Phy

sics

2D

Lec

ture

Slid

esN

ov 1

2

Viv

ek S

harm

aU

CSD

Phy

sics

•r

Mea

sure

men

t Erro

r : x

±∆

x•

Mea

sure

men

t err

ors a

re u

navo

idab

le si

nce

the

mea

sure

men

t pro

cedu

re is

an

expe

rimen

tal o

ne•

True

val

ue o

f an

mea

sura

ble

quan

tity

is a

n ab

stra

ct c

once

pt•

In a

set o

f rep

eate

d m

easu

rem

ents

with

rand

om e

rror

s, th

e di

strib

utio

n of

mea

sure

men

ts

rese

mbl

es a

Gau

ssia

n di

strib

utio

n ch

arac

teriz

ed b

y th

e pa

ram

eter

σor

∆ch

arac

teriz

ing

the

wid

t hof

the

dist

ribut

ion

Mea

sure

men

t err

or la

rge

Mea

sure

men

t err

or sm

alle

r

Inte

rpre

ting

Mea

sure

men

ts w

ith ra

ndom

Erro

r : ∆

True

val

ue

Com

parin

g M

easu

rem

ents

With

Erro

rs

(dis

?) a

gree

men

t bet

wee

n m

easu

rem

ents

Bac

k to

Sha

rma’

s wei

ght :

Mas

s mea

sure

d w

ith p

oor p

reci

sion

10

00 ±

700

kg is

con

sist

ent w

ith 7

0±15

kg

Mea

sure

men

ts w

ith E

rror

s

•If

you

r mea

surin

g ap

para

tus h

as a

n in

trins

ic e

rror

of ∆

p•

Then

resu

lts o

f mea

sure

men

t of m

omen

tum

p o

f an

obje

ct a

t res

tcan

eas

ily y

ield

a ra

nge

of v

alue

s ac

com

mod

ated

by

the

mea

sure

men

t im

prec

isio

n :

–-∆

p ≤

p ≤

∆p

•Si

mila

rly fo

r all

mea

sura

ble

quan

titie

s !

Wav

e P

acke

ts &

Unc

erta

inty

Prin

cipl

e

in sp

ace

x:

si

nce

usu

al

2h

k =

, p =

app

roxi

mat

e re

latio

nly

one

writ

es

In ti

me

t :

si

nce

=2,

.

.

./2

./2

kx

wfE

hft

pxh

px

π

ππ

λ

ω

πλ

∆∆

=

∆∆

⇒

⇒

⇒=

=

∆∆

=

∆∆

≥

us

ually

appr

oxim

ate

reon

e w

rite

latio

ns

./2

./2

Eth

Et

⇒∆

∆=

∆∆

≥

Wha

t do

thes

e in

equa

litie

s mea

n ph

ysic

ally

?

Act

of W

atch

ing:

A T

houg

ht E

xper

imen

t Eye

Phot

ons t

hat g

o th

ru a

re re

stric

ted

to th

is re

gion

of l

ens

Obs

erve

d D

iffra

ctio

n

patte

rn

Diff

ract

ion

By

a C

ircul

ar A

pertu

re (L

ens)

See

Res

nick

, Hal

liday

Wal

ker 6

thEd

(on

S.R

eser

ve),

Ch

37, p

ages

898

-900

Diff

ract

ed im

age

of a

poi

nt so

urce

of l

ight

th

ru a

lens

( ci

rcul

ar a

pertu

re o

f siz

e d

)

Firs

t min

imum

of d

iffra

ctio

n pa

ttern

is

loca

ted

by

sin

1.22dλ

θ=

See

prev

ious

pic

ture

for d

efin

ition

s of

ϑ, λ

, d

Res

olvi

ng P

ower

of L

ight

Thr

u a

Lens

Res

olvi

ng p

ower

x

2sinλ

θ∆

Imag

e of

2 se

para

te p

oint

sour

ces f

orm

ed b

y a

conv

ergi

ng le

ns o

fdi

amet

er d

, ab

ility

to re

solv

e th

em d

epen

ds o

n λ

& d

bec

ause

of t

he

Inhe

rent

diff

ract

ion

in im

age

form

atio

n

Not

reso

lved

reso

lved

bare

ly re

solv

ed∆X

d

θD

epen

ds o

n d

•In

cide

nt li

ght (

p,λ)

scat

ters

off

ele

ctro

n •

To b

e c

olle

cted

by

lens

γ

mus

t sca

tter

thru

ang

le α

•-ϑ

≤α

≤ϑ•

Due

to C

ompt

on sc

atte

r, el

ectro

n pi

cks u

p m

omen

tum

•P

X, P

Y

•A

fter p

assi

ng th

ru le

ns, p

hoto

n “d

iffra

cts”

, la

nds s

omew

here

on

scre

en, i

mag

e (o

f el

ectr

on) i

s fuz

zy•

How

fuzz

y ?

Opt

ics s

ays s

horte

st d

ista

nce

betw

een

two

reso

lvab

le p

oint

s is :

•La

rger

the

lens

radi

us, l

arge

r the

ϑ⇒

bette

r re

solu

tion

Act

of O

bser

ving

an

Ele

ctro

n

Eye

Phot

ons t

hat g

o th

ru a

re re

stric

ted

to th

is re

gion

of l

ens

Obs

erve

d D

iffra

ctio

n

patte

rn

sin

sin

elec

tron

mom

entu

m u

ncer

tain

t y is

2h

psi

n

xh

hP

θθ

λλ

θλ

−≤

≤

∆≅

2sin

xλ

θ∆

=

Put

ting

it al

l tog

ethe

r: ac

t of O

bser

ving

an

elec

tron

Eye

Phot

ons t

hat g

o th

ru a

re re

stric

ted

to th

is re

gion

of l

ens

Obs

erve

d D

iffra

ctio

n

patte

rn

2s

.

in .

2sin

/2

hx

h

px

pθ

λλ

θ⎛

⎞⎛⎞

∆∆

=⎜

⎟⎜⎟

∆∆

⎝⎠⎝

⇒

⇒≥

⎠

Putti

ng th

em to

geth

er

•C

an n

ot E

XA

CTL

Y m

easu

re L

ocat

ion

and

mom

entu

m o

f par

ticle

at t

he sa

me

time

•

Can

mea

sure

bot

h P x

and

Y c

ompo

nent

ex

actly

but

not

Px

and

X

Pse

udo-

Phi

loso

phic

al A

fterm

ath

of U

ncer

tain

ty P

rinci

ple

•N

ewto

nian

Phy

sics

& D

eter

min

istic

phy

sics

topp

les o

ver

–N

ewto

n’s

law

s to

ld y

ou a

ll yo

u ne

eded

to k

now

abo

ut t

raje

ctor

y of

a

parti

cle

•A

pply

a fo

rce,

wat

ch th

e pa

rticl

e go

!–

Kno

w e

very

thin

g ! X

, v, p

, F,

a

–C

an p

redi

ctex

act t

raje

ctor

y of

par

ticle

if y

ou h

ad p

erfe

ct

devi

ce

•N

o so

in th

e su

bato

mic

wor

ld !

–O

f sm

all m

omen

ta, f

orce

s, e

nerg

ies

–C

ant p

redi

ct a

nyth

ing

exac

tly

•C

an o

nly

pred

ict p

roba

bilit

ies

–Th

ere

is s

o m

uch

chan

ce th

at th

e pa

rticl

e la

nded

her

e or

ther

e –

Can

t be

sure

!....

cogn

izan

t of t

he e

rrors

of t

hy o

bser

vatio

ns

Philo

soph

ers

wen

t nut

s !...

wha

t has

hap

pene

d to

nat

ure

Philo

soph

ers j

ust t

alk,

don

’t do

real

life

exp

erim

ents

!

Mat

ter D

iffra

ctio

n &

Unc

erta

inty

Prin

cipl

e

Inci

dent

El

ectro

n be

am

In Y

dire

ctio

n

x

Y

Probability

Mom

entu

m m

easu

rem

ent b

eyon

dSl

it sh

ow p

artic

le n

ot m

ovin

g ex

actly

in

Y d

irect

ion,

dev

elop

s a X

com

pone

ntO

f mot

ion

∆PX

=h/

(2π

a)

X c

ompo

nent

PX

of m

omen

tum

∆PX

0

slit

size

: a

Par

ticle

at R

est B

etw

een

Two

Wal

ls

•O

bjec

t of m

ass M

at r

est

betw

een

two

wal

ls o

rigin

ally

at i

nfin

ity•

Wha

t hap

pens

to o

ur p

erce

ptio

n of

Geo

rge

as th

e w

alls

are

bro

ught

in ?

m

Geo

rge’

s Mom

entu

m p

02

2

On

aver

age,

mea

sure

<p>

= 0

bu

t the

re a

re q

uite

larg

e flu

ctua

tions

!W

idth

of D

istri

butio

n =

()

()

;

ave

ave

P

PL

PP

P

∆

∆∆

=−

∼

L

Qua

ntum

Beh

avio

r: R

icha

rd F

eynm

an

See

Cha

pter

s 1 &

2 o

f Fey

nman

Lec

ture

s in

Phys

ics

Vol

III

Or

Six

Eas

y Pi

eces

by

Ric

hard

Fey

nman

: A

ddis

on W

esle

y Pu

blis

hers

An

Exp

erim

ent w

ith In

dest

ruct

ible

Bul

lets

Erra

tic

Mac

hine

gun

spra

ys in

man

ydi

rect

ions

Mad

e of

A

rmor

plat

e

Prob

abili

ty P

12w

hen

Bot

h ho

les o

pen

P 12

= P

1+

P 2

An

Exp

erim

ent W

ith W

ater

Wav

es

Mea

sure

Inte

nsity

of W

aves

(b

y m

easu

ring

ampl

itude

of d

ispl

acem

ent)

Inte

nsity

I 12

whe

n B

oth

hole

s ope

n

Buo

y

212

12

12

12

||

2co

sI

hh

II

IIδ

=+

=+

+

Inte

rfere

nce

and

Diff

ract

ion:

Ch

36 &

37,

RH

W

Inte

rfere

nce

Phe

nom

enon

in W

aves

sin

nd

λθ

=

An

Exp

erim

ent W

ith E

lect

rons

Pr

obab

ility

P12

whe

n B

oth

hole

s ope

n

P 12

≠P 1

+ P 2

Inte

rfere

nce

in E

lect

rons

Thr

u 2

slits

G

row

th o

f 2-s

lit In

terf

eren

ce p

atte

rn th

ru d

iffer

ent

expo

sure

per

iods

Phot

ogra

phic

pla

te (s

cree

n) st

ruck

by:

28 e

lect

rons

1000

ele

ctro

ns

10,0

00 e

lect

rons

106

elec

trons

Whi

te d

ots s

imul

ate

pres

ence

of e

lect

ron

No

whi

te d

ots a

t the

pla

ce o

f des

truct

ive

Inte

rfer

ence

(min

ima)

Wat

chin

g Th

e E

lect

rons

With

Inte

nse

Ligh

t

P’12

= P

’ 1+

P’2

Prob

abili

ty P

12w

hen

both

hol

es o

pen

and

I see

w

hich

hol

e th

e el

ectro

n ca

me

thru

Wat

chin

g Th

e E

lect

rons

With

Dim

Lig

ht

Prob

abili

ty P

12w

hen

both

hol

es o

pen

and

I see

w

hich

hol

e th

e el

ectro

n ca

me

thru

Wat

chin

g Th

e E

lect

rons

With

Dim

Lig

ht

Prob

abili

ty P

12w

hen

both

hol

es o

pen

and

I D

on’t

see

whi

ch h

ole

the

elec

tron

cam

e th

ru

Com

pton

Sca

tterin

g: S

hini

ng li

ght t

o ob

serv

e el

ectro

n

Ligh

t (ph

oton

) sca

tterin

g of

f an

elec

tron

I wat

ch th

e ph

oton

as i

t ent

ers m

y ey

e hg

g

g

The

act o

f Obs

erva

tion

DIS

TUR

BS

the

obje

ct b

eing

wat

ched

, he

re th

e el

ectro

n m

oves

aw

ay fr

om

whe

re it

was

orig

inal

ly

λ=h/

p= h

c/E

= c/

f

Wat

chin

g E

lect

rons

With

Lig

ht o

f λ >

> sl

itsiz

ebu

t Hig

h In

tens

ity

Prob

abili

ty P

12w

hen

both

hol

es o

pen

but

cant

tell

from

flas

h w

hich

hol

e th

e el

ectro

n ca

me

thru

Why

Fuz

yFl

ash?

R

esol

ving

Pow

er o

f Lig

ht

Res

olvi

ng p

ower

x

2sinλ

θ∆

Imag

e of

2 se

para

te p

oint

sour

ces f

orm

ed b

y a

conv

ergi

ng le

ns o

fdi

amet

er d

, ab

ility

to re

solv

e th

em d

epen

ds o

n λ

& d

bec

ause

of t

he

Inhe

rent

diff

ract

ion

in im

age

form

atio

n

Not

reso

lved

reso

lved

bare

ly re

solv

ed∆X

d

Sum

mar

y of

Exp

erim

ents

So

Far

1.Pr

obab

ility

of a

n ev

ent i

s giv

en b

y th

e sq

uare

of

ampl

itude

of a

com

plex

# Ψ

: Pro

babi

lity

Am

plitu

de2.

Whe

n an

eve

nt o

ccur

s in

seve

ral a

ltern

ate

way

s, pr

obab

ility

am

plitu

de fo

r the

eve

nt is

sum

of p

roba

bilit

y am

plitu

des f

or e

ach

way

con

side

red

sepe

rate

ly. T

here

is

inte

rfer

ence

:Ψ

= Ψ

1+

Ψ2

P 12=|

Ψ1

+ Ψ

2 |2

3.If

an

expe

rimen

t is d

one

whi

ch is

cap

able

of d

eter

min

ing

whe

ther

one

or o

ther

alte

rnat

ive

is a

ctua

lly ta

ken,

pr

obab

ility

for e

vent

is ju

st su

m o

f eac

h al

tern

ativ

e•

Inte

rfere

nce

patte

rn is

LO

ST

!

Is T

here

No

Way

to B

eat U

ncer

tain

ty P

rinci

ple?

•H

ow a

bout

NO

T w

atch

ing

the

elec

trons

! •

Lets

be

a bi

t cra

fty•

Sinc

e th

is is

a T

houg

ht e

xper

imen

tid

eal c

ondi

tions

–M

ount

the

wal

l on

rolle

rs, p

ut a

lot o

f gre

ase

frict

ionl

ess

–W

all w

ill m

ove

whe

n el

ectro

n hi

ts it

–W

atch

reco

il of

the

wal

l con

tain

ing

the

slits

whe

n th

e el

ectro

n hi

ts it

–B

y w

atch

ing

whe

ther

wal

l mov

ed u

p or

dow

n I c

an te

ll

•El

ectro

n w

ent t

hru

hole

# 1

•

Elec

tron

wen

t thr

u ho

le #

2

•W

ill m

y in

geni

ous p

lot s

ucce

ed?

Mea

surin

g Th

e R

ecoi

l of T

he W

all:

Not

Wat

chin

g E

lect

ron

!

Losi

ng O

ut T

o U

ncer

tain

ty P

rinci

ple

•To

mea

sure

the

REC

OIL

of t

he w

all ⇒

–m

ust k

now

the

initi

al m

omen

tum

of t

he w

all b

efor

e el

ectro

n hi

t it

–Fi

nal m

omen

tum

afte

r ele

ctro

n hi

ts th

e w

all

–C

alcu

late

vec

tor s

um

reco

il

•U

ncer

tain

ty p

rinci

ple

:–

To d

o th

is ⇒

∆P

= 0

∆

X =

∞[c

an n

ot k

now

the

posi

tion

of w

all

exac

tly]

–If

don’

t kno

w th

e w

all l

ocat

ion,

then

dow

n kn

ow w

here

the

hole

s ar

e–

Hol

es w

ill b

e in

diff

eren

t pla

ce fo

r eve

ry e

lect

ron

that

goe

s th

ru–

The

cent

er o

f int

erfe

renc

e pa

ttern

will

have

diff

eren

t (ra

ndom

)lo

catio

n fo

r eac

h el

ectro

n–

Suc

h ra

ndom

shi

ft is

just

eno

ugh

to S

mea

r out

the

patte

rn s

o th

at n

o in

terfe

renc

e is

obs

erve

d !

•U

ncer

tain

ty P

rinci

ple

Prot

ects

Qua

ntum

Mec

hani

cs !

The

Bul

let V

s Th

e E

lect

ron:

Eac

h B

ehav

es th

e S

ame

Way

Qua

ntum

Mec

hani

cs o

f Sub

atom

ic P

artic

les

•A

ct o

f Obs

erva

tion

dest

roys

the

syst

em (N

o w

atch

ing!

)•

If c

an’t

wat

ch th

en A

ll co

nver

satio

ns c

an o

nly

be in

term

s of

Pro

babi

lity

P•

Ever

y pa

rticl

e un

der t

he in

fluen

ce o

f a fo

rce

is d

escr

ibed

by

a C

ompl

ex w

ave

func

tion

Ψ(x

,y,z

,t)•

Ψis

the

ultim

ate

DN

A o

f par

ticle

: con

tain

s all

info

abo

ut

the

parti

cle

unde

r the

forc

e (in

a p

oten

tial e

.gH

ydro

gen

) •

Prob

abili

ty o

f per

uni

t vol

ume

of fi

ndin

g th

e pa

rticl

e at

so

me

poin

t (x,

y,z)

and

tim

e t i

s giv

en b

y –

P(x

,y,z

,t) =

Ψ(x

,y,z

,t) .

Ψ* (x

,y,z

,t) =

| Ψ(x

,y,z

,t) |2

•W

hen

ther

e ar

e m

ore

than

one

pat

h to

reac

h a

final

lo

catio

n th

en th

e pr

obab

ility

of t

he e

vent

is

–Ψ

= Ψ

1+

Ψ2

–P

= |

Ψ* Ψ

| = |Ψ

1|2+

|Ψ2|2

+2 |Ψ

1|Ψ

2| co

sφ

Wav

e Fu

nctio

n of

“Stu

ff” &

Pro

babi

lity

Den

sity

•A

lthou

gh n

ot p

ossi

ble

to sp

ecify

with

cer

tain

ty th

e lo

catio

n of

pa

rticl

e, it

s pos

sibl

e to

ass

ign

prob

abili

ty P

(x)d

xof

find

ing

parti

cle

betw

een

x an

d x+

dx•

P(x)

dx

= |Ψ

(x,t)

|2 dx

•E.

gin

tens

ity d

istri

butio

n in

ligh

t diff

ract

ion

patte

rn is

a m

easu

re o

f th

e pr

obab

ility

that

a p

hoto

n w

ill st

rike

a gi

ven

poin

t with

in th

e pa

ttern

P(x,t)= |Ψ(x,t) |2

xx=

ax=

b

Prob

abili

ty o

f a p

artic

le to

be in

an

inte

rval

a ≤

x≤b

is

area

und

er th

e cu

rve

from

x=

a to

a=b

Ψ: T

he W

ave

func

tion

Of A

Par

ticle

•

The

parti

cle

mus

t be

som

e w

here

•A

ny Ψ

satis

fyin

g th

is c

ondi

tion

is

NO

RM

ALI

ZED

•Pr

obof

find

ing

parti

cle

in fi

nite

inte

rval

•Fu

ndam

enta

l aim

of Q

uant

um M

echa

nics

–G

iven

the

wav

efun

ctio

nat

som

e in

stan

t (sa

y t=

0) fi

nd Ψ

at s

ome

subs

eque

nt ti

me

t–

Ψ(x

,t=0)

Ψ

(x,t)

…ev

olut

ion

–Th

ink

of a

pro

babi

listic

vie

w o

f pa

rticl

e’s

“new

toni

antra

ject

ory”

•W

e ar

e re

plac

ing

New

ton’

s 2n

dla

w f

or su

bato

mic

sy

stem

s

2|

(,

)|1

xt

dxψ

+∞ −∞

=∫

*(

)(

,)

(,

)b a

Pa

xb

xt

xtdx

ψψ

≤≤

=∫

The

Wav

e Fu

nctio

n is

a m

athe

mat

ical

fu

nctio

n th

at d

escr

ibes

a p

hysi

cal

obje

ct

Wav

e fu

nctio

n m

ust h

ave

som

e rig

orou

s pro

perti

es :

•Ψ

mus

t be

finite

•Ψ

mus

t be

cont

inuo

us fn

of x

,t•

Ψm

ust b

e si

ngle

-val

ued

•Ψ

mus

t be

smoo

th fn

WH

Y ?

mus

t be

cont

inuo

usd dxψ

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