2
2 22
0
,
):
,
,,
(
&
ˆ 1
nm
L
nL
nL nL spin spin
m m
L
nL
L
Product wave functions r R r Y L
finite
Spherical harmonics eigen functions to angular momentum parity opera
u r
r
Integra
Y
bility R r r
t
r
Y
d
ors
L L
r u r
L
d
Y
q
q q
q
= =
=
= +
→
( )
( ) ( )
( ) ( ) ( ) ( ) ( )
,
ˆ , ,
ˆ , 1 , ( )ˆ :
L Lz m m
LLn n
Lm m s
L Y m Y
Y Y Treat spin lr at rr e
q
q
q q
=
→ = − = −
3D Square Well
Oscill.
Woods Saxon
Single-Particle Wave Functions
W. Udo Schröder, 2019
Sym
mT
rans
& R
ot
III-1
neo
47
( ) ( ) ( )( )
( )
( )
( ) ( )( )
( )2
2
2
2
20
,
:
2 1
,
.
2
LnL nL m
n
n
nL nnL
L
L
mr R r Yr
Simple radial waveequ for funct
L
ion u
r
Y
L
m
d
u L r
u r u rm
Ed r
r
Vr
q q= =
+ − − =
+
3D→1D Schrödinger problem
Square Well
Oscillator
Woods Saxon
( ) ( )
( )
20
2
0, : .
2 ;
( 0, 0) : exp 2
n n
n n n
Solutions for L SquW radial potential piece wise constant
Wave vectors k m E V valid for r R
For r R V E onential decay m E
= −
= −
= = −
Independent of m=mL → E ≠E(mL)
( ) ( )=V r V r
: 0n
Bound states
Single-Particle Wave Functions
W. Udo Schröder, 2019
Sym
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rans
& R
ot
III-1
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48
1D spherical Schrödinger problem
( )
( )
( )
2
: @ cot( )
: ( ) 0,...., ( 1)
: exp
0 )
,
(
= = →
= → = −
= −
−
= = =nn
n n n
nL L
L
n
nL
n
n
n nR
k
For well Require continuity boundary r R k R k
r R R r j k r L n
r R R r r
For infinite well R r R root at boun
finite
Bessel
dary r
functi n
R
o s
0 SW Wave Functions= 0 SW WaveFunctions=
R
EnV(r)
un0(r)Finite Spherical Well
Spherical Bessel Functions
W. Udo Schröder, 2019
Sym
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rans
& R
ot
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49
x-y line graph of jL(x) for
Isometric 3D graphs of jL(r(x,y, z=0)) for L=0,1,5.
j0 j1 j5
x x x
y y y
( )
( ) ( )
( )
( )
0
2
2
0
2
1
2
1 sin
sin sin cos; ; ....
:
e 1 2 e
− −
=
= −
= = −
n n
R
L
r R
n
L
L
L
R
d xj
Integrability
j r
j
x xx dx x
x x xx j x
x
r
x
dr
x
d
Single-Particle Energy Eigen Values
W. Udo Schröder, 2019
Sym
mT
rans
& R
ot
III-1
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50
0 R
EnL Conveniently (easiest) to evaluate from boundary conditions.
Example infinite (radial) square well: RnL (r=R)=0
( )
( )
( )2
22
0
1, 0 1 1
, 0
2 2 2 2 22
,
,
0 2
2
2
( ) sin 0
( ) sin 0
:
(
:
:2 2 2 2
0
1
2)
=
=
=
=
=
= → = → =
= → = → =
= = = =
=
==
L
n
L
n
Ln L nL
L n n
n nn nL
L
th
L L nn
R
l
R 0 k R k R
R R 0 k R k R n
Relative to bottom of V
p kE n
M M MR MR
w
More genera
h
l y
n ro oiEM
ott jf jR
1s
1p
2s
1d
1f
1g
nL
• For each L value there are discrete energies EnL
Radial wave functions have (n-1) zeros.• Each energy level EnL has degeneracy ( ) 2(2L +1), -L ≤ mL ≤+L
Factor 2 for 2 fermionic spin orientations• Higher L –values correspond to higher (lesser bound) levels.
Similar arguments for finite potentialsDifferent boundary conditions (RnL (r=R)≠0 ) → different values of levels EnL but
not their numbers (same number of degrees of freedom, dimensionality).
L
mY
FermiGasy
Hamiltonian Symmetry: Unitary Transformations
Translations and Rotations• Taylor expansion and unitary transformations• Translation invariance
1D periodic lattice• Rotational symmetries
Angular momentum operators, algebraKets for states w/good angular momentumLadder operatorsSpherical harmonicsRotational matrix elements
• Rotationally symmetric energy eigen functionsSquare well, Bessel functions
• Intrinsic spinPauli matrices, spinors
• Coupling of angular momenta, Clebsch-GordanWigner Eckart TheoremSpherical tensors
• Exchange symmetry
Spin-1/2-Eigen Spinors/Operators
W. Udo Schröder, 2019
Sym
mT
rans
& R
ot
III-1
neo
52
Half-integer “intrinsic” spin (S=1/2) angular momenta do not transform in rotations like wfs for classical objects: Rotation about 3600 does not return wave function to original!
( ) ( ) ( ) ( ) ( )ˆexp 0 2 exp 2 0 02
spin z spin spin spin spin
i iS q q
= − → = − = −
Intrinsic spin angular momenta correspond to a different Hilbert space! → Extra dimension→ extra spin wave function (spinor), multiplies spatial component wf.
0 1 0 1 0" ": ; ;
1 0 0 0 1
1 1 0 0 1 1 1 1; ; ;
0 0 1 1 1 1
s x y z
z z x y
iIn m representation
i
Eigen vectorsi i
− = = =
−
= + = − = =
( )
( ) ( )
2
2
0ˆ: ;0
cos( 2) sin( 2) cos( 2) sin( 2)ˆ ˆ;sin( 2) cos( 2) sin( 2) cos( 2)
i
z i
x y
eOperators of finite rotations U
e
iU U
i
q
q q q qq q
q q q q
−
+
=
− = =
− −
Pauli Matrices
W. Udo Schröder, 2019
Sym
mT
rans
& R
ot
III-1
neo
53
q
q
+
−
=
=
= = −
= + = =
→ =
=
,11 ,12
,21
2
,220
( ) 0, det( ) 1
; , 2 and 0 and , 2
. (2),di
:
1. :
2
!
m
i i
i j ijk k i k k i i k ik
i nk
n
k
k k
i
k
Tr
i
mem
Us
bers of
eful properties and relations
Matrix rep en
rep special unitary group SU
11
( ) ( )
( )( )
( )( )
q
q q
q q q q
+
+
= =
+
= =
= +
+
− −→
→
= = +
2 2 1
,11 ,12 ,11 ,122 2 1
0 0,21 ,22 ,21 ,22
2 2 1
0 0
1 1
2 ! 2 1 !
1 1cos( ) ; sin( ) ;
2 ! 2 1 !
n n
k k k ki n nk
n
n nk k k k
n n
n n
n n
en n
connect to trig fun
split into ev
ctionsn n
en and odd n
→ example( ) ( )cos sin
i kke i
q q q
= + 1
0 1 0 1 0: ; ;
1 0 0 0 1s x y z
im representation
i
− − = = =
−
Spin-1/2-Alignment
W. Udo Schröder, 2019
Sym
mT
rans
& R
ot
III-1
neo
54
Construct spinors aligned with any spatial axis by rotations:
( ) ( )
( ) ( )
1 0: ; :
0 1
:
; :
:
; :2
2
2
2
i
x
i
x z z
i
y z z
z z
i
y z z
z zAligned with z axis anti aligned with z axis
Aligned wit
e
ei
h x axis
anti aligned
Aligned with y axis
anti aligned
Undetermined
i
p
e
h s
e
a
+
+
−
−
− = − − =
−
− = +
= −
= +
= − −
−
, . : 1.i i
e factors e e Default set =
( ) ( )
( ) ( )
,
,
1 1:
2 2
1 1; :
2 2
z x x z x x
x z z x z z
Therefore wf aligned with one axis has components parallel to another
and can be expanded in those components
Example and
anti aligned
= + = −
= + − = −
: ,z zBasis
Arbitrary Alignment
W. Udo Schröder, 2019
Sym
mT
rans
& R
ot
III-1
neo
55
x
y
z
S n
→ = →
= +
+
+
=
=
= + →
= =
=
=
1 2 1 2ˆ ˆ; ??2
1, .
: cos sin
ˆ
ˆ ˆ ˆcos si
ˆ
ˆ ˆ2
nn x y
z z z z
z
n n
x y
n n
n z zn z
General direction n S S
Expect since qu axis should be arbitrary
Example S S
S S
Sn u u
S S
( ) ( )
( )
−
−
+
+
− + =
−
→ =
→
−
→
= =
=
( )
0 1 0 0ˆ ˆ cos sin2 21 0 0
0
0
ˆ ˆ2
i
n n i
iz z
n n
z
z
zz
z
z
z
i
z
and MatProject from left with
i eS S
i
e
rix rep
EV probleS m
e
Se
( ) ( ) = + = − 1 1
:2 2
i in z z n z zEV e and e
−
+
−= −
−
→ =
2
1
1i
i
e
e
=
z
z
Have expanded
Measurements will always result in ms=±1/2 !