QUANTUM MECHANICS I : Ph 503

IIT GANDHINAGAR

LECTURE NOTES

B- PRASANNA VENKATESH

⑤LECTURE I : Introduction to Quantum Mechanics

- - - -

⇒Course : Prasanna . people . iitgn.ac.in/ph - 503 - fall -2019

Website

(a) .Preamble

(txt AM ultimate Theory of nature in the sense that

everything at Ultimate level obey QM - as far as

we know no violation A AM laws n'hawse!( * ) As a calculation at tool - extremely successful( * Reno;sngheni.mn Ii:whfYm"iy9"" "n¥

- unification with gravity ?

( * ) As an ultimate theory QM applies not just to

the microscopic world → but to even regularmacroscopic systems

manifestations of RM mom appeal- in microscopicWorld

many"

regular"

-impenetrabilityof solidsPauli exclusion

.macroscopic phenomena

→

-magnetism

→ " spin"

have

Iguana:mmman

:/ae-spni.ua÷::::3:L . ,

cannot explain !!

( * ) Historically it were macroscopic manifestations of

Am effects led to its discovery- Black body radiation

- Photoelectric Effect- Stability of atom !

( see Chapter I Puri for Historical introduction ) ②

Cbl.

What QM concepts do you know So far : Discussion

Some topics

0 Uncertainty fnincipk- statement accurate Atrrecise determination tf

even InfooinciblePosition t momentum not possible

#

① Classical vs QuantumI I ① wavefunction is✓ V

I riot the most

Phaser wavefunction-

space.

} → ( ? ? ) fundamental description-

Xi to Probability tf a quantumdistribution particles

- trajectories,in

→

Schrodinger equation"

Slate"

vector

① Entanglement (? )

CAT Slaty

① Measurement inquantum mechanics e ? ) .

(e) .

Two Experimentsillustratingthe fall ' ' weirdness

"

of AMCThought )

③f I ) .

DoubleHittExperimentForecho "

@ounce : Feynman Lechers Vol I ]

Consider first the experiment

Foodshown to the Left

n

Se.•Gun that sprays bullets

in a random manner over

a large angular spread••In front an armor plate with

two small enough holes

as to just allow bullets

to pass through .

•

Attsome fixed distance a backstop wall that can absorb the

bullets l made of thick food for example )

• Movable detector - e .

g . filled with sand can slow down A catch the

( along X ) bullets . Later we can Count the number Sf bullets -

-

Observable - What we are interested to measure-

Probability of bulletshitting at posh"

X"

on the backstopthat

/ (Meningeal

probability Since there is no Sim ble wayto say

where each

bullet will so after bouneiy off the edge of the holes for AS '

Easy to measure this : no . of bullets arriving in given ④probability detector box 2h Sometime-total no . of bullets

- bullets always arrive in lump ) or whole no

Results Discussion :. '

half" bullets

- fi.

. At any instant only one bullet hits backstopfi . If rate of bullet emerging from

gunie made

Small → At any given intact - Either no bullet

hits or ones bullet hits

Let us call Be the

forohabitychest

j ifhas the shape Shown

( bullet could have )in Fig I - i (c)

Come from holeII

or 2

only concerns why Max at x-

- O ?

So Consider distribution P,

Cx ) A Pz Cx ) see Fig I - I Cb)I I

hole 2 hole Iclosed closed

P,

= Rex ) t RH )no

" interference "

t

probabilities add

-

Double Slit for Waves-

Setup: As shown in Fig I- 2

,

we consider water waves produced* source

by a motor moving abt down → circular waves like

with a stone in a pond -

• Way with two holes A aback wall that absorbs waves

①Movable detector which measures intensity A the wave .

intensityx rate if energy I power L we

Carried bythe wave

ght2

• Interested in measuring intensityas a fu Of position

on screen x .

ResultsSTDscwsion

: ① first thing no Lumpinee to the' '

wave"

-

as

-

intensity if Source charged observed intensity at

any"

x"

an_y charges !

① In ex ) in Frg I -2 Cc ) is the observed intensity←

" interference Whereas I,

HI, Izcx ) intensity when

pattern"

hole 2,

I is shut off !

① Standard wave Behaviour :

instaafhasnegeuenheishtdgoumbhowaue, alone

-

-

red,

e'wt ]

h,C-

¢ it howzabnethfhzeiwtff

,

complex I.= If,

eiwtl ? 1h,12

Is = 1h ,ze wtf 's 1h42interference

In -

-1h

,eiotthzeiwt 12

qterm !

= lh.it/hzl2t2lh,llhzlcosCf )I

In I I,

t Iz'

phase difference!

( IT ).

Electron Double Slit ⑥- -

-

Set-up-

Consider as before a double

Slit experiment .

Electron gun → heated tungsten

② wire in a box - box at layert ve charge - Is accelerated

out.

As before metal sheet with

①

two holes in front of thegun

① Behind the double slit we have a Screen C another metal sheet )

Htta detector → e- multiplier attached to a speaker

C At the time of Feynman - this experiment was only a thoughtlxerimeut → needs a very small - scale System . . see end! )

I

We justsit

A listen to the" clicks ' '

on the detector / speaker

-

ResulksDisausion

① we first notice that each if the Clicks are identical A sharp- no half clicks !

① Clicks Come erratically - not in a Simple Continuous steam

Cro- pattern

① If we court no . of clicksover along time [ o,

T ] = Rt

. and. , it [ T

,2T ) = RI

an averageht E NI → we can talk about rate at

Which clicks arrive

① As we move the detector along *,

we find different average rate Of ①

clicks but still - but thefsioeudneslsfchicks same=

① If we lower the temperature of Tungsten A reduce rate of e- emission

from gun- the rate at each x becomes lower the

detector clicks very rarely but still each click is same

as before Cin terms of loudness ! )

① If we put two detectors - at Kuo different - X - only one=

Clicks atta time ! ( for very high intensity we may not be able

instant to resolve )

: We conclude e-

s arrive in lumps ire lone - by - one in whole

to the Screen ! Thus as with bullets we can ask what is the

relative probability for e- to arrive at dilteseat positions along x ?

Piz C x )

Answer is : Fig I- 3 Ces → There ie aninterference pattern !

e- s behave like "

waves"

in tame of this

probability !

How to understand this behaviour ?-

Proposition At ⇒←S that maketo the screen haveeither

gone through 1 or 2 .

if this is true the distribution with hole 2 closed P,

Cx )

ee tig I - 3 Cb ) ] I closed Put )

hastoSatisfy Pn =p,

tpz

but clearly Piz t Rtp ! !

no way to reconcile proposition l lait para) t e-

s arriving in lumps ! ⑧

I 9 9 on aconto lated explanations such as e 's travelling in weird pathsAoexplain the

interference will Jail !

But mathematical we already know from waves how Such

interference Arises . If must be that

p ,= 16,12 172=110212

for Some

Piz = 10*+0212 aamplitude" fry

to,

. Ok .

.

: we conclude ⇒ e-

s arrive in lumps like particles(82rad

Bob . of arrival of lambs - like the

intensity distribution if a wave ⇒ duality.

particle A

wave nature .

Proposition A is Clearly false .

-5 To illustrate this

to the QM non - intuitive features :

Let as watch the e-

s ?

Put a light so that when e- scatters

off the light we see which hole

the e-

comes through .

What happens then ?

every click of detector ⇐ flash of light near

hole I or 2

Inever at both places ! !qq.jn.yqmeywgga.ae

cog , ,

P,

'→ by keeping it there was a click A flash near hole I ⑨

back

pal → u 4 ' ' 2

gP,

'

=P,

s Piz'

=P,

't Pz'

! !

I blocky hotel ) ↳ just the prob . ignoring the Adhd

.

'

. when we measure e- s come'

throyh I or2

switch off the light → p, ,

binchhefenais

So the interaction with light measwny The e-

S Completely

Changes Piz → Pn'

what happen it we use very weak

}.

.

O flashesdqflisahtpwg.hn?s-ssn.Y

'm

light Sourcesame

=size as before

So weak that the

' '

perturbation"

is small① only charge we have clicks

light is also£ sometimes without any flash

" lumpy"

ffho=bnsD ① so consider three casesi

11P

,

→clicker flash near

I

p,

"→ same shape as P

,

'

Pz

"→ a " near 2

Pj'

→ " a Pz'

p

"12

→ Klick but

Piz'

→ a 11 as P, ,

no flash

.

'

. unseen e- s

'

Still Grodno interference ! !

logical Unseen e- s undisturbed ?

So can we see e- s without disturbing them ?remember that photons have the E = HV =

hey , so may be

foyer rwarelhyh low lreyy Photons will disturb Is here !

So let us keep making X layer A layer !

When t > d ④

D= Sep!

ration of suits ,

whenever Is scatter off the photonsnear

we Cannot distinguish a

which slit I or2

the flash came from !resolution is at best

I

A ! ! Dilhaaion limitv

it ie precisely when lad that we again See the

11interference fringes for e- s !

✓

Manifestation Gf ) ..

It it impossible to build an apparatus

that tells us which hole the e- went through

Uncertainty principle t at the same time not destroy

interference pattern ! !

Generali sing lemons we learnt :"

ideal experiment"

-

eevveerryyas beet Controlled

no uncertain ex atonalInfluence

event → set of initial A final conditions

e- leave gun → hit positionX on

Screen

① Ir 9M probabilities of events P = 10/12↳ prob . amplitude → complex

number

total① when an event can occur in multiple ways , the probability

amplitude of event - ¢ of to D= 14+012/2-1interference !. an result

e.gg.

2 ways

① If in addition in the experiment we can Say C"

measure" )

the two ways has the system taken,

P =P,

t Pz = 14,12+101212 → probabilities add . ④

: Additional in - class points :

④ what about bullets → built out if many"

e-"

- like small

particles

wahbaotut non - ideal → decoherence .

④ Feynman's idea → Seen in a lab e show ?

II ) Stern - Gerlach Experiment-

.

.

( Sakurai Chatter 1) abbreviation : SG

\ f z-axisya

Apt

.

¥7

"

Flemish Stern - Gerlach

¥¥¥÷i€¥÷÷in ferment )

"H

-

in:i Is

Ag atoms Fig#

. toile.

mating- magnets slit

Experiment

⑦ It Iver atoms leave an oven A are collimated into a beam.

through a Shall hole

Beam is passed via a region tf inhomogeneous magnetic

field produced by shaped magnets .

following this the atoms are observed by taking a facture of them④exiting the magnetic Held region or usinga screen to accumulate

the atoms .

Observation A Discursion- -

We need to work out how the magnetic Feld a Heels Silver atoms .

( Silver )

Ag atom → Collection of protons , neutrons A e } Cneutral ]

magnetic Held can apply force on a neutral object ?' FEE 9mi !

only if these are"

current"

involved or in other words

we need a magnetic dipole moment !

In a verysimplistic model - we can

ignorecompletely the magnetic moment

①

if the nuclear particles- they have much Smaller particles .

① So all of the magneticmomeet Comes from the electrons

claieically me -- I Ari

howcan e- s lead to a → They have tomore!magnetic moment form current

iT AAT

loops 'I

or in other words '

l-

angular momectum → magneticAg - 47 e-

s moment

446e-

s form Eclosed"

shell structure with 2€ ttogalarmmuhem

All of theangular mometum comes from 4Th electron occupying

a Ss orbital → SS

→spherically symmetric orbital → orbital{ Emonon this later ! ! ) angular momehum ⇒

(spin - )

→so only

' '

spin"

e- - magnetic moment

I = angular

magnetic moment Amomlhem of final e

-

Ag atom ⇒trash !

it = e- 5 eco ④

Mec

interaction potential = - TT . B C dipole in a B - field) -

.: 2 - compared. if force :

Fe ⇐Zz fit.B) I.µzdgBsz KishoreBn, By ]

I

atom as a whole in"

heavy"

-classical trajectory of the atom is

meaningful.

With the. away meet ie Fig II; we have an upward force M¥20( Saco )

downward it Maco C Sato )

theBeam is expected to split up according to Mz Values .

: . TheS - G apparatus"

measures"

z - component of or8)

-

Result :- >

beamX

beam 7Screen Screen

expectation ! Result !

Atoms in oven orandomlyorieekd - So we expect a continuous

-

distribution of Mz → single beam A one blob on the Screen

-1µLMeLIM"

Claes i Cal angular momeehem"

- Instead , we See two distinct Components inthe beam A two-

bloke on the Screen .

Thus,

it looks like µz or Sz Can only take 2 - values !④

e-

① let us call these two values Sz t

Sz -

① The two possible values C isotropy) are also multiple A

a fundamental quality . It hens out Sz =t the or - the !

⇒ Firstlemon from e

-

spin is quantizedSG

note nothing special about z - direction ⇒ same result A SG

apparatus applied field ahoy y

oorry!

two beans( Sx'T S

-

es.

. ,( g

':) !

Sequential SG exist

Sz # comp SztOven SGI

, Sai ⑦s

E - . - - - - - - . No Sz - compz

-

comp

.

Sz t beamS

,t

Oven SGI, SGI beams ③% S× -

Sz -

beam

. Szt Sit bumSzt beam

oven SGI Sai Sai ②% E E - beam

Sz - beam S× - beam

Scgoewrial Sa : atomic beam Soa through more than

I SG apparatus !

See ① , ③ .CO ⇒ SGE → Sa apparatus to' '

measure" Sz

B- field .

alongI

59 I → a sa ⇒ Baby I

① : I : measure Sz A block Sz - component of beam ⑤

2 : send Szt beam through SG I s only one beam !

Result not surprising all alone in Sat state !

⑤ : I:measure Sz A block Sz - component of beam

2 : send Sz t beam through S G IResit

: Get two beams → atoms in #It state .

Hoeyggtoain?

: Say we propose so 's. of alone It are made of

I

so . I.

are made up of atoms Set A SS,-t with

SO -1.

I , it Set A Sx -

This proposition runs into trouble !

② . I:measure Sz A block Sz - component of beam

2 : send Sz t beam through S G I A block S,

- beam

a second att output3 : Send Sit beam through SG I set up !

Reset : We get two beams out → Set A Sz -

from above proposition we expect only Sat ! !

what gives ?

-

:O. assigning Sat A Sx ± simultaneously is wro# / not possible

in AM !

① S* measurement in step 2 destroys all Sz information

:.

In AM we cannot determine 5×,Sz Simultaneously ! ! ④

Compare to .

cm.

Spinney top I = Iw→

thecan measure simultaneously Wx

, Wy , wz

-

by obscenity how fast top spins about each

axis

-IT computed from . shape A mass density

-

.

'

.Lx

, Ly , Lz Can be Simultaneouslydetermined !

Note the inability to determine Sz, S×

, simultaneously as with the

double slit example ,not a limitation of experiment - we cannot

make Sz - beam in set up ③ disappear by matayabetter more precise experiment ! !

-

Analogy with Polarization of Light-

Maxwell lens in free - space A EM waves ( guide orecap ) .

. .

- -

7. E = o cause law 5=0 ] EXE = - JBL C Faraday law )

atF. 5=0 o-xis-foe.ge

Empire's

law

. . .

c-{

→t - maxwell

correctionit = I Csi

,

E ) -02 E = 0×1-25/2 t ) F=oyo →

apply . i -fzlEx5 ) = -Moesof 2

:for' E = Moe .

'

/ Moa -

Ie to -

.me .III :-

' ⇐

mop"Iw÷Yng"gun : E-

- ET e''

Che- wt '

Be=

BY eicke - wt )

Complexnotationprob ahoy

"

z"

F. E = F. 5=0 ⇒ 2,

EE,

eih -

+2 ,

Eo!!ein +

2ztEozlike ④

" "

° ik ZO

wtft ikEr e = o↳ A

⇒Eoz=ot Boz ⇒

FEE = - fi ⇒ -

kEog=w③o×A K E

ox= w Boy

Or

=

- BT = ICE x ET ) f .

-

: I Boll IEOI typically .

So we focus on

in free space just E field of

more generally : E =µg§Eo eickz- wt ))

Plane waves .

with possibility fEs f ¢with Eo EIR

→ E.I =o & ⇒ Polarization !

consider

realI first : two ( lignify.naiedependat) Choices

E =

in- x polarized right

& =Ey

- y polarized lightwalk !

→E =

Eoolncoslkz- wt )

} → Linearly polarised→

E = Eo ly cos Chez-

we , Ight

letting unpolarized light through × - filter polarizer → polarized

philter right.

a - filter → rotate by goo → y - titter !→ z

y filler

light#¥wT] - nought !

↳ " Malus law"

¥4 I = IocoioO angle between

truthsecond filter !

+17,4k¥?

,iED¥neon

'⇐

④

lightyour

"

97:b . . " !I

rig € '

--

I ↳ffiidhterslight with pot along Ex .

once x'- filler selects a

'- polarized light beam

,

it is immaterial if beam was a - polarizedanalogy before !!

Very analogous to

gSz ± atoms ←s x, y polarized beam

setup 3 of|s×±atom ←s si, y

' polarisedbegin! ' ④SG abbasahes

now the x'

, y'

polarised beams in Em theory can be written as :

Eo II Cos ( Kz - oof ) = Eo [ Ygz Excos Ckz - wt ) t yr ,Eycos Ck 2 - WH)

F- o Ig

'

cog ( kz -wt ) = Eo [ - lgz In " + they " ]

In fig 2 above → stage I x - polarized → linear combination if a'

, y' light

I tall #'

-polarized ) → n' polarized linear combi

.

light ugh .

=

.

" AYpolarizedabsorbed

beam .

v

I →iy polarized light Selected !

him that

From Correspondence Cl t the discussion n we might beable to represent

spin state of a silver atom by a vector in new

"

2 - d"

space

- not the usual x. yEuclidean space where Ex Ey live !

Just as U'

. y'

polarization → linear Combination of n A y

en = exteylrz ey'

= - lxtey If

if I sx,

t ) → denotes the state of silver atom after SG I ④↳ " Dirac Ket

"apparatus

we

canspegcaeigatel }⇒

' " it > falsest 7tflsz ; - >g , ①

lsyjt s I-1gal Sz

,t 7 t 'Gal Sis - >

I t✓

"

superposition"

we will Show

( this clearly later ) .

'

.

the

unblockedB×t out of SG I apparatus= superposing

Zt A z - !

Now,

how do we represent Sy ,I States :

subject to

By symmetry we can expect : Sz ± beam -

g%yy × .directSG g- apparatus

Ill

S2 I gong aly y- denn .

on - subject to

Sai

aphcsehy

.

.

. Sg ± → has to alds be a linear combi of SztSz - !

but we have exhausted possibilities in ⑦,

in fame of

linearly Combining Szt A Sz - !

what Can we do ?

Analogy to circularly polarised light →

+ right circular

EEEihe #ilrzey - left circular

E±= the Ceo either "EI ) =

Egg REcos cha , wt )Polarised .

+ e-yagl.kz - wt ± MED

:.

Sy t

asEt

Sy - ⇒e

.

anahoy !

:. Isg ;I >¥ Kz lsz ,t7 till Sz - 7 ④

we

proposeI Thus

we need a 2- d to complex space on

which the ismtatesWilFLThis will be

a linearvector space !

=

We conclude by stating a central

populate of QM ,from which we will

launch the full formal mathematical description Of AM :

Quantum States are represented by vectors in Complexlinear vector space .

Summary :

* am - ultimate theory of nature with Some veryCourter intuitive features

* Description of quantum States as vectors in

a complex ,linear vector space !

x - x