-

:

①correction Exam PA 2019 - 2020

Note:this correction à fally detonled . Unter amwas could be gum.

1. Hyperfine structure ef the atomic ground state

a)baxpnopertesefangul-armomentumo.ttlà, Û ,

E- dont commute-

for instance [ là , Lj J - il LÎ ( other relation au obtuimdby circulas permutation !

⇒ then it is not possible to defme a common basis forthe three component of the angulm momentum .

- IEÎ¥ (see lecture en TD ).⇒ We can defme a common basis for I

'and Îe

→ Ne dénote this common boirais le, m )

îlabels

[labels eyenualuesugmvalnes of I.

g E

Q2 I I'll,m) - hielexDK.msÎz Il,m) = m k le,my

Qs for an orbital momentum LEYQh millylues

④

as : Foa spin half - integer ) .0,6 For electronic spin e- Yz . Degenesag 2 ¢)

-

-

tellureQt : m = 5 ⇒

l-qt.43and4a-repnbleyfhmgromdotakQ.se: .

For the ground lpo and a- % ⇒ dimension 2p-

dimension dimension1 2

. Ground date basis 10,0) @ 1% , ± =

lqYy9_t4z7ieIimetmoQ9.lzflp.mqmsI__lxm@Hoime.ms) .

-

for l ⇒ me -0 → Îz 10,99ms ) = 0

Q.IO In the some wayt'± 194,0, Il = O

since Ito, = o

-

îîl me

We have l'a= ÛÊ and Îy =

⇒ t.lqd.com#goO

③

→ the action of l'a , Îg and Ia is null on hello)-

QI Woo"

= Aj I. J' = 0 : No adûn .

î -

l-0

AI Î : Itô → from the lecture fpjmm = test1 d

e !-o a-%dma = Hal .

since e- 0 ⇒ en lg one possible value j - so Ye$ and mj - t Yz

otdptignmdaEQI.frRubidium ahans ( "Rb) , we have

i.sk#---Y-gmnag4).QI . Dimension of What space : nucleus ② Reçu)î

degmnagh degeneray 2

⇒dmenaorighale.at#.Basvili,j,mi,smjJ=lm.,mj ) fa smphuty ..

.

1%4" l'fils : Heidi# 4) :$ 1%-47: l'h,-49 : l - ki -ki ; t -% -

'a

④

916

Ê¥Î- - - - - - - - Call Imi,mis au

~ Mimis degmnated ) .f-"

finedûictme "

e) PropicesFÎ

11 Same as Qkh with j --% and i - %

→ f = % ± IL =2 or 1

µFoi f- 2 mg =

-2,_

l,0,tl, +2 ( s'values )|f-tmf-tQHC3valn-QIsuppoxthatlmi.my) one eyenstates of À ' with

ergenvalues tij :

Ê 1mi, mj ) -

- tij 1mi , mj )

→ then Être 1mi,mj ? à Ê(kmjlmi.mg#hmjxijlmi,mjSaIbonlj,mj) .

- In the same wayÎe t' lmigmj ) = aij Îlmi , mjs-bmj.xijlmi.mg)

⑤r r

→ Them fa all states Lm, ,mj ) ÊÊ Im, mis _- E F

'

Imi,mjs

Since 1m, .mg ) constitue a complet basis → ÊÎA = JZÊZ

→MautfKpgùha! !

QI . Ï and Ê don't act of the same veetaialsubnpaa.

{ Îauts on nucléon variables .

Î ado on electronic variables.

⇒ Thus Î and Ê commute lire un delire at the some timethe electronic state and the nucléon state )

then Ê = +Î) ! Ï . +

ÈÏ+ËÊ-

→ ÎÊÊwith ËÎ

_

-Î

.

È .

æotr: États", +LËÊI +2Ff, à ]c- -LElu f0 since Î dont

commuté since commuté since commuté withÎ is an angulen I and I commute Îx and Tymomentum

Game as ⇒⇒ )Ghia pneaedmg question)

⑥

→ Fiat 2 [Eia .ÊÎ + Ê - Î,

Ê ]= 2¥. îa , + EÊIJ , ) L a

- -

= Etta , Et = ÊTÊ ,⇒c- LE

= - il Îy = ik Ja

→IIÊ, Et = dit fÊÊ - ÎÊ ) ¥0-

-

i fo and j' f0

→ [Ê, Îe) f0 since Îy ,Î , Îa , Îy have mon zero- action on the thlbnt

space ( i. % Foand j --kf0 ) .

QI Asa conséquence : 1mi, mj ) states are not engendra tes of Ê'

(see Ql9)

0¥21 In the same way as Q 20, we have :

[FÎÊI .- 2f Îm + ŒsÎ , )-

⇒ à IËÊI,- -

- ils Îy ihÊ

ÆÎ. )

⑦then Ë

,Et _- EFIÊJRÊ . Et

using Q .20

Remonte : this result could have been .

gwen dnectlg , rince , from thecourse

,we know that Ê = Ê + Î û a sum of angulo momentum

and is has an angular momentum.⇒ EÊFIËIIÊ ) = JIE, ⇒

QI All the openalors Î, J"

,

Ê' and Ê commute

* Possible to refn directey to the lecture : sum of angala momentumis If unclean for you , you can venfy all the commutantas

| [Î ', ÎY = o lance EÊÎJ --0,see Q -

191

I. ÎËT2F, ÊÊ]-

Gee Q. B)

'since IÎÎ)-0

' Some naming gwe [Ê,Ff 0

t , = ⇒ +ËÉ-

- To because Î and Ê commute→

(properties ofangular momentum).

- Same way L ⇒

⑧

- Lost [ Ê, Fa ) ⇒ Gee 9.227-

⇒ Then it is possible to defme a common basis composed efergens Tate of this 4 opera trois . We label it .

§ , j , f , mg ) . = 1f , mg ) fa scmphutyA

la!:b eafglteab.es Ê since i and j

j ,labels are fcxed on the

F2 problem .

QI .

there is 5 stats for f-2 and 3 stats for f- t

→ then the dimension à 8,that is exadly the dimension-

of Halla) ( see Ql 5)

dHypnfineægym

QI . The energy Eg à fally degenerated Exe QUI

→ then we have to dragon alze the restructurai of the observable

Ûpg = App ÏÊ to the atomic snbspaa Halla )→ The ergnnvalnes will correspond to the l' tender energy correction

Qd We have ÊIÎI J'+2 ÏÊ⑨

→ Ûp - Afd ( É- Î- F)⇒ the dates 1f

, mg ) are engendrer tes of FÎÎ'

and Î'

j :* 'ou

⇒ This they are engendrâtes of Ûhf with agenvalues

EË Afl k'[ ftp.D-ilii-D-jljtt) ]- -

%t % Utah)

→i¥Ë= "

fr2010171- 142)12,1 )

pü÷¥÷÷÷÷÷÷÷÷.L

c-

f- totale

⑧E

Q 28 my valuesmm-2

- t o tt t 2

Ep -

IË ¥ = f- a Grata)

EÉÎ !- - - f-- t ( 3 stats)-

t 0 tl .

-

my values

QI → The hyperfine perturbation split the initial dgenonyis two énergies , depending on the fnumben .

f-2 : degeneracy 5| fil : degonna ey 3 .

-

2- Effet faweakstaticmagnetefuld.am#gnomdotaEQ3oWz--- Bo (M'* +

M'çz +

MI,z )

= + Bolyflms) Û. +21rad Ê - gpµnÎ )p

cornes from the Lando fadàggz

①

→ ¥z)←WnËQI Yao = 94% = 9pmt-m-5.SI?'«1

167 10-27-

QI .Since the ondes of magnitude of electronic spin Êe and

Maclean spin Îz sis the same (v k ). and un « wo→ thoughbible

Q 33 .

From Q.tl we know that Ê-o for the groundstate thebest space .

⇒ Ee --0

⇒ Nghdmg the nucléon moment,we have :

~.

Ûz = 2 wo Jz-QI Ûz is diagonal on

the { lmiimj ) } dewupled basis

smulmi.mg ) à agenda K of Ie with

eyemvaluelxmj-nwzlm.mjt-2kwomjlmi.mg?-

⑧Son the découple d basis ( bewaè different from If, my ) basis)-

1"

,

- 1%471- : O lkiks

'

t-Kiki¥! . ±- ¥÷"÷.| O I - t 0 % ,

- tu- l l - ki - K )

\ O- i

- t t -% , -47--

mg=%mj-

ibahypafmmngyqeuumQ.ISSec 9.26 to 29 : the énergies are

¥4 pa f-2 - degenaated × 5

- - -- - - -

- -_-

|

Eg" tu fa - degennated XJ

④

Q.bg We have to consola the restriction of the interaction Ûh"

to each f level (f-2 af- 1)

-

Then the dragon alyati on of this restriction , that we callÛ!! gives the Mader energy uneehà ( the ugen values) .and the new engendra tes ( the associated ingen vedras) .

QI

:l% : on ly one date with my = 2 .

. l'4 , 4) = dit k¥712,17 + Htt %,4) Hits

inthe ong two stats with

with my = Yalla = 1.Looping at the décomposition of lf.mg ) , we have

44114,47 -- 4,41 ? D= %{d-Hyuk ) = L'441 ! D= - %⇒ l'hits = Élus - f141?-

Using the same procédurefa all the other 1mi, my) statswe have :

④

-1%47--12,27 (my _

- 34+4--2 )

1% , 4) = ¥1417 - 41717. (mg.nl/ztk--1)I-Yz,Y7--¥ 12,0) - ¥11,07 . lmgi-4ztk.co )§::÷:÷::÷÷:::Ë÷:÷:l'h , -47--4,407 +¥1,0) Cmg _- K - % - o )

1- % , -47 -

- tzl , - 1) + { 11, -D. (my .- - % - Yi - t )

1-% -f) = 12,-2) ( mfr - % - K -

-- 2)-

d. 38

ÎLE}VK.lz.ks-jkwok.lk?--kw.lIIseeQ37see Q 34

a Wal - % -4 > = - kwotz.EE -ka I

* WIKI ) : ¥ Waltz , 4) + KAI %, -47.-

htxwoIfk ) - Karol 3f , -%) .

- ④

→ Welsh-

- Kuo ( ¥11,47 - { 1%-4) ).

- -

H "

États -444 4141k¥11 " .

IÀkD=kwof%)* best Û Hot - YEÛI-4,4 ) rt ÛIY , -47.--

= Kwok, %) = - kwolk.tk ).

= -kwo ftp.l-k.lk) +¥14 , -497

4

Êtes - Yates ÉbattisÛho) = - hwa 12,07-

¥4 -- - -

- - - - ←wiki

| c- 12,22

% f-20 -1ûüf⇒±Ï÷÷÷ - ËÏÏËËI'

i i o

y [ ¥?, } fitcalculatedaotàn-% ! .

.- -t-efwz.vni

'

p p

a. 38: WÎKR wftqpwzpz.ywdzoz.TW?lf--tl.

④

QI The restrictions of Û to each f subleveb are then diagonal.

"" ii. ÷:Ë÷ai:*. ¥" . ËË

.

QII The correction to the energy at the 1st ondes in Û is thisthe engen values of Wolf) (fa each f value ) .

.

For f-2 ,the engen values one k¥0 mg ,

with ergenreetas 12, my?

- f-1 , - - kf-0 mg- 11mg>

→ The totalenergyarethus.ES=2 (ms ) = Efs + ¥ Ahghi t k¥0 mg .|

Egal 1mg ) = Ep - % Aff ki - kf-0 mg.î î

fine structure hyperfine interactionÎ Magneticinteraction

( full g degenerator) Wtf Ûz

⑤

QI The engendra tes are unchanyed →strlltheffmgotàhsnnàthey are agendaG- of the restriction ÛV!)

Impact the If, mg )

state are not ergenstatêf ÛGee Q -39) since the agenda k cf Û are the 1mi,mjsstates .

QI The energy dragram.is presence of magnetic fieldà bras :

A EO -

¢ t 2

:#§ -

- ¥.

Ego -:

- 5hAM}o

v- f-1 states

→ The energy are ohfted proportion nally gwen .

to the magnetfeld → Linear Zeeman effet .

⑧

Energy shift with respect to magnetic field:E dopes :A +2 IMBI .- ÊMB ah Kam

¢tl 1MHz-- O O

""¥¥: ÷::- Inktthf ) -

i dopes : InkV.# o o

H- 1MHz

- Bo

Beware . prete onlyvahd alloue magneto field : Ûz must-

be small compare d to the hyperfine interaction .

+ Fastrong magnetic feld → Ûedommahs and the ergastulesare the 1mi

, mj) states since Ûz = 2woÎ

but this is an other story . .

.