Quantum mechanics 1

handwritten notes of quantum 1 book by Dr.shied Hamad Bukhari, Dr.R.M.Arif Kalil, Malik Shafqat Hayat, Hafiz Umer Farooq.

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QUANTUM
MECHANICS- I

2nd Edition
For Students of ‘ Salient Features
> MSc Physics > Essential questions & answers
| > MSc Applied Physics > Fully solved problems
> MScElectronics > Multiple choice (MCQ's)
> MSc Mathematics > Link to online videos Ei
> BS Physics : > References and bibliography
> BS Mathematics 2 > List of essential formulas
> BS Chemistry > Alphabetical words index
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Coal anne)" PP
À e 2 &
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“Corby Amar

Dr. Syed Hamad Bukhari Dr. R. M. Arif Khalil
Malik Shafgat Hayat Hafiz Umer Farooq

2. TOPIC: INTRODUCTION TO QuñWTum
_MECHANIC-

= DEFINATION®

is He stud:

PC debraatie's hisPaths
associated with mauelenth)
a Pacticle have dunl_nature

) ——ExPlaratiens

a
+ Pesticle have dial
__natafe tt behave

_Busticle a6 well os.

this ox, Shaw thaks- == ety

> Micsescopic —Posticles have lower warelenafn.
toctescopic Parties have smaller wunuelenath-—

3. Pasticle duality. =

a SIx THINGS (QUANTUM MECHANIC) =
1: Everything is made of wave! end
2- Quantum Physics. is discrete (enecss toval)-—

32 Quantum H3Sic5 iS Probablistic- (Ya 2) =
y= Quantum Physics is —non-local- _

5- Quantum Phäsics_ is mostlà. Concerned
_Smali_Pacticles.

—£- Quantum Phsics is nt magic.

with weed.

+ AAA
=> DIFFERENCE Blw_ CLASSICAL MECHANICS AUD —
_ QUANTUM _ MECHANICS 30 u
Qualls MECHANICS _|

CLASSICAL MECHANICS. |
+ Deals with macroscopic | 3. Deals with _miciase Pic
Puxticles. = | Particles. = ‘os

2- JE concesred with the |
Study of Particles whose ___
velocity Compare with —____|
|_vetocity of ignt —
velocity af light

22 ft conceso withthe _}2=
_ study ok Pasticles — —
| whose velocity ComPar

withuelocity of li pit is |

___Zeln-— — 4 —____,
@ioches
de À —— -
E
MAA —

Ii based on MAKWEL — 3. E based on MX Plans

eleckxomagnatic wave. Eheor3. theoch.
__(engcas_emit oc absafoe — | (ened emil ot absosne in
Senna Our LPacuets))

7 DANS) —
nn Ben _ >>

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4. Y Based on EsTa |y- Basel on Heisenbexa ———
— Ll luncettainitss Prince _
HS also call by Debxode MI thesis — —
— Name of mEwTonIon —| #Shoraleatex wave e4uistion

MECHANIC. | e Direct_evuation —————
5- EMED RESULT | Prvabviits of Ending __
—_— Au Forces | the article
—__+ Position B_Velocits

Classical domane is
A
= ano —
a

_ Ese =
hy hc/>

met bel I
——_ meso : —

Bann > a: Wp | ———

« Planks. constant 6:63. x10" 38”
_e Direct evuation —-motion_of elections =:
Le Clagical__mechanic fail to discuse .. —

_> Velocity of Night = =

> MictoscoPic_Paxticles.
» 1905_-» Lidht have _Parkicaı nature _

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5. 8-21-2019 QuaurumMechaures

ToPic:- "Block bodY rodiatian" - —
a Ware Parhcal_dunlit :- Clormak down of clasical_mechonic) += ——
Ecol? Copiado pas

wove is. —diserioe 64. one vector À —
ze Ko ths 20 = Pen
a bis Choi that” if ive increase the mar lensth

So the wae vector wi decr
THERMAL RADIATEON __. = en

hueso -

Te_Seliotim. enitecl Goan a material
thermal Cadistion, ——

temPesstute is cal

When _.se_heat_the Solid =

[radiations eisen te he E
form of electíomaspetic_ ena | = mae ven

(because is ae to the
_electin§ that Present on she
di

xtonce which is crent bg hent sat
the Salid- So,) electrant at the Surface
Start do tadiate energy due to heat

—— Intensit3_of radistion_depend_ufon

a Temperature — TAT (int ola bina

— ws inczesse the temfetsture ine _

of distin ols increase
e Frequency
a AAA FE —
intensita of Cdi

> Classical (alure _
Closgical —mechonic explained that ener emit_or
—olcemed_in—the —ContinowS... forme ane mate.
Ve continous
aut im cealite

crense

Lee tram.

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6. According "; Moe form.
—— Chunta y _
—Kandized-

— So; Calossical \thesty mechanic foil

| BACK BODY + —_
black ood is the opone abs

so a pad € mile
— 0. STEFEN_BOLTZMAN LAW: ____
—_——Intensit: — Power Pezunik_ Area is Knoon_ as Intensitd- ——

if we increase the pee 2=kime | [Lea = wy
—_ khen intensity will Change (incesses) |
Ibatime { wa T*

| 2-_K vector (wave vector) rente io
r

a w ET ae |
Wert“ _ |
is Seen Waltzman Constant: — [unit — |

7777727777 … © | ru … |
| __eMcasi= _ ERS |
|

frevuenc 3

N
f Q#2-_What_is_ ut Belskion? Estotolished the 7
_Plonk's__ formula to_er Plain to_enertt _disteikutin
in black od radiation Pecirum:

7. o - 4

»_ Wiens form an _aPPlicable ons

for_Shostes wavelength and.

— aid Io —erplsin the tesjon_of à
cuve _fes-\onget_uauelengin.
_» RAYLEIGN UAW

I lam Sr KT Leaman Consent
Vaan Sw
e This tonos opine on forte longer wavelength.
but foil to_eaPlain the Geier cane, Fur —
hoctes wavelentih= —
——2 PLANK’S RADIATION LAW:
Planes —Pubtisheol the theor:

axe of Clasical mechaic) __ —4

in tho Sian op. ce the Porm of Packets) 0
_ ——

PODA na |
Ai is applicable (os bath louer ar Inner |
souelength=— — _
> PAQTICAL REHAVOUR DESCRIBE BY + |
+ PHOTO-ELECTRIC ERECT. 1
© ComPTON_ERECT_ - a
PALR_PRODUCTION.

PATR_ANHALLATI0
| = PHOTOELECTRIC. EFFET

DEFIWATIOV- =
The_converSion of Ant _ener&% _ into 4he

eiectulcal eneses under Same sPecific Condition
lied Photo electric esjecé-

ig

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8. |» The_deuice which is. Ue do Convert the

hhto.clecbrical enesgy—hcaled Phat cell —

+ The_elechsons threat oso emited Gram __cathad _svst-nce —
age caused Phehs electron =

___ when light with (Sutiabie) freyaenez Fall =
_sunface’(cathodé) the current is estulisnad-

I is cated Onoto uen
_THRESHOLD_FREQUENCY _
IE is the minimum frequency of

Inceclent An
| umich con couse Phob-electric effect =

E= ho —

OR _
Enesgy_Per unit Plenks contant_is cated

__Hifeshold_ frequency ——
WORK= FUNCTION

| Minimum Sense es _to_enject tho __ —
_ electron from the Surface metal is
__calted Sort function

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9. KE MAXIMUN +

———The Pioduct of _ Sais GET, Du EN e é
——Lction=— 15 _Coned maximum k:E_ (Of Prato electrons), AE
_ __kE, se = _ u A
Kinotic_eneteyy of election. Er vn ———

1- if we increase the intensit' of waht ____.___

.

—— Hen it increase the amount of.
-Photon_in_ Fhe _beam of \iShE = +

meee we inctease Me —freouenco of —inceclent
___NiGht_if with_imesease the energy aj
A AA == o

—Eshv

of lidht-
2 GRAPHICAL RERESENT ATEO 2
—_ y FREQUENCY. VARTATIONs-

_ Ks tobi in flecks al _degond-
__uPon_ Frequency ef
it
a Number of Photons Ve
__ remain constant; it
___ Sxeuenta os \ighi inerense
election Alto inusesse ———

En hy —_—

10. ——» IMTENSITY vARTIATON:

This _gaph _shas that

—Stoping Potential doesnot

dePenc_uPon the —intensits.

3S_cotteh_St0Ring,

_t0_Z000.

Rokential..

Rckeritio’ ot Lahid endo Eee ae ——

11. IS-1-209 QUANTUM MECHANICS LS

— The _Scattecind of
2-5 —vohen they,

—— fe incedent _on tne. Socedent _ ——
——teyQet_moteriat. is. Pi Scatter Proton
Called_.ComPton_effect a

= Scottered_2-30%_(Photon)_haue larger —___
——Wave_length and less ‘enes4_anel Freyuench-
| —=*Compien_effect_conferm_the particle _natuse—
ee e
> Classically,_The_wavelerathof_incedent-and
Scattered _wave_Shule_be Same
=> Quantum MecHAnics; 5 di

—Eshy a he —accosdiny_to_mordien_

lin a a
|____wiavelength_of scattec photon is enter
[then the wauelensih_of-incedent Photon.
= Dna ER EDEN _

isis encinas PC
Compton Shift eruation-

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12. > hhen_0.
—— ax hf1-cos0"]

mec

When 9.90?

ef G5 act am
— Ace 43 MO m. qna
AXIS Inczenses with.

Pair PRODUCTION :-
__When__hiâh._ Frevuencs
|___clecttomasynetic
——Sadiation_ Pass oda.
| gil, the indiviclual_Photon_iS_abseroed
| bo the muciess of foih-ond_the Pair of Pastis.
(ie electson—and Positron) is Produced. _ Ñ
> Minimam_enexog_cesusied fos Paix production iS
0D Me — =
+ Paix Production_is_occas_in Vaccum.
Bs me?» oosime =
+ Photon_= Election +_Position _
Foca net) tect net)
| + m,.is_the zest mass of elechron- and Pasitren
___ hve _2mc ene ener _
|. ke" and fe are the kE op Pp
Positten=

__23= his Known as _Compton. wovelenäth-

13. — PAIR Annihiliation=
The_inverse of Pair Production —

——i8_called Pair _annihiliation- | - -
I—When_elechzon and _posikton. a a ~~
~Come__close__thes_omihiliate— 4 AA ————+

i #e*__, 27 — E

— + gama_ toys ase also the Packet of eneldies-—

WAVE ASPECTOS.
> _de-Bmiie_ h¥Arthesis- = — +]
0 _RAdiatin_haue dial noture- # be have |
like_waue__When_it_trauel,_wave note —
_are_exPlained in _clffraction and.

——intexfesence _Phenomena.. ==
29 _behaue_like_particle_vohen it Colide-
Rasticle_nature_Can_be_exphin ey omPion.
isPsoduction_and Photo electric

Aa ss
= >
_mc=2h£/n ee tessa _|

| his relation Show hat mattes _a/S0__haue |
notuse, 7 _beha dike wove __
____when_4ravel_and__as_oparsticle when Collide

This is known ale-hrogle hy Pothesis =
| obout_ the du 1 nou Of radiation.
and matter |

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14. —de-bsogie_hyfotheses ose also comfermed ——
———b9,—G_thomson. -expeximert_, Davison_——
——GEBME Expeximent= |]
according ie |, PEA — -
——————h wouelensth is_associete:
TE it mass,
> Micioscopic —Porticles __have_ Laser “wave erat

—= BOHR “MODEL: OLD QUT THEORY +
>Bohe model dives ine accurate
| —account_of the otomic_shouchute
— Os wellas_Convincing,-
——L£aplonation —for_¡Ys stabilit-
——= Rohe _ossumed that electron
———-0f_each_okom moves in_cisculac orbiWs, rl
——— Henucleus under he influence. of electrostatic
attraction_of the nucteus- _ u
———=-eleckrons_ate_Couoluine around tne _ a
n the orbit, Each _oxbit hove aie on
4 u ee fir, Ey, Epa

-anguls momentum of Me elechrn
SE np h/ar

15 4h e cadius of nth orbit.

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15. —>—AS_lone, _as_on_electron_cemein_in_orkit., it_—____—
——doesna&_rmadiate_electromaanetic _eneneyy=—______———
—When_etectan__aain_Some__enetryy it Sum

—— $om_\owes_enesxg State _to_\sther_energg_—__—_—
Site. pte ons, Hme en. elechren —__—_——
come ck lo is orinal place it emit
— Cheesy inform _of electro magnetic nes) —
o Energy of hi + En ————
> Energy of loo “on -

— AEs Ey=Ën = =

|__=» FRom_hkohr modie we Come to caos
Hot _Cadius and enecdy of electron is.
= -Vuonti2ed- — —

En== Mee”
A BRC EK
e Quontized vadius-
= HT ER

me

Ge yeah = osa” Om 01M Am

> WAVE EUNCTHOUs A]
_—Wwouefunction_olencted samba of" D Sal —

16. | > Ph¥sical_interPretation= __ A
Wave funcrion_ot__a Pswticuloc Hime Conterin.
|_ All information {hat _ontlood4 ok that Hime.

Cah_hove_ about Porticle._
Waue- Funcrion_oaly represents the - etude. a
Ehe Cidd and has_no. direct _Phissicel__|
meonin’.

hove_.a..phöfsical_meonind- D
= OBABILISTIC INTERPRETATION eee
| The probobilitW density and _\apitol*r_as_the
| Probalollitts of findio3_a Particle at time E
| in the volume element dr, ____
The_total_prolobitity of findin the “Particle
| Somewhere in_Sfoce must _lbe_e% ual to one
I > Probcbilits __olway. given in
I yl Pedoilit) desity —

Es Volume _

_ Psobabilit} E qe pu de

— fier a

[> ETA he. Maximum. Probabilikg.
| PROPERTIES OF WAVE FUNCION:

a mr finite. This__mean_Hhat_ con net |
_infinite. nun
ALCA leans _t that fae any,

Value: eee must | have om one — — |
er A 2 “|

Be ae E O

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17. The_derivotive ok 4 Should _also_be continues.
— This _imPlies hot _discontinuites — in the —
= —lerivative ob y. _is_not_allawed-

5 Par

Sei

Pr

et
Po —

= HEISENBERG:
—— UNCRTAINTY PRINCICOLE + E u
Uncestaints_PxincipleexPloin_dualit3 „of matter
cnd_—matter_and_codiation-. al
— POSITION AN MOMENTUM = =
The_Position_and —momentum_ of miccoscopic Particles.
Such_as_electron-cannot- be measure with lent
ot_the Same time or-exoct_fsition. =

Change _in_momen-tum:~ = >

An + E —

Ah
ax

A ar —
a apt — E

> ENERGY AUD TEME à —
_ The_amount_0f-enesgydlusing_de-excitation —
____and-time-éntemal deexeitalion Can-hot

Te mensuse with Stent accus at the Same 1

__time-——

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18. | —_multidly both_e-9 —D ond —D
LEA he XK

AE At > h. >
—these_tusro_ famous felations..shere Uncertaintg.
Celationfs re ne

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19. >-An_exPeximent_wilh_waue +-—An_exPeximent with.
hove S__can_be__Seon—in__£:3_(0)_,_____—
——t_consist_of a _wiave-source_,Slit's Sand 52
——e_Wlave fronts_ form from each Sit and
———inbesfaso with each other os the vesult —
dark _and_bridht feinges are A OS
———Mhis_wW4y maxima ancl minima app esting, —————|
on_Seceen
> _An_exkasiment with electrons.
——An_exPeximent —usith—elecifons —.Can. cab be —
—— amrded_ in order to_-obsasue_how_they —____—§_
—__behoue——So,— instead —of Solid _otÿects — deu]
——_ Ol wove, we noid have a funda menti —
__Pagticle oS a Soufce-
____e-Wavelength— associated with the elecron- —
—_x=h/mv ——— e
if this condition Satisfied , Hanns enna ved
____on interfexence_ Pattern on the Screen, Just
_we observed with wave. This how
4hot_electtan behave like wave whenito |
4xayel aS the result inter serence Paktren
_ ofe _ Shown onthe Een. - mr
__ When we place a detecton bus SS
electron come out either SiS, IE will
detect from wheaïthez it is going from
which Slt: AS the —vesult on the _ _|
screen inkiference _pattren disappears-

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20. 1 SRE SCHRÔDINGER WAVE _EQUATEON =
Ihe wave equation _uhich_desasibe the Fielel_ be hand ——
— _a_a,uontum Particle 4#hiS_equation_is called
_Schtadinepe wove equations _____—
—The_sole of Schrodinger wave equation in —
Quantum mechanics _is_analoyousto_that_Newtots —
Jaws __in__classical_mechanics.__tooth_descaibe_metion-—
| __ > NEwlowu's second _law_is o differential eayustion
—_which_cleseribes__how a Particles_Moues.
|? The_Schtodinger wave equation iso Partial

| difroxentical evusttion which describe how the —
wave function representing a Iuantum.
Particle _vises and False — = -
The most gemal_form. + ‘eho wsaue_t UE —

ae: ane Gaine —— +
Js_the feucied answer-

21. > PROPERTIES OF THE SCHRODINGER ——
-—WAvE_ EQUATION
|-—4=Solwin’s_the_Schrottingec wave equation — mean |}
— —how _Wrt)__depend__upon_2¢_and_t_——

—— rt Theze_iS_ mixed _olerivation wet to SPace-
ae time 2— —

a Pain Aifferentinn La Om —
HS a complex equatioh..—— = 2

a COMPLEX NUMBER : a
—e_Quantum_mechonics. is Sui Of complex

(c),_namber’s_invdlving, À
omplex conjugate of cis denoted 103 c

TN — en
che CeCe.

C,is_colled the seal past. |
— Called _the_imaginary Past |
2 EULER IDENTITY — _ Y
E Coso+ eine =
a. exponential function is a monoatomic_ Function
| thot changes steadlÿ_in che direction

|_= FUNCIEONS AS VECTOR ©
__In_mathemalical _idea__q,uantum_mechanics is. Bat

— functian__can_be_treated__ in diffatent_ form af _

_ Voctos- _

_SPike_diafami-_Plor the voue in huo ~dlimention
__#_ con _also- represent

22. > COMDITION OF MORMALIZATION «=

if _total_probobility of Finding the particle ot ——
Position x’ at quen time Eis units, called —

notmali zation- = _ _
u — pen er ‚Such.o_woue Sunctin "|

__is_called_nasmalized. =

> CONDITION. | OF CRTHOGNALTTY =
i£_total _Probabbililty of Sindind the particle ot — —
Position ‘x! at Quen time't! is Zero, called _ nes
____ Orinoajnalit =

ARE

+0 nn
Le ¡Uan! de = o _ =
> CONDITLON_OF ORTHONOR MALIZATION « —
"The condition_of notinelizationanc_concliion
of _orthognalits together is. called
_ condition.—of _ortnonormatization -—__— ee
ik iS given o's un

eu
= E LPC GE dE =

More ote "Rossiilitie En
i) For n= nig e
pont

=_2 condition d_Notmalizshon- ——

23. 3 LINEAR VECTOR SPACE =
-—Deggination
_——A vectoc_Spoce which ist ta e ena

—ond ollo two fides. q mathe medica is lied
——lineas_vectos_Space.—.
-i)_A_Seé_of wave function:
—— MPA 0,
il A Set of_ejden_molue-

denen urn —

The_fules of. athe matic Hex — st
—— Y) Addition_tule. it) Multiphcation rule. —
_i)_ Addition Rules: Ar
—1: if Pond Y ose the members of. ieac echt _
sface_then_W+h is also member of ineat
- vector — Space. -
—2- Commutative — Propertey Aye: li
—_——held_i-e;___ >

ces due to addition
pW +0) AR = pelo)
_—u- Zero oc_murtal vector exist ie-
W402 Ory _
2 irenicirics lunes Oe eats wollen de me ad
Ya) 20» CA) ‘
_ = W-is_the_additive_inverse—of
_ fi) MULTEPLICATIOV_RULES=-
___ The multi ponian as lo ar
ATA LES

24. ———SPace_then_theis__lineax__Combination__ayy + bp is ___—_——.
also_ member of linear_vector_SPace_ |
— 2 _Distsibutive _Proßectr Existe —
— AY +6) = ay +ap
—Co+bly #+bp

—5- Associative _Propests ws
hold __ice;_____
AM) (ad)
_Unitasy_scolat_also_exist.
ETEC ER ES ‘a
o A (+ _ = — —

Sense mern ae

_— LECTURE # 5:- THE HILBERT SPACE, |
__ a THE HILBERT. SPACE + =
Product _uectos_sPace—Hy—in addition to
anothes_Propest's.callecl __completeness..is „alte

Hilbest Stau = =
Hilhert__ Sace canbe finite _ dimensional de
__infinite_ dimensional ( both_ Case we call it Hillbeck _

_ Sfaces) — — _
__» The Hillbert Stoce_: PEOR too _set_of elements
o —> Al set of wove-function ie; VQ) ts ==.
> ASet of einen values. ie;_avbe,d, =
_— Pilllbark_sKace_is_olenoted ts "W_— — _
Hillbetk Race follow Ihe four psoperties-— _ _
> His alineat Space: — — EIER
_All_Mingat vector Sfaces_. ose the past of hillber

ose and follows the properties of

TL inear-vector Space.

25. 2H has_a defined_Scalor Product Mat is Strichg —____—
— Positive: - -
—a Properties:
1) The_Scalar_product_of Y with o_is_ a tobe clea —
Conjuspte_of Scalar_product_of with Y —
LOTS EN Co,
—2)The_scatar_Product_of $ with Pis linear wet
——_second fochod_ if Veafi+bY, _
— (01.0402) = afp) + (810) un _—
———ond_ anilineas srt fisst factor if
———L00 4h) 1) = A, ple (pv)
—8)The_scalac_product of vectut_y withitselg is
Positive Leal__numbe:
(9) = 19”
= His SeParable
Thece__exists__a__cauchy Seyuence Yn- E H(n=12,)-.
Such that _fos_each Y of H-and_E>0., these exirks
ok Least one Wy af Sequence _
i ET
HS Complete
re ee Wn EH converges to on
element. of H., hat is for_ang- Un,
the. Cela tion es —
Lim Ly-Wl=0__ en
fis ie reed A in ‘complete. _

=> Dirac Notation: ___
_ = Dirac._Nofatione_ ——.———_—_——— _
e psc sleep a System represented ia

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26. -L-vantum mechanics by elements of Hilbert space: |
— these element —ase_caled state_uectors= -
— Me con @present a s3stem_in_cylindrical .ond.
SPherical _co-osdinate «System.
——Dirac introduced a-valueabple notation_—for_Stete
———urentum__mechaics-
——>Dirac introduced _the—ket_, bras. u Li Kos
———Nototion-
=> kets:- Element _ of vectos_ ir _—
—Dirac_denote__the_Stste_vector Y by the symbol
— IN) Which is called a Met vector, or simply
— Het, __ket bebng to the Hilbest Luectox) Space-
E A —
=> Bras:= Element ek. chi. 30002.
_In_quntam__mechanic_euexy —State_uectos has
dual sface- + is demoted y Symbol
—<Sl called toas vector or Simph og.
—MAHE:
For_evosy— “Ket. les these 2 exist a unique bag 40
OMA vice versa.
> BRA-KETS NOTATION.
_Ofac denoted the scalar produét of ket a BER
kof of Siete veckor BY suymiool <> Called —
—_ bea = kel’s notation. —
he _lora=Ke¥'s_notebion of Strike Keane — ——

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27. = The proloabilitty of particle — in tora=lUek _natedtion___|
—i8_votiten as ea re

J (oe) Mond? = HER —
~->Com dition __of _natmoliZation.

—<OCab Wot) 21 _ -

Condition of. -otthoapelitf).i- in tora Mat Notation
— M0) | pUnt) = 0 _

—=> PROPERTIES OF BRA, MET AND. BRA- KETNOTRMON:-
Every _vectoy. space has. i's_dual... In_.netation.
adopted _in_quantum_mechanics, A_mamber |
-of A vector _sface is callad._ket (13). and_on the
——othex hand, the element of the dual Séaco ——
ae denoted toy a tom (el) in_dicac_notetion, —__
The __Notaten_adopted—is_callect_Hra - Ket_notalion =
m EVERY KET HAS_A_CORRESPONDING BRA:- |

To_evexy Ket 1Y>_, there Coœesponds a

a unigyue tora <pl_ and vise_ Yesa

|_æ RRA-KET TUER-PRODUCT RULES: a

_ JU quontum mechones, Since —the scalar product —
1s_o complex __numives the ordering, matters.

oo |

coms A PATES

Kg <vie
ORTHOGONAL_SIATES: un 8
| Two kets, (1w> and 10»), ase -Saidl_tobe_
__osthogenal they have a sah
gcaar product ie
__<ylé 20

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29. 26-11-2019 Quantum Mechanics. Lec #5

Tepic: "OPERA.
—>-PHYSICAL_MEANINO.0f DER PRODUCT
—1-To_evecy_Ket_theic. (exist)is_a unique —b6a_andl_

— far. every bias_Hheis is o unique Met

i A]
|——2- These_is a me-toone_ cres pondence bw. _
——8%S_and Kets. _ IIA

——— Hip) A boy) e le a
~———Whese,_a.ond_b_ isa complex number. — Ser
3: Te _follwsing_is_a_cammon.nebakion. |
—lo)= ay), <avl = dc] a
——t-A_Salar_Product of (Complex —conjugate is — =
I dien by, __» state veer
= — pps Coie _
5 =The_addition_propotly of Scaiar product is, —
SWlaWy+ bYs) ALU ly + bel)
Kah +a) | bib; 1). u
—= 152 MI EL SL I DER ERC HT + FD, Cot,
— 7 <A, +00) = OFC O40) + CP, dp) —_ a
8-_ SCHWARZ INEQUACZTY:- u
Fog am 4uo_Stote_lY> and ID of the Hillbed _
Space, we can Show that _
|< plo)" <¥1>< O19». ¿e
TRIAMGLE IMEUALITY- 7
_For_any toy and $ of Hilbert Soca. __.. _ u
ER eye ojyre_< Ncuie> egy
> OBSERVABLE = _A—
WHAT IS... You MEASURE TS WHAT You Know _

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30. |=» OPERATORS» =
— A_operatoc A when applied to te Kat Ip is —
Im Hong forms _into__anethes ket 1Y> of ine Same —
Pace and_wWhenit act on ara _<hl-it
I trans fosmed _in_to_cn_othec ta <q"
Aal ___
<A BEE: -
A_wove function Comain_ol_+the inpormation—aloowt
Particle. and_wecan_9et the information.
by Aling Some _Sitabe_ cparator-
= Ayzay
a —is_the_eigen. value which is.
basically
=> EXAMPLES OF OPERATORS=.
412 Diffexential.opetator=

PEER
da ot e

out énformation

2- LAPLOCE OPERATOR
pe D ae De
2x? y.

Jeust-vclh = féxê. dA
__3- UNITY OPÉRATOR:-
leave any Mer unchanged..
_ Ama _ _ _ _ u
1- MONENTUN_ OPERATOR:-

vihon_we_afpiicd a: mathematical expression
which when

2ePlied_on_a_wWave Cunction we
can find Inponmtion clcout_momentum_a _
__Posticle is Caled mopentum operator:

31. The_wove_function— for

| Mumbes of woaues_ Par unit

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32. | This. Con be_vutitn ob= — a

Re tha
dx
k 2
25
AA,
+R LR 7
fa aa]

A —

Bo tha ie _ q _ |
2

P_is the momentum ope ror ince __

2- ENERGY OPE RATOR- _ a
_An_opesalos_ which will _giue ¿nfoxmation about |
__eneray, when use apply iton soe Gunckion- |
y= AEE EE) se
___-diggeventiaking wird —
aug [aN Ese) u
dt oe

E ne RE
ehe > Pal Leen] ————
Ay = Wied leso\= WE

de Pb lesa ¡ES .

Drop Ypo boobh Silt _-z> _ — —

33. “Be “HOMILTOUTAN OPERATOR WHICH Is FREE
—— ENERGY__OPERBTOR:- (Brel cnexas_oPetohs)

——The_Sum_of "WE and_P-E_is_coted_tcko\_mechomrical
—— energy — 05_hamilitonian enezgu

ee -assecialed wi aia.

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34. la LECTURE . #
|» LINEAR OPERATORS 3 |
__ An _operutor À" is_sotcl_49_be_lineat if it obeys +
___the_chighCiloutive law, __
Operator A_is lineoc if, fran ae ES ll
12) omd_om complex number, we have. moe
Alay 1D) = Aly + bala». une
_ 3 HERMITIAN ADSOIWT OPERATOR:- mn
__Hexmition_operatos_1S_define_bJ relation.
_cylAtior 2 <piatl uy sun
_Hhis-operalot is Known as on jugate Square —
. _ond._is —denoled _bj_i At a
m PROPERTIES. — ai
RePlace constante _b9_ their Complex — erie PS)
CS
_RePlace Ket (bas)_bY_apsrespondiing bins (els):
land (en? 18
CA by sheis adjoint:

Y congugale

Tat at

o. laced Cae nn
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35. => HERMITION OPERATOR: a =

> SKEW- HERMITION OPERATOR
An_opesatos_&_seid._to_be shes. Hesmition_operta 4

msde Fellomı _ the. Condition. o _ ieee zZ de

- Ê

—— embase = = nd a

36. 0

5212-2019 Quantum MECHONICS Lec ##.

ToPIC:- PROTECTION OPERATOR ————
PROTECTION OPERATOR® _______
—An_operator_is said. ul à pujadas operator ———
if_its_Hesmition_edjaint ond. it’s squase is
—24ual_to tise.
pr = B_

A

—— Gesral. it_is_o ule, oP etalon;
ef I
e PROPERTIES, Fr RR OPERATOR
The_Predict of ba
À and À

commuting..prejection —eparatery =
piejectioncpesstor., since

tôt PA - AA E _
[—2) The sum of hao. Progechion_céerbts_is_ gemally nata.
| — pi) ection_perahee —— aa
B)_Thle_ Projection ppetaors.(Cttonctmal if thie product 15 —
Reto q
4) Poseo op projection operator's Bra À + ----
be waste ortlogerl_a. Piejection opetador, If

[_—— Projaction operates este matusliy_osthagnal-.

|_@ PARITY OPERATOR: _

¡ The_Spacefeflection_about Hhe_erigin_0j_the cooxdinate _
sústem_I5 Called an Inyersion os a Pariit_opzintr-

37. Bly
Drop nen. on_both_side;—

__TPES_of OPERA En
—There_ove_tio_types_of aa aies
)_Eveu PARTY ———— =
ODO_PARITY _ =.
à. EVEN PARITY= _ -
Parity is _Soid._tobe-euen.paritY # Em |
the Conditii

TA

2 YE) UE nn

m) ODD.MRITY _
Porit_15_said._to_be. 5 ila wit if it follows —
the condition- un it _
By = yet) = =e a
EVEN AUD_000 OPERATOR :-
— EVEN _OPERATOR = —________—___———
| An eperatos Is Saldto_be_eucn epexatos _if_iE follows.
the condition, — = —

__An_opesatos is. soil a be on add pene |
follow s. the Condit, $$
Pe A nn

38. -B.COMPLETE SET OF ETREN FuneT Lone
— DEFIMALIO=. ss ms
The_tineas_combination_of oxthogonal. igen function called.
~--~—-Compleke_Set__of eigenfunction. "|
—Y2_ 2 +422 + 055 =

yee By i

Gor

ab. DEGENERAT E_EIGEM_. Fuld pr
if there is _only_ one eigen fonction —cossesponding. to.
euetlyualue,_then_the eigen voue — is

|-—Len=aegenerate_ Cien fuorton ____ =

y LES O

| —DEGENRATE _ETGEN. FUCTIOW= 5 po rel

it luso digerent_oigen__functon_giue_same_eiyen vale

| under the Same operator that cigen value is

| Called —degensate _eigen_ function. =

fee Aber ER

| os2x Fr gen vaues

| d’/dz? _is_an_eigen_opetatos- _

wd fix, = dl? Sinan» -USin2x EN
de de

ee — eigen_ function |
With 2igen_value=¥
la COUDITIOU_OF_HERMICITY
| considex_on_operates Ht, we Said Hat opera as
___Hesmision operator —Y_ it_ folows He Candition: _
IA A dt |

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39. —____< o)Aiy> = <Agl>
_Te_cpettes which sy 8 _above condition fs called — =

hesmition__cperatas.__—

AAA ————
_ PIC :- "POSTULATES OF QUANTUM —
| MECHANICS" u
POSTULATES OF QuANTUM MECHANICS:
Tie_foumalism of Quantum mec Hoani:s 5. based _on—
a_number_of postulates, These. postulates_cornat—_———]
be devine, there sesult get fsom- experiment
Postulote #1: STATE_OF SYSTEM: = a
JIn_Hihert_space,theStateof each phil sical_quantit) —
_is_xepresonted.b_a—state_uectos Una and time E
Postulate #2: OBSERVABLE. ANO-OPERPR-
—Everd_ physically. measusable_ysontit Ais. colled_obsevable —
__ variable fas_evert_obsensaldle. these is_cbasses paneling. —

_hincas_pesmitien_opelatox-— eee =
Postulatese3:= MEASUREMEMT AUD EIGEN = VALUES OE OPERATE
" Measulement_of -obsesvable—A_may—be sapresenteel ____
fosmoll_b} theo tion of. Aono state vectos_|¥.tu>. —|
Alyce -=On}Y0)-———— — = |

amis the eigenvalue mhishie out hfasmatin al

hh tale vache Ink ne get is tine —
— independent —— a
on. is-2igen valueof-opesatat A_—

Postulade.- #4: Time EVOLUTION OF. SYSTEM = ==
= The time evalation of 4: tem_is_gaverned by

7 schrädinger war equation Eu

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40. = Kulm) = EJ — =
— When —hexmition__.opexator _ofply on (neue function) Stet
Actos, then the state vechs_tepente and we
—Set_eigen value m enesgy=

AQ] yy — __ ___
AO |e. —

sae ron PROBARIISTIC OUTCOME Œ

MEBSUREMEM =
When _mensusing_an_obsexuable _A_0.n System in state
— 1) -the_probabilit¥. -of Obtaining. nen-degexnated_
Eigen sales an,

— Tie above Yelotion_ 35 vsti fe discute facts, —
which_con_be extended to measurement À
_Yeilds o value ba one arda ona _suskm |
which is initials in state [yy
dPlar = we
da zum _
_ a UNCERTATUTY PRINCIPLE + u
__Uncex tainty _.Peinciple__ exPlein_the_ oduolity _nalare of.
___ matter and -raddiation-

Miss an 2 ue Pchameteiied He dunes —— —_
__ Wu —mmicioscopic gsm ot -macmepie— System ___

41. >
m POSITION AD MOMEUTUM:-— ——

—The__position__ and momentum _ a} mic Sapic— pos tides ——

— Such_as__elechent__Connot be _mersuse nit ———

— Meat accuracy at the Some A
Position,
CHANGE Ivy MOMENTUM:
——4p_s_h —

— — en = ==>

Parr 5 en e]
A E a =

Ay system those alterne is q the cides E —
ka quantum Sistem; if the indetesminacy |

Product Is much longer, the System is dessicon=_

ENERGY_ANDTIME = el

TRe_omount genes using _—dé-exeitation and time —

__ interual__0-de-exciation— cnnet-be_ mangue |
uith_gueat_occulacy al the Same time ———————

whe = he —@.

> ax

Cat ax 0 =
by 04 =0 ond = D = >

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42. —How MEASUREMENT DISTURO THE SYSTEM:
—-2=IU_Classical..physics it is_possible_to._peiform:——————
—_measuse mentS_on_a_systom_wsithoutdistucbing it —
-- 2-10. Quantum. (physics) mechanics, He _measutemeat —— - —
——Piotess_pettush's the system Significanty-————=="
—B=_While_Carsing_cut_mansurement$_on_classica! ———
Systems, this_pextusbationdoes exist, but. itis ———
‘Smell_enough_thet-_con_be_neglegeble- ———__——-—--—
IN_atomic_o0% _Subofomic. System. (Quantum.
Mechanics). the_ack_of- measusement_gexnsiiy — -
——Changes_the___stote_of the Systeme —

—CouSIDER AN! EXPERIMEN. that_measure the position of
— an_eleckson in an atom fox this wo need _to_bombard
the elechen xi electfomagnatic cadiation (protons)

lf we _detexmine _ the Position Aturatels , the wovelenSth
_———_0f Xadiation_must_toe_Shosier Meno! m. mn

Photors have WiShex enesd3.- e

Hhis.not om pextuse_the position op electa de =>
lll Knock tout. completly from ¡ÉS —omoik, anal _
alam become. —ioniZed= _

| a

cad aj AE Changes Eee

stem

_m GERNALIZE PROOF OE UNCERTAIMTY PRILCIPLEr

a. The tal numbexs of Pesticies axe.
2

NE ¿My —

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43. —2=The__ptobabilit's of _mumbess_of —Pasticles ove

EM. condition. y nesmelization Hetol_psobabilita
to_units > € “ PE

—_4-_Me_auctage-ualue -o$ Dis Bon as,

<i> = Zinc - 2.300, —
N do

L ASSINGMENT
«PHASE VELOCITY AND_GROUP VELOCITY
-b GROUP VELOCITY = _ ———

» The speec of the overall Sapa ga iad inked A
= {envelope i5_colled_gyoup velocity

|p PHASE VELOCITY 3 _
me speed at which a a GE ING of en > > _
through Space is called phase velocity
__ BY GRAPHACALYs |

D
_>EXPLANATTON
h Equation of cer: ia;
Y, 2 Sin 2x| a ve) ss

ongulaë-waue-rumbex; Ke AT/x _
A wa

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44. Vie Sin (Kost) —
——oue equation of Second wave ——
— o Sinf Kadi) = (os edialt)
there is__Slightty_ -dhjfeïence _ in. wove umbet_andonguled — —
———#requency from wa yy" >
——-RESULTAUT_WAUE —
ee 000
in ua stat) + S [Ud 2 (sort) ——
As se Know Hal; Q_AA > —Á

Had = on

= — dote de galena fe

_e_Cosine term decide how the group is. wii
_Sin_term_indecate the wave mare; it indecate____
the phase velocity

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46. ©

GrENERALIZE PROOF OF eee —
PRIACIALE __
ta numbers of —peaticles axe — ———

een NES - _— —

_The_pscbabilits af numberof |
Ad. m
y

e conditic of. meal zation I probo TS
A
= eo.
|The ouende uote of 3 in Sven as, neers
Ao S722. ue = Zn citi

u DEVIATION _
Tre digeserco op ench_unkie_of -5-ftom_awetage
vale ef ¿dis called deviation,
es nn o
Tako_overage of deutation Value.
<a> < A= <B> = x m
OR ty 2 & 5 ec
naiss A PU =
_= IRFERY LIE

S0;_L a Nas constant.

Change —— - = |

47. | VARIANCE +
— Auazage of _ ee ere - ee
_Lasiances =
ca a ee

— 0 = Etap = ETRE PG) ae A
£ {5% $52? ES DE
Ex Pe) E SPU) + INER —
= <> = 2E52¢5> TA SE
<> = „ae e 25>
ESAS =
| is is the fastest to compute he varie
_Simpl¢_calculate_<.3* >_and_<32*_anal_Substiect-—_——
a STAUDARD. DEVIATION: — —
___ The__$q.uase_xoot—of variance. is cole pe

___ deuiation._— PP. —_

ee is ESS = = a ==

alias
| THE UNCERTAINTY PRINCIPL + ——— _

— In1922_Getman_Phtseist Heisenberg_Inttaduces tra — —

A ___________—__— _
Be gemeizad fam of uncestointS —psinciple of ha a
obsesuabie And Bis — A

Er mo |

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48. She << (Acá
> = 1 (4-<A)> 2
£le <y(A-<Ay)> nn ee

—m_FoR observable A the uasionce is --
592 <(8-< ANY) = La] —_—___——

—<o}-s_<(6=<8>)y)
J9>-2\(a=<8)y> _

Now-use _the_Schworz_in eat
— LE rd) lot A o
ch SCS A

alar 5

Foc_the_complex. number _z.
121? = CAES EN (Ina

ERO ah Tall pg A

“a a-éb - _

À = dtib=préb EE
yo o A E E -

Magnitude_af Complex number

le, te SS
ize ig 0

za b* ie aso

A ie ve i

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49. a
—2* = <a» _ BEER

en

— E22 = <A PPAYS
= <y\ Cz <= ON
So ÄB-Ä 68> Bears Ama) >
= kW Á By) =<8> y] Au>-<ardy By + LAK BI
—<H93-= SÁB) <B> >= 095% + SC =
SEI) =_<ÄB>- <AX8> se
| STIMTLARLYs
a <(8=<.8>)y| (A= art

— Sal €: <8. <AILB> _

| Put the values of < £19> one <1 hey 8,
50,2 ate -[eáb>- <a) (<BÄ>- a

a & pes cases = Ahr ré}
= fi b> cat]

LA BL= <A8>-<3A>

= ——
aa] 2 —_

50. 42019 Quantum MEHANTC IE #0_

—ToPlic- _Sciwadindec_EQuATION—one ——————
—_ Stationacy Stokes ——_—_—. =
La SCHRODINGER EQUATIOU.- IE
—Schcodingec_eayiuation—i& very — important equation
in quantum mechanics because very difficult —
HaSkKS con eig Saleh 183 this equration. Be
——de..Breglie's__pacricies is Siuen_ a, —
O TO 7 cel
Wwhece_ox)_iS_a_porition ren Pera
%_khe_time dependent -
ds _Homiltonian_operator_on—wave function
Wat) then, a
Az t) = EY.

¿Rp —
ot

po = Wofín_= Ey.
ES

: 7 ee
Hs DU Zum. V peu. Et.

wies. Loy De ergo - u
“then we get —

> Ten + Vena CK Bft)
ENTRE fa Dt

51. giScanned with CamScanner]

52. 2
ES ESTAN = Beh nn a
_2m so: a

7 the. era “pi

= DV — Epo)=0.
am ae

Do) + E2M Vos 20
2x? K -
PU + Jo ——_—_—
a = 5 =
Oy +key =o —
(2D Y =o — -
m4 = 0 -
me Kk CIS
Ve nn —
met K = 7 - ~

53. ©
Quantum MECHANIC HEC RS

TOPIC RAISING AND LOWERIUG LANDER
— OPERATOR.
m-Calculate

how» —much.encr99 _ increases or decreases.
—due io the _oPPlication af towering and ising
lander. -oPerator-

— Cosider a —porticiein_Scound Siete having ener,
E_,this__Patticle_are_xeloted with wave function “y
—bY_Schtdingec weve eyuntion..
fips ey
“His_c_caising operator”
wihen-raising_lander _ operator
Wit step upit's state the
represent as) qu”

we—hove_to find the ences of Particle

applied on-Me_wswefuction.
mas Site can be

— FIND THE ENERGY oF 4 PARTICLE I TMS MEW.
SITES, =

Alga)

À is a ing. lander_oferator

se]: aa au +Lhyo,y
= (ga + ho _ E

_ 2(99 +45) y BE
u

54. «0a Knows Mati,

Lees te]eu)= edo] siya
_ fiveew 7

[32 &ho]@e)zgeu hos
= fesse
=festubgp
His Shostak when mue pied the anising
Lander__ of rator Particle with ste? uP Potential
seve function. becomes_it's_enex8s incre

—LOWERTUG ANDER
OPERATOR =

mils
[ga + he) (aw) = [E tw) ay

Ibis es show that ishen_we oPtlied the tours
Jandec_cPetatoc Particle with Step down Catential

sanction ecomes. its anes deccanses WA

55. = QUANTUM =
MECHANIC
TOPIC: AUGULAR

——— MUMELTUM.
> LINEAR momEnTum:

Te 2uontits of melon aboli ia caled moment
53 neston_tmos ie Demi. it is a uector—
— Mantils _and is unit is King” -
=>-AUGULAR MOMENTUM a e
—The—duontits anol awualtt\_of the motion of the bod —
iS Called _momentuan
QuotitY__depandl_ fon mass
Quant detencl_ufon—vetocit's
* The vector product of
——fediuis_vector E is cal
—2rfloc_enomentum
#5_linese_momentum. $ is _atonse.
fhe_tongental__ a9 Lo

liocac —romentumn

—pÎi

odias.
vector À so 8.2 %!*

is ¢x6_« cpsime - Cosinac

Co = cmu_ -

jr Le mur it_is_o vector _ ant
And is Si untis komis”
Nel bebe medal Mur: m/e eshece Miso DlankS constant.
m ANGULAR MOMENTUM IN. SRRIESTEN,
- COORDINATE SYSTEM +
AS_election_revelves.ofeuncl_Uhe_wucleus—
in—cisculos—écbits,

we-haye to develop _opetatacs
for calculatind the -obital
an Sata mementum_ of eleckrons= __

56. |___ComParing_|

|< (Ye, =2P,) = (00-28) ————
BEITRETEN) 2
LE -S Px) — À
k= @ =
the ae of Oand 0
A Na of —petator—

oo

La 8x8 = 4%) ¿ft _
> ANGULAR MOMELTUM In) SPHERICAL _

COOROINATE - SYSTEM = 1

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57. —elechton._con__algo,-be_describeol__

<ootdinates- >

FE rr $
CA

lb A)

2
sr 2

e y Copcdirate-
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58. COMPONENT OF AGUILAR
—MUMEMTUM e E
—The _unit Uectors (21.4 Xecms op Sphecicol

Coocainate system IS Siven 54 —

¿e sino sinpêe ccoccsg 8 = sino

i= Sine sino + cose sing + (hd

Reset ned...

Compa. cli ug)

29 Sino

Sing

= ie 2-+ ei

Her + COSO cosp. 3) ==

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59. I7=1-2020_____ QuontumMECHAMICS _
TOPIC: AUGULAR MumenTur
-6:3-—COMMUTATION RELATIONS.

Since _À.3 nd mutuo commute ancl Sa do —
Lx By and Ez

16 two operators computes
— they om hais Some eiñen A
2 Commutation_relation
—momentum $

between Position © ond \ineor.

id

A A hi

Hhis_sesulk _Shons._thot uve _ore_nct_alole_to_ determing

— Position and_lineat__ momentum _Simutanenual.
dtop_ bothside fem will Jet=
ERA

Sum -

ee, ALA à Cube
a_Comnutation_reletion blew Position? and anular

(Bi - Pa)
= 81e 8 612.9) - 212, Gb
Ps 2L — 29 —0 ASIA (A B)-Caser_
alió Abe Pa _
ap = Row = CET

60. A AA AA AAA u

ae vas sep ue ea

—— and „linear. _ momentum__Simultane ous - zus |
—2ut_all_uolues_én_eg,
- Lila) =0x0

MERMA ns
EICHE LEE Bt BEER.
_ = o+0-Glech)=

(ca. —

3- EN 12,20% CES -

= (%, 201-122 Bal _ e
2 SLA Ba a Bul 23] - 212,801 ~ ICH IE
___ 2 2{ék)+0-0-0 — a
— Calle ESPE 2 = ee

TABLE — _ =>

~ Gylaleo [la dt A
Trade tides — izle CS
tad = hi ll _ alla net

Hence, Commutation _cetation. blo Position TÉ and anfulor_——

_momentum_T_represents thet Anese_two operators — —

_commutes when both operators have the Same

_ components ———_—_—-—
teo - do not compute _ratnen ath. Pernt haut

he different Components

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61. @ Commutation elation. jus. linear and. onaulor
2 (Brsinls (6, 8-28) 1
[BGB] (61,581 NE

SR NEN ENTE TEEN WENE —

9+0-0-0 20

— nana Pa (ét)
CAN ES x8,

= Ba Le (Ba, 28, 58)
CATA KENT E
= ALB ER I A
+ byàr) =0.-0.

| ie 12 Bi) ¿eh
Puri) > is by

I These tecate Shows Mob =

|} —1=-these_tuso_opesators_commutes._ this shows that i

|= comfbnent_of both _momentum_P_and_anSular
——mumentuin..L__can__be_mensute_Simuttaneous-

=— these to opecslods non commutes, Haig shows
Aha. x Component. of momentum and. 4 -conponent —
| of anaubr mamentum_L cannot be measure
Simultaneously -

3- these two oPeralors non commutes, this shows, Mat

—%_comPonent,_of _momentuny_onnc_-comPonert _

~ — ef _andulorc momentum L_conck_be —mesture _
____ Simultaneous =

62. eR Pa Cote] o
——this__ Shows hat ff two (both) oPeretors ace

——-then_.commute _other_ wise not commtite ee

B_Connutotion. ‘elation au “comPonen’s Ps EU :
momentum- E. Lu: iba 28,
vs BEREITET
—[A.8,c.0) = fls.do+ clAı0lae ___ un

Lusty) [987 204, 2002260)
— (58,204) - (48,26) - (28,,28]:(28y 28).
2 3B. 2] A460, Al? (BAT - Pola (bd
— fy(2bd2- +2 (By) 8+ REALS a
Sito = -0-0-0-=0 +0+Py HZ a A —
— 9G.» By)
ir =
NARA —

| {iy tele kl A o

_ lr tale fly A EB
this _shass thet component of —arfulec__momentum 1 _donct
Commutes —___ ps =

Brenn {Scanned with CamScanner aa

63. 2
1-2. 2020 Quart Mers hex WS.
TOPIC.» "ZEEMAW EFFECT
o_ZEEMAU EFFECT;-

Splilind af Spectra lines

(esoctarsPoR TAN THA

into_Secunt__components,
_the_qesence—of-—smaqnetic_field (Best) is ——
— Known 08__Zeemea._egect.
e STARK EFFECT

Splitina,

Of Spectral lines _ into.

seca components.

—hthc_prerence of elechtic field (Gore) 13360000,
Stack ego es en

Reemon_effeck iS analedus da Stack

> ZEEMAN. EFFECT à
de

egect-

5 Strond_conkridcitier
ZEEMAN,

— °

— Morea ZEEMAN.
EFFECT
a NORMAL ZEEMAN. EFFECT:-

ANOMALOUS. ZEEMAN
EFFECT

y SPi0 of election _i$_Rera— Ira + 1 +0

i_Each_enesos _leuel._ SONES. iodo an edd number (2441) of

Euas_Sfaced level. sa

it)_tormal_zeeman_ef fect is ooserved when Spectcal
line of an atom split in do Hate lines under

a madneric field (Sem)
DER, den 4 2 —
dot mex 0.44

een manes
ad

64. © ANOMALOUS. ZEEMAA EFFECT

D_SPin_of electron is non Zero 1 #4.
| Each_enecs._level_SPltt_in_to_an__even_number =

(232) of on unevualY SPaced _tevel -

ii] Anomalous Zeeman effect _is_observed when-a
‘SPecha\_tine_of an_atom__<Alit into mere_then.
ihre lines under_a_meanet field (Bent)

nee.
401,2

Lena 2
Mp=-0,11,22

pren external madnetic_fiete

PAS

ne manebie fed
pire

Des

mend

Meer

me
e_ZEEMAN. ENERGY. (PERTURBATION) =
HanhurBen- Fs Bo one mem
Hz (elie = fs) Bent. E -
a. la
Bong 6 Beng PER;
Nesative San means that- His \
alas opposite_to—S_oc-L. £
L «Bent chacde on
Han [E Ea): óun ds

£l+2e
am im

fürs)

À

DE Bent

BA, magnetic dite

mo

‘momar

65. >> —Reemen__sPlitiss.dePend___upon_the went.
—Crtecnal_masnetic field (Be) is_compacision
ssith_intecaal_maanelic_tield= —

——___He He He ents
THREE CASES:
4-Weeks= Field Zeeman effect = (Bent cc Bint) —

Best-<< Bim — then fine ruche dominate and He
San-be_trested 05 man Peharbalion.

Strong fied_Xeeman_esgectre (Bera » But)
—Bent-33_8,, hen — Zeeman. effect is dominate
—And_ fine Scrnchute_ con toe centred as Sat

Puros tion.

3-—IMtermidicte. field Zeeman. egpecte (Ben = 6,

Boxy By 50 100 nel Abe pal manbinen

f-degernate —peturbation Aheoed-

LECHS Tr Bozous, FERMIONS. we
> SYMETRIC .AVO_AUESYMETRIC WAVE. Function.

For_ideritcal _Bartices_ there oe une different tone

Sanction.

2).84metric wave. function.

2)_ Anti sYmettic_usave function.
— 4) SYMETATC_wAVE_EuNeTION

We ms Hu + won "ence ga

= | ca Wace Yue sul

Warren]
% + Wanna) |

Hhis_ksPc ob come function Is Knaus as

Symeitic wave function =

66. ~2)_ ANTE SYMETRIC WAVE FUNCTION +
——Wa-=. = Appa sal

|_repiace 12 —
> 22

A | at ya = Vorl ac.

cs = cuco]

Hhis_t4te_of soaveganction_is called amti=S metro
weave function:

i)_fecmions.

_ofe_onti_sumeltic_wove function
ii)_for_identical__pacticles;

Wy = lef Uc Yat) tr tH

Ya.2-0. ——
this Show Hat. two. identical Pacticie

cannat be
_exist in same State.

_ ji)_Fetmions have half integre Spin 3 eleciton |
Ki pe A

Eier a

_iv)_FecmionS—obe¥. Poli's exclusion Peincipie-

__Two_ identical —Pactitles_(fermionS) election in
__an atom. an not bein. same_Stote-
e BOZONS=_. A
il Bozons oe _SiSmettic we function
fi) We = + Walt) RCA) + pet Pat —— PP.

_for_identical perds = i
E [+ Pps) Yl) + Pac) po al ef an)

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67. also readl_ trom _book-

this show that hoo _ientioal__pattides... Can. be
exist_in Ihe _Same__state,
il Bozens have ui) integral spine: —Photon-oyalo 23h
iv)_Bozesdoes_not_cloeyPoli_exclusive „principle
becouse_— tuo_édentien particles con be
exist in Ihe Same State. — =

e GEOMETRICAL REPRESENTATION OF ANGULAR —
MOMENTUM _ + :
Eor_a_Sixed_ value of), the total_andulac_momentum
À mai be cepresented_by_a vector whose tenth
E KGa)
— 2 Component af an. anular _—momeetum_is_ presented
Such ce
Ee ee na seen 16
2-3 ane. = =
Classical; represent a vector Loose
end_point's_.\ies_.on the cirele of —cadius.
ATI cotatin3 _alondp_Ahe-_ Surface of cone
> Ris 2 BE
I HRG ARG = Ek Fes min
Ip 2 IA 99 26K
Nie 0,21,22,216. e

Si Saxo

68. ¡Scanned with CamScannen

69. -4:2-2020 Quantum mechanics.

TOPIC: Autre momen ser —

— GERNAL FORMALISM . 0F_AUGUUR. MOMENTUM» —
—andulac_momertumopetator_S=f+8 thor define

44 _three_components Sars Fyand dey which
} ——Salisps_the flowing —commrstation relations

Li. E ES
= pe

3 a, E
15% Ju]=o
wheres Ing, e and Tse
lali Hecmition.. > a
EIGENSTATE _AND_EIGEN VALUE =
OF Je:

rn let ly) is he -ei3en she and
Bis she eigen vue for-Z=
=combonent of -ondular momentum.
Falu>=Bly>
Äh dp BY
de

_ du > PB dp >
WW SER
inteatate_locth Side-
= diya. _<B{d¢.
WR

70. A-is just à cyçecert term _
— if pe) is the point ina wave u
Aisa Reciod-_cepest Liye 22m) ZZ
— piers Iwtgs2ey — =
| TRE TE

ache ace ORE L
ee

Mj>0,1,2,——' So compare _vailn_ above eq uation.

E =
Er

= dB a = impar
K

B- mi
E
= mk eiden ane = =
eisen state em ice be rien inthe em aj m
MD Acme lan.
x

RAISING Au LOWERIUG OPERATOR:
Vaisino oPerator 3,
LouecinS_ Pacstor À

—= ER

Sie nei 3-255)
La ¿bue

Js
(Bis dba
isn il —
——— "à RE
ust lll IS +5, +hjz
tds SHE

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71. Is O —
Tsd ARA =~ any. ==> mn
rtfäh- dé -
— J and on ons

Tr: +) +3
is AAN
is Mb

(5051+ (5.32540
— el

a AS AN
ail = ohn

72. 6-2-2020

Quanilans mM. lec ws
MECHAVICS.

—RPIC:-_ MATRIX REPRESENTATION of _—
ANGULAR Moen iT

— ud mb ide

— 05 we know thet Mp) 2 AE
e_magnetic_quantum mmber—
— dependind_on£—
So.

ch rhone for —

»2j41_also

te,

lp
ML ii | prom Lt a a

al mjsAitle All 3

this is cur matrix

SW. Se
rol eli)

Pl Ar) —

oC What Set Wy, 01)
<Wuol-Jeh-Yrr0)

PA

rod Teh py)
o) Sob Par) 5

nous concluce values

73. 2) da miel, del miso 000.00. - —

2 O)K Bund

_MATRLX _RERESENATION =

Tuy =i S41)

AS_we Know ther WI = Ae, agent upon the
| magnetic Sonn number me 2jsl dus afro

depending on hu So, 112 con nie

Fim: EL jay Luo

ni both

2, fo Leg

u ld LEW Citar |
Unold — UT Apo pol ar
Ley Sg |

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74. 6)

Moro conclude the..values; . .— ——

itl FLY gis EN

——See_fi9_in___Quarkum mechanics -1_ Page Hg _

75. 82-2020 __ Quorum mechonics =
TOPIC :-_SPIV AUGULAR Momentum _ 6
BSPIN_AVGULAR MomeuTUMs af
—When_a_ articles _Tevowves _in_ciecular Path _ about its
—— an axis is caned Spin and _momentum Sp ———
Particle due to_sein_metion._is__called Sin anular

momentum.

e EXPERIMENTAL EVIDENCE oF SPIVt

—mza_Xeto field Pattcen..

a EXPERIMENTAL
Resutt

À oven —
Co = >

PhoteamPhic
- = Pte —

___S seins (rest)

se +» incon Anke om
too orientations
| Steca_ond Gertach in 1922 petçorm an ———

experiment: to verify the SPin_of_electfon. —
They have_use_ siluec_alem_ocem. - SilveC_stom —

_contoin_47_elecians, #7" elec ion Semaín_uapbic |
this_is_occugies_in_6S_ottoital — accacclir on
Schorodingee case ec his leer Naue to seit |
in hong one component,

76. font esas hy Change... bee wa spNt
— ndo hoo comment
this Puel save ti Goudamit and embeds, —
4e 3 ostia hat thece da alo ON AS _——_____——~
— ANAL _ Momentum. ed as sein _anSulat.
Momentum. -
a > a
nihece; me
“is _an_an3ulac_mementum..
Vis ne charge
+m is_a_imss,cis_o SReech of tight -
E _ _

where Bg TC
—is_coted she lande factor,
—»—Those_pasticles which _ have (Spin) inteScasQio.

| gs 011,2, <=: called Bosons —
——> Those acticles wuhich- have helf dd Integral Pin
oe Sr —are_caled_as _Fermions. —

a PAULI SPIN MATRICES:
FOR Pacticles hauins Fin dy Paul Ideen

____~ Mateices _Matices such x E90. ae
| which are tlotedt - spin. MA
BA on
I Commutation. elation —
eee KS, 184.5.
IN CES AS

DETTE mE:

Suns Km [06m —

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77. Ihe _Pauii_sPin _mmpadtices are _

EEE

78. NA-12-2020-__ QuAVUMMECHANEES ————"

ToPIc__ QuauTum_ NUMBERS.
QUANTUM MUMBERS:-_ Quant numbe(s —afe-
the Set of numerical Values which Nes ——

ne complete information „about nn „eleriten
sitar mar hen

non atome =
numbers —io-clesctibe.

— Thee ore four Iuantum-
90 electron—Ín an atom
i) Principal Quantum number Co).
1 AZi mutha _ Quantum number 60).
ii]_tAAGNETIC Quantum number (ne)
Spin Quansum numbec 5)

i) PRINCIPAL QUANTUM NUMBER:
accor ding to — oht_Iheoch _energs of electrons.
in there orbits are YuamiZed mean each
orbit have fixed_eneray, a =
Enz=18:6.eu
racliu Sof elöchton From Auceus also ponia.

Yo" don? where 99.2 O SRAXAS ra

Principle — aan namie is ted Wh
PET FRE FL anne
— Deil_+.K:Shell_
eB» Shell
etection_ shell formula;
_+ Ke shell
2612 electóon

ee A
2ta1*> elecion _

+ = Shen nes __
23> 2 I elecivans

— NN shell m
264}? = 32 elections ___

79. Li AZIMUTHAL QUANTUM NUMBER: —
——The_elecicons-has-—ardlac onomendumn hen ils —
—Sevolues _orouncA the —nuclems- The angulac mamentum
of an electron ar douldY QuentiZedl in mañnäude
and_citection- _madntude of angulac_momentuna
-is_dlenoted 4." ¢!' AZimudil quantum number —
Lis use —to_olescaibe...the. Shape- of arhitals:
ond._it!s _Ualuec_clepond._upon Piincple quantum.
_num ber-

ii) MAGNIT1c- QUPATUM_NuME: — mn.
Te __enSular_momentum of an elechon is —doublY
_quanti Zeck —in_maSnitucle_anl. ob@ction_. the
quantize d.._olivection...of. orbital__anQulac momentum
is discribed b4 maSnetic Quantum number —
db is oemted bd "my" — _ —
magnetic —yuan: _ number use 1 enf lai

_ the magnetic Properties of electrons ol

AA —

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80. ms 041,42,

iv) SPIU_QUAUTIUM NUMBER:

—The_electsons_in_the_exbitk hag hao ties of

——MotionCalled__oxbltal_motion— MA SPN_fnotion=___

Spin quantum number _is use to descúle

— the _Sein__motion of an electron. _
2 is Jlenoted_bY_.ms-

with _ posite. “sin.
—— SD, A, 0. em

ely 120, My 0 y m6.

See he table fam beak Quart MECHANIC | (2) Pye
A148

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