C4 Lecture 3 - Jim Libby1 Lecture 3 summary Frames of reference Invariance under transformations...

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C4 Lecture 3 - Jim Libby 1 Lecture 3 summary Frames of reference Invariance under transformations Rotation of a H wave function: d- functions • Example: e + e - μ + μ - Angular momentum as a rotation generator Euler angles Generic translations, conservation laws and Noether’s theorem
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Transcript of C4 Lecture 3 - Jim Libby1 Lecture 3 summary Frames of reference Invariance under transformations...

C4 Lecture 3 - Jim Libby 1

Lecture 3 summary• Frames of reference• Invariance under transformations

• Rotation of a H wave function: d-functions

• Example: e+ e-→μ+ μ-

• Angular momentum as a rotation generator• Euler angles

• Generic translations, conservation laws and Noether’s theorem

C4 Lecture 3 - Jim Libby 2

Frames of reference• Consider a frame of reference O in which a generic state is described

i.e. the H wave function ψ(r) of an e- in the 2p state.• If O’ is a different frame of reference connected to O by

where G is a group of transformations i.e. translations, rotations or Lorentz transformations of the coordinate system

• The wave function in O’ will in general be ψ´(r´) with

where ψγ is an orthonormal basis

GggD where)( rr

)()(

))(()()(

O and Oin same theiesprobabilit as )()(1

22

rr

rrr

rr

a

gD

C4 Lecture 3 - Jim Libby 3

Example: infinite 1D well

• For the observer O the ground state is given by

• Now translate x´=D(a)x=x+a. In O´ the ground state is not ψ(x´) but

• In analogous fashion in O´´ where the translation is D(a/2)

-a a

0 2a

-a/2 3a/2

ψ(x)

ψ(x´)

ψ(x´´)

x

x´´

axBx 2cos

ax

aax

axaD

ax

BB

BBxx

22)(

2)'(

2

sincos

coscos)()(1

a

xaxB

ax

aax

ax

BB

Bxx

222

422)2/(

2

sincos

sincos

cos)()(

The physics is invariant

Different eigenfunction

Summation over eigenfunctions

C4 Lecture 3 - Jim Libby 4

Some atomic physics

• We will consider the H atom in one of its excited states• The eigenfunctions are:

• We define a new reference frame O´(x´,y´,z´) which is rotated about the y axis of the original frame by an angle β

– r´=Ry(β)r

• The Hamiltonian and L2 are unchanged as are their respective eigenvalues n and l. In other words they commute with Ry.

• However, the z direction has changed so m is not the same, it is now projected on a new z´-axis

)( ofcomponent theof eigenvalue theis

)1( momentumangular of eigenvalue theis )1(

eigenvalueenergy theis

where)(

22

21

mLzm

llLll

En

u

n

En

nlm

r

2p wavefunction

C4 Lecture 3 - Jim Libby 5

Rotation H wavefunction• The new wave function u´nlm(r´) can be expressed by a superposition

of wavefunctions with the same n and l but with different m´

• Now we will use the 2p state (n=2,l=1,m=0) as an example:

)()()(

tocompared becan which )( )()(

rrr

rrr

a

uRduu mnlm

yl

mmnlmnlm

,

:satisfy tscoefficien The

sin),( and sin),( ,cos

are thesecase 2 For the harmonics. spherical theare where),()(

and on dependenceonly consider thereforecontributenot doescomponent radial The

1

1

1

1010

10

83

1183

1143

10

21

1

1

10210210

mm

m

m

ii

lmlmnlm

mm

ym

YdY

d

eYeYY

pYYrRu

uRduu

r

rrr

C4 Lecture 3 - Jim Libby 6

H continued

),(),(sinsincossin and cos

harmonics spherical theof in terms cossin and cosfor sexpression find then We

cossinsincoscoscos

:givesrotation under

, modulus, theof invariance given the which s,coordinate spherical tomovingby

cossinsincoscossincoscos

relation thegets one

sincos

implies Noting

sincos

sincos

:rotation theofn descriptio a is require n weinformatio of piece final The

111132

21

1034

YYeeY

rr

rxzr

xzz

R

xzz

yy

zxx

R

ii

y

y

rr

rr

C4 Lecture 3 - Jim Libby 7

H continued• We use the previous results to express Y10 in terms the β and

spherical harmonics in the rotated frame

• Comparing to the general expression

we get the d coefficients

• In a similar fashion all dlm′m coefficients can be calculated

– Somewhat labourious – neater method later

• Work out the probability that rotated state is in an eigenstate

),(),(sin cos

cossinsincoscos

cos

111121

10

43

43

10

YYY

Y

,, 1111011

11010

10010 YdYdYdY

sin and sin ,cos2

11102

1110

100 ddd

210'

2

111

0'

2

101 , , , , mmmmm dYYdYYP

C4 Lecture 3 - Jim Libby 8

Tabulations of d functions from the PDG

http://pdg.lbl.gov/

C4 Lecture 3 - Jim Libby 9

Example: e+e-→μ+μ-

e+ e-

Spins in the relativistic limitμ+

μ-

Only photon exchange in relativistic limit (Mμ <<CoM energy<<MZ0)

z

z′

θ

Left-handed electron annihilates with right-handed electron from helicity conservation.Therefore, final state particles have opposite helicity as well

Two amplitudes must be of equal intensity, ∫|A|2dcosθ, because of parity conservation

e+ e−

μ+

μ−

z

z′

θ

Initial state Jz=+1Final state Jz′=+1

Initial state Jz=+1Final state Jz′=−1

RH

RH

LH

LH

RH LH

LH

RH

221

22

211

1,1

211

1,1

cos1

cos

cos1

cos1

LRRLRLRL

LRRL

RLRL

eeee

ee

ee

AAd

d

dA

dA

C4 Lecture 3 - Jim Libby 10

Example: e+e-→μ+μ-

1+cos2θ

Weak, Parity violatingeffects distort the distribution

C4 Lecture 3 - Jim Libby 11

Angular momentum operators as generators of rotations

zx

xziJJ

xa

za

xa

za

aaaaa

aaaaaR

R

xzz

yy

zxx

R

xzz

yy

zxx

R

Ry

yyi

zxzx

xzyzx

xzyzxy

y

yy

y

because 1

expanding)(Taylor

0)(for ),,(

)sincos,,sincos(

point specific a take As

coordinate same at the and between iprelationsh a find want toWe

sincos

sincos

inverse and

sincos

sincos

'

:axis about therotation aconsider willAgain we

1

1

1

aa

aaaa

aa

a

aa

arrrr

aaa

rrrr

rr

Jy is the generator of rotations about the y axis.

Similar results for Jx and Jz

C4 Lecture 3 - Jim Libby 12

Angular momentum operators as generators of rotations

• If ψ is a solution of the Schroedinger equation is ?

)(

1

RU

yi J

0, ifsolution a is Therefore,

)()(

)()(

)()(

left with from Operate

)()(

isequation er Schroeding The

HUHHUU

tHUUtdt

di

tUHUUtUdt

di

tHUtdt

dUi

U

tHtdt

di

RRR

RR

RRRR

RR

R

Requires [Jy,H]=0

Unitary operator

C4 Lecture 3 - Jim Libby 13

Angular momentum operators as generators of rotations

• We will now consider time variation of matrix elements of Jy

• Then matrix element is invariant with time and the eigenvalues of Jy are constant. This implies:

– the y projection of angular momentum is conserved– wavefunction is invariant under rotations

tHJHJti

tt

Jt

tt

Jttt

JttJt

ttJt

t

yyy

yy

yy

0],[ and timeoft independen is If

and , used have weWhere

y

*

yJHJ

tttHtdt

ditHt

dt

di

C4 Lecture 3 - Jim Libby 14

Finite rotations

mjJi

mjd

mjdmjJi

Ji

U

Ji

Ji

Ji

U

yj

mm

m

jmmy

yR

y

n

y

n

y

n

nR

,exp,

,,exp

Recalling

exp

Therefore,

expexp1lim

rotations malinfinitisi of nsapplicatio successive

by generated becan angle ofrotation finiteA

0

J

iU R

ˆ.exp

General case where α is a vector in the direction of the rotation axis

with a magnitude equal to the angle of rotation

C4 Lecture 3 - Jim Libby 15

d matrices from rotation operators

1,11,11,1 and 0,11,1

1,10,10,120,12

1,11,11,11,11,1

again and

1,11,11,11,10,10,11,1

again with Operate

0,11,1

2)1()1( where0,11,1

fact that theuse and examplean as 1 Using

tscoefficien findcan we

,, and ,,for relation findcan weIf

1exp

)exp(

212

2

12

24

2221

213

211010

221

221

2

2

2

11112

2

212

33!3

22!2

1

ny

iny

yii

iiyy

yi

y

y

iy

jmiy

iy

jmm

mm

ny

mm

ny

yi

yyy

yj

mm

JJ

J

JJJJJJ

CCJJJJ

J

J

CmmjjCCJ

JJJj

d

mjcmjJmjcmjJ

JJJiJi

jmJimjd

C4 Lecture 3 - Jim Libby 16

d matrices from rotation operators

cos

1,1)exp(0,120,1)exp(0,1

sin

01,11,0 and 1,11,0 as 0,10,1

1,110,11,1)exp(0,1

cos111

1,11,-1 and 01,11,-1 as 1,11,1

1,111,11,1)exp(1,1

cos111

1,11,1 and 01,11,1 as 1,111,1

1,111,11,1)exp(1,1

.calculated becan tscoefficien d the1,11,11,1 and 0,11,1 relations With the

100

213

!31

21

2

2

123!322

33!3

22!2

1101

214

!412

!21

21

212124

!41

212

!21

21

33!3

22!2

1111

214

!412

!21

21

212124

!41

212

!21

21

33!3

22!2

1111

212

2

12

yyy

ny

iny

iii

yi

yyy

ny

ny

yi

yyy

ny

ny

yi

yyy

ny

iny

JJiiJid

JJi

JJJiJid

JJ

JJJiJid

JJ

JJJiJid

JJ

C4 Lecture 3 - Jim Libby 17

Euler angles

• Generic rotations described by Euler angles

• Define three successive rotations:– angle α about the Z axis

– angle β about the y0 axis

– angle γ about the new z axis

• Can be recast as first γ about original z, β about the original y and α about the original z

• The rotations of wavefunctions can be represented by D matrices

• Using angular momentum operators as generators of the rotations and

(z0)

x0

y0

jmm

zyzj

mm

dmmi

mjJi

Ji

Ji

mjRD

)(exp

,expexpexp,

C4 Lecture 3 - Jim Libby 18

Euler angles

1) γ about z2) β about y3) α about z

C4 Lecture 3 - Jim Libby 19

Euler angles

1) -γ about current z2) -β about current y3) -α about current z

C4 Lecture 3 - Jim Libby 20

Translations

• Similar analysis can be applied to translations:

• Assume a is infinitesimal:

)()()()( arrar STU

11)(

:asresult general thecan write We

:isoperator momentum that theRecalling

1)(

),,(),,(),,(),,(),,()(

paaa

p

aa

ar

iU

i

U

zyxz

azyxy

azyxx

azyxazayax

ST

ST

zyxzyx

C4 Lecture 3 - Jim Libby 21

Translations

• So for finite translations:

• Invariance of the wave equation under translations ↔conservation of momentum

• Similarly, for time if Hamiltonian, H, is time independent:

• Invariance of the wavefunction with respect to time translations ↔ conservation of energy

H

iUTT

exp)(

pAA

i

U ST exp)(

C4 Lecture 3 - Jim Libby 22

Symmetry Principles

• An invariance or symmetry principle exists for a physical system, S, and transformation, g G, if the physical laws, expressed for S by the observer O in his coordinate system, also hold good for the same system S in the coordinate system of the observer O’

• Noether’s theorem:– Symmetry principle ↔ Invariance of theory ↔ Conservation law

'

O')(

O)(

:same theare nsHamiltonia then thesame thearemotion of equations theIf

on.ansformatiunitary tr induced theis )( where

'

OO

gUgD

HH

gU

OO