hamiltonian symmetry: unitary transformations€¦ · translations and rotations • taylor...

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

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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










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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

qq

−+ = −

+

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

qq

+

−− −

−

= =

= −

=

+ +

= ,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


Sym

metr

ies

Tra

ns&

Rot

25

q

q

From Euler’s equation

H-3px R(r)·Y1(q,)

http://mathworld.wolfram.com/SphericalHarmonic.html


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


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 = − → = −


Sym

m T

rans

& R

ot

III-1

neo

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


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|


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


Sym

m T

rans

& R

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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= +


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


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


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


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










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!


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”


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


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

qq

qq

q

q

+

= =

−

− −=

=

− = −

+

−

-

FermiGasy










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 =


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:


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

− + = → =


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


Sym

mT

rans

& R

ot

III-1

neo

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


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


Sym

mT

rans

& R

ot

III-1

neo

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


Sym

m T

rans

& R

ot

III-1

neo

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


Sym

mT

rans

& R

ot

III-1

neo

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









Spin-1/2-Eigen Spinors/Operators


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

qq

q q q qq q

q q q q

−

+

=

− = =

− −

Pauli Matrices


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


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


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 !

hamiltonian symmetry: unitary transformations€¦ · translations and rotations • taylor...

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