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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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