I. Nuclear symmetries and quantum numbers

I.1 Fermi statistics

Antisymmetric wave function Fermi statistics

neutron 1proton 1downspin 1 upspin 1

numbers quantum orbital },,{ state quantumeach on nucleon oneOnly

33

3

zz

z

nni

Fermi level

i

N

Second quantization:

excitation hole-particle ||

state ground 0|.....|

],[

nucleon a anihilates

nucleon a creates

nucleon no - vacuum 0|

1

''''

hp

N

iiiiiiii

i

i

ccph

cc

cccccc

c

c

Fermi level

Multi configuration shell model

Complete basis

...|

|||

''''

''

hphphhpp

hhpp

phhpph

cccc

cc

Big matrix diagonalization

I.2 Interactions and symmetriesInteraction strong electromag. weak

Exchanged boson mesons photon W,Z

Translation yes yes yes

Lorentz yes yes yes

Space inversion yes yes no

Rotation yes yes yes

Isorotation yes no no

Time reversal yes yes yes

I.3 Translational invarianceSpatial:

kkkkkk sspparr ,,

conserved. momentumlinear Total

0],[,1

Nk

kpPHP

Time: tt Total energy E conserved.

I.4 Lorentz invariance

Low energy – Galilei invariance

kkkkkkkk ssumppturr ,,

energy otal t2

mass ofcenter 1

mass total

],[

2

intrinsic

,1

,1

MPEE

rmM

R

mMMPHR

Nkkk

Nkk

High energy – Lorentz invariance

MMcE

PcEE

i massrest

energy otal t2

ntrinsic

222intrinsic

Mass spectrograph

The rest mass and rest energy 2McE

Creation of rest energy (mass) from kinetic energy. A high energycosmic sulfur nucleus (red) hits an silver nucleus generating a sprayof nuclei (blue, green) and pions (yellow).

I.5 Space inversion invariance

kkkkkk sspprr ,,:P

kk ll

:P

state theofparity |||

0],[

PP H

Quantum number

1 D

N

zyx NnnnE

)1(

)2/3()2/3(

3 D

E1 M1

Parity of electromagnetic dipole decay

I.6 Rotational invariance

kkkkkk sspprre

)1(,)1(,)1(:)(

R

conserved. momentumangular Total

spin

momentumangular orbital

0],[

,1

,1

Nkk

kkkNkk

sS

prllL

SLJHJ

But not spin or orbital separately!

3D rotations form a non-Abelian group

cyclic ],[

cyclic ],[

cyclic ],[

zyx

zyx

zyx

jijj

siss

lill

Lie algebra of group

2SU

1|)1)((|

|)1(|

||0],[0],[0],[

2

22

IMMIMIIMJ

IMIIIMJ

IMIIMMIMJJJHJHJ

z

zz

Spherical harmonics eigenfunctions of orbital angular momentum

),()1(),(),(),(

cyclic )(

22 lmlmlmlmz

x

YllYlmYYl

yz

zyil

lml

lm YY )(P

Spinors

10

1)(down spin

01

1)( upspin

1001

00

0110

matrices Pauli

2 particles 1/2spin

z

z

zyx ii

s

I

IMIM

I

P

notation picspectrosco

quotednot usually substates magneticenergy same theall have , projection m. a.

A odd .... 3/2, 1/2,or A even ... 2, 1, 0, momentumangular

)1( parity ginterestinnot momentumlinear

:statesnuclear of numbers quantum good

Spectroscopic notation

l

l

)(by parity changes

away carries0 has

Way to measure spinsand parities of groundand excites states

Alpha decay caused by strong and electromagnetic interaction

Angular momentum couplingBit complicated because of Quantization and non-commuting components

||||

rulesSelection

21321

213

IIIIIMMM

sljjIIMI

z

2133

numbers quantum

Clebsch-Gordan-Coefficients

||||

||||

||||

21321

21333322112211

321

22113322112133

3

21

IIIII

IIMIMIMIMIMIMI

MMM

MIMIMIMIMIIIMI

I

MM

Spin orbit coupling

Spin orbit coupling

l

j

l

l

(-)

.... 5, 4, 3, 2, 1, 0, .... h, g, f, d, p, s,

notation

Particle statesHole states

Pbch208| Pbcp

208|

Two particle states

Occ Jpp16

' |}{

Selection rules for electromagnetictransitions

Multipolarity of the photon – its angular momentum

|||| ifi III

The transition with the lowest multipole dominates.

ons transitimagnetic )(

ons transitielectric )(1

Pure M1

Pure M1

Pure E2

Pure E1

No transition

For alpha decay hold the general rules of angular momentum conservation too.

I.7 Isorotational invarianceStrong interaction same for n-n, p-p, n-p –charge independent.

Conservation of isospin (also for particle processes caused by strong interaction).

cyclic ],[,,0],[ 321321 TiTTTTTHT

1- 1 meson

0 1 meson

1 1 meson

1/2- 1/2 neutron 1/2 1/2 proton 0 0 hyperon

particle

0

3

tt

10

1)(neutron

01

1)(proton

1001

00

0110

matrices Pauli

2 particles 1/2isospin

3

3

321

ii

t

2/)(3 NZT Same orbital wave state

Total state must be antisymmetric.

HeNHO 414216

2/452/45 3 TT

2/432/45 3 TT

2/432/43 3 TT

Isobar analogue states

209

I.8 Time reversal invariance

kkkkkk sspprrtt ,,:T

nconjugatiocomplex : KKi yT

number. quantum aimply not does 0],[y.antiunitar is

HT T

angle in center of mass system

diff

eren

tial c

ross

sect

ion

Reaction A+B C+D has same probability as C+D A+B“detailed balance”

Random interaction

}4

exp{2

)(

ondistributiWigner

2

22

2 DDP

}

2exp{2()(

on distributi ThomasPorter :yprobabilit emission neutron

0

02/1000

)(n

)(n)(

n)(

n)(

n ΓΓΓΓΓP