Problems
The
human
can
awill
signal
100toofavery
u sec" with
radapproach
iationMaxwell's
canto
bequantum
neglec
tedmechanics
in these
time
delay
haseye
been
observed,
wremrn
ithoflight
ther. Such interference in thielectromagnetic
s sum-aver-paths
surface of the m
equations
in quantum1.5.
electrodynamics
(QED).
Weeven
discusThese
individual
amplitudes,
each
of
which
has
a
magn
itude
and
a
phase,
canliofght
interfere
photons
per
second.
How
far
away
can
a
I
OO-watt
circumstances?
low intensity. In classical physics. the energy
the
or photons) ca n be used to derive suc h fund amental
resuits
of
optics
as
the
of
wouldv
Fromit these
Verify
to the rules time
. (a)we
£o e;(h-Wlproof
) is a solution
£ = without
havethat
stated
some simple
sion of QED in Chap tcr 10. For1.1now,
with each other. Such interference
in be
thiseen
s sum-aver-paths
to quantum
mechanics
by a dark-adapted
eye?out
Assu
me the light
bulb
electromagnetic
wave isapproach
spread
uniformly
over the binding energy)
f least time.
wave
equation
we canausc
to calculate
probability
of detecting
a photon,
incl
uding
the fact that frequency v,
1.5. The human eye can that
of 100 canthe
u sec"
bulb
istoinderive
oof
uter
soIn
that
thepicture.
light
is calculate
not
scattered
forwith
li ght signal
(for photons)
be used
suc
h fund
amental
resuits
of
optics
as the located I m awa
surface
metal.
this
the amount
thespace,
cise form for these probability amp litudes fo r the
lightphase
comes
from
solving
,
kd
, where
dAlso
of th
is equal
is the
distance
traveled
by the photon
byofto
the
atmosp
here.
assu
me
th
e
bulb
is
that
photons per second. How farprinciple
away can
aeleast
I amplitude
OO-watt
of
time.li ght
I A-.
time it would require for I cV of energy (a typica l atom
1.3.to be
What
m
equations in quantum electrodynamics (QED). We
to
will
remrn
a
discusand flitudes
radiat
es fo
atfor
wave
length offrom
550
between
theAssu
source
andlight
the
and kenergy)
=icamp
2;r
propagation
in a11111.
medium
A; that
eye?
me form
the
bulb be seen by a dark-adapted
flu x is a
The
precise
for detector
thesemonochromat
probability
ralight
solving
binding
to be absorbed
by comes
an atom in a meta
l incident
of
minimum
D in Chap tcr 10. For now, bulb
we have
stated
without
proof
some
simple
rules
a reasonable
est imate
theindex
of th e
than
vacuum,
suchinasquantum
glass,Use
Aelectrodynamics
AII m
where
refraction
for
n isfor
II away
is in o uter space, soother
thatMaxwell's
the
light
isequations
not scattered
to aarea
(QED).
Wediameter
will
discuslocated
from
a the
I-walt
bul of
b.remrn
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ofthat
the w ith light of
Physics
240/250,
Spring
2022
,
dark -adapted
pupi
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eyc.
1.1 3. The maxim
usc to calculate the
probability
detecting
photon,
incl
uding
the
fact
that
=
provided
The
wave
equation
in
a
medium
w
kc.
by theof
atmosp
here. aAlso
assu
me
th
e
bulb
is
that
medium
and
thatinthere
nge
in-.stated
the
amplitude
for fracti
reRection
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light
10. Forcha
QED
Chapistcra phase
now,
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proof
some
sion ;of
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such
as
g
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of
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is
give
n
by
,
d
f th e amplitude is Problem
equal to kd
where
is
the
distance
traveled
by
the
photon
monochromat ic and radiat
at a we
wave
ofcalculate
550whose
11111. theindex
is es
reflected
bylength
ausc
material
ofAM
refraction
is greater
that
ofkthe
material
that
can
to
probability
of
detecting
a photon,
incl
uding
the
fact that in 100 Mm. Ho
incident
flurad
x is
by ththan
e atom?
1.6.
An
io absorbed
station
broadcasts
1000
Hz
with
at
0.82
eV
for rad i
e source and the detector and
=
2;r
that
for
propagation
in
a
medium
k
f
A;
Use a reasonable est imate
forthe
the
diameter
th e
, where
da'£
phase
of th
etraveling.
amplitude
equal
to kd
iswill
thewatts.
distance
traveled
by the photon
which
the
light
isof
Inis the
chapter
we
start
to
see
howtheresults
such as
an next
output
power
of
Assum
ing
50,000
a'£
II'
width a?
determine
Planc
=0
pupi
finvolve
theindex
eyc.between
electrons
1.1
3.
The
maximum
kinetic
energy
ejected
of
rcqu
ire
pe
r
second?
per
cycle?
Do
the
answers
(d)
ManyA ofdark
these
problems
formula
plugging,
so
it’s
not
as
long
as
it
looks.
1.12. The photoc
vacuum, such as glass,
AI-adapted
of
refraction
for
that
n isl othe
II where
the
source
and
the
detector
and
=
2;r
that
for
propagation
in
a
medium
k
f
A;
anten na is
located 100mechanics.
km away fro m th e
these can be derived for the case broadcast
of non relativisti
c quantum
sodium.
and
indi
cate
at th
e granularity
o f th e radiates
to (b)
fro
is
1.85
for
radiat
ion
of300
nm for
andthateffect seem to ap
nd that there is a phase cha nge OrIT in the amplitude for
reRection
when light
receiver
and,
ease
ofntheV
calculation
, the
other
than vacuum,
such as glass,
Am sodium
AI(c)for
is the
index
ofantenna
refraction
II where
1.4. A radio
1.6. An AM rad io station broadcasts at 1000 k Hz with
electromagnetic
rad
iation
can
be
neglec
ted
inswave
these
delay has be
What
must
be
the
re
lationshi
p
between
the
length
A time
0.82
eV
for
rad
iation
of
400
nlll.
Use
thi
data
to
1.14.
Show
that
isotropica
lly,
estimate
the
number
of
photo
ns
per
cubic
by a material whose
index of refraction
is greater
thanmedium
that 1.5,
of;the
material
OrIT
and
that1.11,
therein1.12,
is a phase
cha nge
in the1.25,
amplitude
for1.28,
reRection
when light
Problems:
Townsend
1.1, 1.2,
1.4,
1.8,
1.18,
1.20,
1.24,
1.27,
1.33,
9intensity.
1.5
MHz
w
circumstances?
low
an output power of 50,000 watts. Assum ing the
and
theatfrequency
inlocation.
order thatand
[; the
v 's
£oei(kx
-function
II)/) is a of rcst cannot beInac
=
determine
Planck
constant
work
metcr
the
receiver's
is reflected
by a material
ight is traveling. In1.34,
the next
chapter
we will start to see
how results
such as whose index of refraction is greater than that of the material inelectromagnetic
Problems
1.36,
1.40
What is w
t
(a)
broadcast anten na is located 100 km away fro m th e
solution?
sodium.
1.5.next
Thechapter
human eye
a signal
100 such assurface
which the light is traveling. In the
we can
willu sec"
startwith
to see
howof
results
is
the
ene
rgy
e derived for the case of non
relativisti
c
quantum
mechanics.
of th e me
receiver and, for ease of calculation , the antenna radiates 1.7. The output power of a d iode lase r in a DV D player
photons
per
second.
How far away
can
a I OO-watt A,
li ght
these
can be derived
for the case
ofWhat
non
relativisti
c quantum
mechanics.
of time
it would e
are
emitted
From
these
values
determine
the
wavelength
the
Verify that
=
is
a
solution
to
the
1.1 . (a) isotropica
£o
e;(h-Wl
)
£
k
1.2.
are
the
numerical
values
for
and
for
the
w
is
50
milliwatts.
How
many
photons
per
sccond
strike
1.14. Show that a photon that strikes a free elec tron at
lly, estimate the number of photo ns per cubic
bulb be seen by a dark-adapted eye? Assu me the light
binding
energy)
in each
cyc
let
frequency
and
evaluate
v,The
Exp
SlIggestion:
i-.),v.
thrcst
e DVD?
wavelength
= 660 nm.
wave equation
electromagnetic
wave
metcr at the receiver's location.
be
abso
rbed:
bulb cannot
is in o uter
space,
so that the light is not scattered
located
I
m
away
2.0 form
microwa
in the
J,
ax'
lution to the
n a medium
give n by
c' aI'
by the atmosp here. Also assu me that th e bulb is
atom
to be I A-.HS
Problems
1.8. A helium- neon lascr (i-. = 633 nm) used in a
reception.
to conserve
ene
1.7. The output power of a d iode lase r in a DV D player 1.3.
What musticthe
idthesa atofa awave
slitlength
be soofthat
monochromat
andwradiat
550 the
11111. first incident flu x is ab
of
the
doubl
e-sl
it
experiment
has
lecture
demonstration
Such
a
reacti
rcqu
ire
pe
r
second?
per
cycle?
Do
the
answers
(d)
1.12.
The
photoclc
From these
determine
the wavelength
A, the strike minimum
is 50values
milliwatts.
How many
photons per sccond
Use a reasonable
est imateli for
the diameterpattern
of th e observed
of values
the single-s
t diffraction
From
these
determine
wavelength
A, the
an outp
utand
powe
r indi
of 5cate
mi lli
watts.
How
manyo fphotons
per
1.1 . (a) Verify that £ = £o e;(h-Wl ) is a solution to the
to (b)
th
at
th e the
granularity
th el-statc
(c)Exprcss
effect
to appe
effect
for maxim
an
elec
the
encrgy
of
theocc
fina
ron
SlIggestion:
dark
-adapted
pupi
l o f the
eyc.
1.1
3. seem
The
i-. = 660 nm.
th e v,
DVD?
The wavelength
frequency
and evaluate
),v.
w
ith
light
=
of
wavelength
).
550
nm
urs
at ane lect
angle
emand
ittedevaluate
by iation
the lascr?
second
are v,
frequency
),v.
electromagnetic
rad
can
be
neglec
ted
in
these
delay
bee
= kc. (b) The wave equation in a medium
provided
w equation
wave
Why
can has
absorp
fro
m sodium
is 1
+ 1I/ 2C 4 and see whether it is po ssibtime
in the form J
le
of IS'
? The
diametcr
of a broadcasts
human haatir1000
is typica
ll y
circumstances?
low intensity. In cl
k Hz with
1.6.
An AM
rad io station
such as g1.8.
lassAwith
index of refraction"
is nm)
give n by in a
helium633
0.82 eV for rad ia
to conserve
energy
andofmomentum.
1.3. What must
the w neon
idth alascr
of a(i-.slit= be
so thatused
the first
A
beam
ofUV
wave
length
),ing
=w
197.0
nmfirst electromagnetic
1.9.
1.15. Supposewa
a
an
output
power
of light
watts.
Assum
the
50,000
100
Mm.
How
does
this
compare
in
size
ith
the
a
1.3.
What
must
the
w
idth
of
a
slit
be
so
that
the
determine
Planck
lecture demonstration of the doubl e-sl it experiment has
Such
reaction
ta ke
place
inpotential
th of
e photoelectric
1.5.
The
human
eye does
can uThe
with
a signal
100needed surface
sec"
falls
onto
a ametal
cathode.
sto
pping
meta
of the
t diffraction pattern observed
minimum of the single-s liII'
Compton
wavele
broadcast
anten
na
is
located
100
km
away
fro
m
th
e
width
a?
of
the
single-s
li far
t diffraction
pattern
observed
sodium.
an outp ut powe
r of 5 mi lli watts.
How many photons per minimum
photons
per
second.
How
away
can
aanode
I OO-watt
for
anfor
electron
in an
atom
or isinradiates
ali ght
solid.
-ofelectron
time it would
re
toeffect
keep
any
electrons
reaching
thc
2.08
V
initia lly
receiver
and,
easefrom
ofbound
calculation
,nm
the
antenna
w ith light of wavelength
). = 550 =0
nm occ urs at an angle
w ith
light
of
wavelength
). = 550
occ
urs
atlight
an angle binding energy) to
by
a
dark-adapted
eye?
Assu
me
the
bulb
be
seen
second ware=em
by the
lascr?
provided
wave
equation in a medium
kc.itted
(b) The
absorpt
occur
in of
thiofthe
s photo
case?
W
Whacan
t is lly,
theestimate
workion
funct
ion
cat hode
sur
face,
(a)Why
1.14.
Show
that
a
photon
irthe
sca
isotropica
the
number
ns
per
cubic
of IS' ? The diametcr of a human ha ir is typica ll y
is in station
odiametcr
uter space,
soathat
the at
light
is is
nottypica
scattered
of IS'
? The
of
human
ha
llvy =
1.4.
radio
broadcasts
a irfrequency
Abulb
located
I m away
f
such
as
g
lass
with
index
of
refraction"
is
give
n
by
u
in
eV?
What
is
the
velocity
of
the
fastcst
electrons
(b)
metcr
at
the
receiver's
location.
rcst
cannot
be
ab
,
energy
is
transfe
What
mustHow
be the
re lationshi
p between
the
wave
length A
by
theSuppose
atmosp
Also
assu
me apowc
er bulb
is =
that
100 Mm.
does
this
compare
size
w ith
ofUV
light of in
wave
length
),the
= 197.0 nm 9 1.5
1.9. A beam
100
Mm.
How
this
compare
in thsize
ith
the
MHz
wit hthedoes
ahere.
total
radi
ated
ofwP
1.15.
a photon
with
wavelength
equal
to th eatom to be I A-. Su
Kmax/
1IIC2
emitted
from
cathode?
Since
Note:
«20 I,kW.
and
the
frequency
v in order that [; = £oei(kx - II)/) is a
monochromat
ic
and
radiat
es
at
a
wave
length
of
550 11111.
incident flu x is abs
cathode.
sto pping potential needed (a)width
width a?falls onto a metala'£
a'£
II' The
a?
1.7.
The
output
power
of
a
d
iode
lase
r
in
a
DV
D
player
Compton
wavelength
makes
a
collision
with
a
free
What
is the wavelength
ofthe
this
radiation?
the
nonrelativistic
expressionA
for
ki netic
energy (b)
can What1.16. It takcs 3.
- - - - =0
Use a reasonable est imate for the diameter of th e
solution?to keep any electrons
is
50
milliwatts.
How
many
photons
per
sccond
strike
from
reaching
thc
anode
is
2.08
V
What is th e max
electron
initia
lly
at
res
t.
W
hat
is
th
e
energy
of
the
final
be utili
Avogadro's
number
of photo
(c) lIfophoton
ax'
aI'
is the
enezed
rgyhere.
of each
How
manynsphotons
dark
-adapted
pupi
f the eyc.in eV?
1.1 3. The maximu
SlIggestion: Expr
i-.
th
e
DVD?
The
wavelength
660
nm.
=
W of the catvhode
t is the
work functationa frequency
sur face, 1.4.
(a) Wha
radio
station
broadcasts
at
a
frequency
A
v
=
1.4. AWhat
radio
station
broadcasts
=
stri
ke
s
each
square
mctcr
of
the
surface
in
one
hour,
? How much
photon irthe
ttering How
angle many
is 180 photons
kinetic
fro m sodium is 1.8
each sca
second?
are emitted
must
be the re lationshi
p between
thewwave
A are emitted
k and
1.2. What
are
the
values
foru of
for length
the
in
the form
J
1.17.
The encrg
Hz 20
with
1.6.
Aniswit
AM
iointens.ity
station
broadcasts
atr 1000
1.5
MHz
hrad
a total
radi
ated
powc
of, P
kW. 0.82
in eV?
What radi
is the
velocity
the=fastcst
electrons 9what
(b)a numerical
I of
is the
average
th e beam
ink=
units
energy
transferred
to
the
electron?
9 1.5 MHz
wit
h
total
ated
powc
r
of
20
kW.
P
eV for rad iati
1.8. cyc
A heliumneon
lascr
(i-.
= 633
nm)receiver
used in arequires to
and
the
frequency
v in order that [; = £oei(kx - II)/) is a
A
in
each
le?
pa
nicular
radio
(c)
conserve
energ
(as
found
in
livi
electromagnetic
wavethe cathode? Note: Since Kmax/ 1IIC2 « I, (a)
an output
power
of 50,000 watts.
ing the
ofW/ml?
emitted
What
is the
wavelength
of Assum
this
Planck 's
is the from
wavelength
A of this radiation? (b) What
(a) What
of the A
doubl
e-sl itradiation?
experi
ment(b)
hasWhatdetermine
lecture
demonstration
Such
a
reactio
solution?
of
radiation
2.0 microwatts
to
provide
intelligible
we
be
worried
a
broadcast
anten3.na1 is
located
100 km away
fro m
th e
takcs
10
di ssociate
aHow
AgB
r molecule.
sodium.
thergy
nonrelativistic
expression
the many
ki neticphotons
energy can
is 1.16.
the
eneItutrgy
of reach
photon
in How
eV? many
many
photons
an
outp
powe
of 5cV
mi
lli watts.
photons
per
is the ene
of each photon
in eV? for
How
effect
for an elec
tha t operate
in t
receiver
and,
for ease
of1.5
calculation
, the antenna
radiates
1.10.
Use
Mi
llikan's
data
on
the
photoelectri
c
effect
reception.
How
many
9
MHz
photons
does
this
li ght requi
What
is
e maximu
m wavelength
of
red?
utili zed
here.
number
of photo
(c) If Avogadro's
aresecond
emitted
second?
How many
photons
are emitted
em
itted
by the
lascr?
aretheach
Why Show
can absorpt
k and
1.2.beWhat
the numerical
valuesphotons
for
for thens
1.14.
that a
are emitted
eacharesecond?
How many
arew emitted
isotropica
estimate
of photo ns
per cubic
(Fig.
1.14) 10lly,
obtain
a valthe
uenumber
for II , Planck's
constant.
stri ke s each square mctcr of the surface in one hour,
A
in
each
cyc
le?
pa
nicular
radio
receiver
requires
(c)
Suppo
(a) be
1.18.
metcr at the receiver's location.
rcst cannot
abso
electromagnetic
wave
in each
cyc le? (c) A pa
nicular radio receiver requires
1.17.
redwave
to break
bond 1.15. Suppose a
beamencrgy
ofUV requi
light of
length a), chemical
= 197.0 nm
1.9. A The
what is the average intens.ity I of th e beam , in units
photon to arrive
radiation
2.0
microwatts
to provide
1.11.
The work of
funct
ion of potassium
is intelligible
2.26 eV
of radiation to provide intelligible
2.0 microwatts
falls
onto aoutput
metal
cathode.
The
sto
pping
potential
needed
(as
in
living
tissofue)
typically
few
Shoul d 45
Compton
wavele
1.7.found
power
d is
iode
lase
r inclec
aa DV
D eV
player
Proble
ms
ofW/ml?
probability
that
reception.
many
9 1.5a MHz
photons
does
this
What
is The
theHow
maximum
kinetic
energy
of
trons
ejected
toiskeep
any electrons
reaching
thc
isstrike
2.08
electron initia lly
we
be
worried
abou
tmany
rad iation
damage
from
cell V
phones
reception. How many 9 1.5 MHz photons does this
50 milliwatts.
Howfrom
photons
per anode
sccond
is the probabi lit
from potassium by ultntviolet light of wave length
a'£
ax'
a'£
c' aI'
c'
wave length A
kx - II)/)
is a
nd w for the
answers
o f th e
in these
l of 100
OO-watt li ght
me the light
ot scattered
ulb is
th of 550 11111.
of th e
effect
seem
to
appear
instantaneously.
In
particular,
1.8.
A
heliumlascr
633
nm)Kmax/
used
in a « I,for no
1IIC2
emitted
from
theneon
cathode?
Since
Note:
Suppose
that
th(i-.
e =probability
amplitude
a to conserve energy
(a)
1.18.
by the atmosp
assupicture.
me thatcalculate
th e bulb the
is amount
surface
metal.Also
In this
of the here.
monochromat
ic and
radiat
es IatcV
a wave
length(aof
550 11111.
of
time it would
require
for
of energy
typica
l
Use
a
reasonable
est
imate
for
the
diameter
of
th
e
binding energy) to be absorbed by an atom in a meta l
emitted
cathode?
Note:
i Sincefracti
TheIfrom
work
funct
ion of1+
potassium
isKmax/
2.26
eV
A
-.the
atom1.11.
to be
What
on1IIC2
of«theI,
Suggestioll:
the nonrelativistic
expression
for the ki
netic
energyejected
can
What
is
the
maximum
kinetic
energy
of
clec
trons
incident flu x is absorbed by th e atom?
be Show
utili
zedthat
here.by
If Avogadro's
photo ns
(c)ultntviolet
(d)
from
potassium
lightnumber
of waveoflength
dark
-adapted
pupifrom
eyc. bul b. Take the area of the
located
I m away
a I-walt
, l o f the
atom to be I A-. Suggestioll: What fracti on of the
1.6.
An AM
io stationby
broadcasts
incident
flu x rad
is absorbed
th e atom?at 1000 k Hz with
an output power of 50,000 watts. Assum ing the
broadcast
na is located
100 km of
away
fro m th
e
electrons
1.1
3. The anten
maximum
kinetic energy
ejected
receiver
and, for
easeeV
of calculation
, the
antenna
radiates
fro m sodium
is 1.85
for radiat ion
of300
nm and
isotropica
lly,rad
estimate
the400
number
of photo
ns per
0.82
eV for
iation of
nlll. Use
thi s data
to cubic
g the
metcr at thePlanck
receiver's
location.
determine
's constant
and the work function of
fro m th e
sodium.
ns per cubic
energy is transfer
lecture demonstration of the doubl e-sl it experiment has
1.16. It takcs 3. 1
the
nonrelativistic
expression
foreven
netic
canvery
timephoton
delay
has
beenat
observed,
ith
light
of
to
arrive
a detector
isthe1/ki(1w
What
is the Such a reaction
+ i).energy
1.11. The work funct ion of potassium is 2.26 eV
anutili
outp
ut here.
powe r(c)
of 5If mi
lli watts. How
manyofphotons
Whatfor
is th
maxi
be
zed
Avogadro's
number
photoof
nsper
effect
aneelectro
circumstances?
low
intensity.
In
classical
physics.
the
energy
the
probability that the detector records a photon? (b) What
What is the maximum kinetic energy of clec trons ejected
itted mctcr
by theof
lascr?
second
are em
stri
ke s each
square
the surface in one hour,
Why can absorpt io
1.12. The
photoclcctrons
in light
the photoelectric
electromagnetic
wave
spread aout
uniformly
over the 1.17. The encrgy
is the probabi lit
y 0 1·is
detecting
photon
if thc probability
from
potassium by ejectcd
ultntviolet
of wave length
46 is the
Chapter
1: intens.ity
Light
I of th e beam , in units
what
average
1.5. The
human
eye
can
of 100 no
u sec" with a signal
effect
seem
to
appear
instantaneously.
In particular,
A the
beam
ofUVi ?In
light
of
wave
lengththe
), =probability
197.0 the
nm amount
1.9. of
(as found
in livin
amplitude
equals
(c)
Determine
of 1.15.
Suppose
a ph
surface
metal.
this
picture.
calculate
200 nm?
ofW/ml?
falls onto a metal cathode. The sto pping potential needed
we be worried
ab
Compton
waveleng
photons
perhas
second.
How far away
I OO-watt
li ght
time
delay
been observed,
evencan
w itha light
of very
of time
it
would
require
for
I
cV
of
energy
(a
typica
l
detecting
a electrons
photon iffrom
the probability
ampli is
tude
isV
to
keep
any
reaching
thc
anode
2.08
electron
initia
lly
at
tha
t
operate
in
th
1.10. Use Mi llikan's data on the photoelectri c effect
dark-adapted
eye?
me the
bulbintensity.
be seen by
light
low
In aclassical
physics.
theAssu
energy
of the
binding
energy)
towork
be absorbed
bythe
ancat
atom
in face,
a meta l
hode
sur
(a) Wha
photon irthe sca tte
(Fig.
1.14)t is10the
obtain
a funct
val ueion
forWII ,ofPlanck's
constant.
bulb is in o uter space,
that theout
light
is not scattered
electromagnetic
wave so
is spread
uniformly
over the
I u of
located
I m(b)away
a I-walt
bultheb. fastcst
Take electrons
the area of energy
the
Suppos
1.18. (a)
in eV?
Whatfrom
is the velocity
is transferre
00 k Hz with
tenna radiates
(a) Wha t is the work funct ion W of the cat hode sur face,
photon irthe
sca
SlIggestion:
thte operate
DVD? The
= 660 nm.range?
tha
in wavelength
th e I to 2 i-.gigahertz
amplitude Exprc
equal
200
nm? (b)
1.12.
The
photoclcctrons
ejectcd
photoelectric
u of in
in eV?
What is the velocity
the the
fastcst
electrons
in the form J
1.10.
Use Miper
llikan's
data(d)
on Do
the the
photoelectri
rcqu ire pe
r second?
cycle?
answersc effect
(Fig.
1.14)
II
,
10
obtain
a
val
ue
for
Planck's
to (b) and (c) indi cate th at th e granularity o f th e constant.
Proble
ms
45
electromagnetic rad iation can be neglec ted
in these
1.7. The output power of a d iode lase r in a DV D player
is 50 milliwatts.
How
many
photons
sccond
strikeat
1.14.
Show that a
photon
that
strikesper
a free
elec tron
th e DVD? The wavelength i-. = 660 nm.
,
--+i
photon to arrive
It takcsthat
1.16.
3. 1 ct
probability
What
th e maxim
is theisprobabi
lit y
stri nm?
ke s each square mctcr of the surface in one hour,
amplitude equals
200
1.1 3. what
Theismaximum
kinetic energy of electrons ejected
1.17. The encrgy r
the average intens.ity I of th e beam , in units
Figure 1
-- iion of300 nm and (as found in living
fro m ofW/ml?
sodium is 1.85 eV for
radiat
of
1+ i
we be worriedglass
abo
0.82 eV for rad iation of 400 nlll. Use thi s data to
1.10. Use Mi llikan's data on the photoelectri c effect
tha t operate in th e
from potassium by ultntviolet light of wave length
is the probabi lit y 0
1.24. (a) S
determine
Planck
's constant
and the work
function of
10 obtain
a val
for II , Planck's
constant. What
Suggestion:
is(Fig.
not 1.14)
a va lid
probabil
ityueamplitude.
Suppose
1.18. (a)
wavelength
sodium.
would
be the probabi lity of detecting a photon for this photonthickness
to arrive atd
1.11. The work funct ion of potassium is 2.26 eV
amplitude?
probability that the
is the
maximum
kinetic
of clec
tronselec
ejected
1.14. What
Show
that
a photon
thatenergy
strikes
a free
tron at
rcst 1.19.
cannot
be abso
Express
therbed:
complex number I , = (../3 + i)/ 2 inamplitude equals i
200 nm?
the form r e;". What about "2 = ( I + ../3i)/ 2? If these
where )c ' is
complex numbers are the probab ili ty ampli tudes for
A' = )./n,
photons to be detected, what is the probability in each
Note : In th
case?
Exprcss the encrgy of the fina l-statc e lect ron
SlIggestion:
axi mum
ource if there
nce? (b)
ngth 550 nm,
t and Maller
lex probability
revoluti ons
Feynman
deed
54 em.
of a thick
located
glass reflects
e glass, as
nitude of the
photon?
y ampl itude
photon of
plitudes th at
urface of the
Express your
kness d and
hat is the
ry to produce
nd
A in the
sa
t reflects
. The two
t. and the
alf-silvered
ansmits
oton in either
e position of
ne the
obability of
visibility
Pmin
are the
1+ i
wavelength
reflected.
Assume
that
amplitudes
th2atin
1.19. Express
I, =
the
complex
number
(../3
+ i)/and
surface?
(b) A
In isQED:
The Strange
TheO/}'
oj
Lig/II
+
r
e;".
form
What
about
(
I
../3i)/
2?
If
these
=
"2
the
involve
multiple
reflections
at
the
bottom
surface
of
(d) Show
that states that as the thickness of a thin the
Feynman
Maller
film
can be
neglected
yoprobab
ur calculat
Express
your
complex
numbers
are inthe
ty ion.
ampli
tudes
layer of glass increases
from
zero ili
thickness,
the for
A and r what
answer
as well
as the
thicknessindeach
and
photonsintoterms
be detected,
is the
probability
probabi lity of refl ection fi rstreaches
- i a value of 0.16
the
index of refraction 1/ of
the
acetone.
What
is
the
case?
layeri of glass is 5 mi lli onths of
when the thick ness of the 1+
minimum thickness of the coating necessary to produce
an inch. What index of refraction is being assumed? Take
1.20. refl
Rewri
te the
fo llowing
numbers
NOle:
zero
ection?
For thecomplex
air-acetone
and in each
the
o= rthe
light
in air
torebei ¢,Suggestion:
the
samex ,asy,What
you
iswavelength
notforms
a va lidz probabil
ityand
amplitude.
x
z
=
r,
+
where
of
the
iy
0.1.
acetone- glass surfaces r
determ
ined
in
Problem
1.22.
What
is
the
minimum
(c)
would
be
the
probabi
lity
of
detecting
a
photon
for
and ", are real. (a) ( I + 2i i (b) 1/ (1 + i) (c) )3 - this
4i
valuc
of
to first
produce
reflecattion?
d necessary
amplitude?
1.26.
Assume
that the
beamzero
splitter
A in the
/4.
eirr
(d)
Mach-Ze hnder interferometcr (Fig. 1.23) is a
1.25.
Suppose
that
acomplex
thin filmnumber
of acetone
1.19.
I , =(index
the
(../3
+ofi)/of2 in
1.21.
AExpress
certain
photodetector
resolve
thereflects
time
"thi
rd-sil
vered mirror,"
that is, can
a mirror
that
=
d
refractionn
1.25)
of
thickness
is
coating
a
thick
8
r e;".
What
( S.I +
../3i)/
2? Iftwo
these
= two-th
"210the form
of The
these
arriva
orathe
photon
to about
within
Two
one-t
hil rd
light
and
transmits
irds.
of glass
(index
ofanrefractio
n =ili1.50).
Take
the for as
plate
complex
numbers
are
the
probab
ty
ampli
tudes
detectors
are
used
in
anticoincidence
ex
periment,
mirrors at Band C reflect 100% of the light. and the
magnitude
of
the
amplitude
for isreflection
photons
to
detected,
theisprobability
in each
described
in be
Section
1.4.Dwhat
the ofa
ma xiphoton
mum
(a)
second
beam
splitter
at
is aWhat
traditional
half-silvered
from
the
top
or
the
bottom
surface
of
the
acetone
case? that
average
ralereflects
emission
fromand
source at
of photon
the transmits
if there
mirror
one-half
the light
normal
incidence
to be r and assume
that there is(b)an
is
any
hope
of
demonstrating
anticoincidence?
one-half. The probability of detecting a photon in either
1.20. Rewri te the
fo llowing
complex
numbers
in each
additional
change
of IT light
in
the
reflection
from
Asslimingphasc
the source
emits
Ofw3vclcngth
550the
nm,
photomultiplier
PM I or
PM2 varies
w ith the position
of
z = xsurface
z =acetone,
re i ¢, where
x at
, y,each
r,
forms
+
ofand
the the
iy and
top
bottom
of
the
since
what
is
the
power
output
source?
of
the
the movable mirror, say mirror B. Determ ine the
", surfaces
are real. (a)
2i i ing
(b)from
1/ (1 a+medium
i) (c) )3
- 4i
ofand
these
light( Iis+pass
with
a
maxi mum
irr / 4 . prObability and the minimum probability of
e
(d)
lower
of refraction
to one
witho/
a higher
index
of
Th e Strange
TheOlY
Light and
Maller
1.22. index
In QED:
obtaining a cou nt in, say, PM I . What is the visibility
Feynman states that the phase of the comp lex probability
refractio
Calcul photodetector
ate the probability
that a photon
ofof
1.21. An.certain
can resolve
the time
v =makes
Pmax about
- PTIlin
ampli tude for photons
36,000 revoluti ons
8
wavelength
is reflected.
Assume
that
amplitudes
S. Two
of these th at
arriva l oraAphoton
to within
10+ ?lIlin
Prn ax
per inch for red li ght. What
wavelength
is Feynman
involve
multiple
reflections
at the bottom ex
surface
of the
detectors
are used
in an anticoincidence
periment,
as
in this calculation?
DoesPmax
th is and
indeed
assuming
Pmin are the
fringes,
where
of
the interference
film
can be in
neglected
yo(a)
ur calculat
your
described
Section in1.4.
What ision.
the Express
maxi mum
: I inch = respectively,
correspondand
to red
light? Note
2.54 em.
maximum
minimum
that
answer
in rale
terms
and r emission
asprobabilities,
well asfrom
the thickness
and
average
of A
photon
the sourced if
there
a photon is cou nted by the detecto r, as the position of
the
index
ofphotodetecto
refraction
acetone.
What
the
1/ rof
(b)
is any
hope
of demonstrating
anticoincidence?
1.23.
One
is the
located
in front
of is
a thick
the movable mirror varies? Note: In the experiment of
minimum
thickness
coating
necessary
to
produce
of
the
Assliming
theand
source
emitsphotodetcctor
light Ofw3vclcngth
550 nm,
another
is located
piece
o f glass
Aspect et al. described in Section 1.5 the visib ility of the
NOle:
zero
reflis
ection?
For the
air-acetone
what
the
power
source?
ofincidence,
the
within
the
glass.
Atoutput
normal
the and
glass reflects
fringes is 0.987 ± 0.005.
0.1.
acetoneglass
surfaces
r
4% of the light. A photon is incident on the glass, as
QED: Th
e Strange
TheOlY o/magnitude
Light and of
Maller
1.22. In in
(a)a W
indicated
hat is the
the
Figure Fig.
1.431.42.
shows
Michelso
n in terferometer
1.27.
1.26.
Assume
thatthat
thethe
first
beamofsplitter
at A
inprobability
the
Feynman
states
phase
the
comp
lex
a/llp/ill/de
for
refl
ection
of
the
photon?
probability
refractio
n.
Calcul
ate
the
probability
that
a
photon
with a movable
mirror AlII, a fixed
mirror
Nh . and a of
Mach-Ze
hnder
interferometcr
(Fig.
1.23)
is arevoluti ons
Problems
47th at
ampli
tude
for
photons
makes
about
36,000
A
wavelength
is
reflected.
Assume
that
amplitudes
(b)
Wha
t is theM"mag
nitudeis of
the probabi
amplthat
itude
beam
splitter
which
a half-sil
vcredlity
mirror
"thi
rd-sil
vered
mirror,"
that
is,
a
mirror
that
reflects
inch for
red of
limultiple
ght.
What
wavelength
is Feynman
involve
reflections
at the bottom
surface of the
foper
r transmission
thelight
photon?
transmits
one- half
the
and reflects one-half the light
one-thi
rd the
and
transmits
two-th
TheExpress
two your
film
canM
be
inmirror
yo
ur th
calculat
ion.
in light
this
calculation?
Does
isirds.
indeed
assuming
Movable
I neglected
it independent
of
thewell
direction
of the lig
ht.
incident upon
A
r
answer
in
terms
and
as
as
the
thickness
mirrors
at
Band
C
reflect
100%
of
the
light.
and
correspond to red light? Note : I inch = 2.54 em.the d and
The source
emi
ts
monochromatic
light
of
wavelength
A.
thesplitter
index
ofat
refraction
of the acetone.
is the
1/probability
refractio
n. Calcul
that What
a photon
of
second beam
Dateisthe
a traditional
half-silvered
There
are
two
paths
that
light
can
fo
llow
from
th
e
source
minimum
thickness
coating
necessary
to
produce
of
the
A
wavelength
is
reflected.
Assume
that
amplitudes
1.23. that
One reflects
photodetecto
r is the
located
front
of a thick th at
mirror
one-half
light inand
transmits
NOle:
zero refl
ection?
For
the
air-acetone
and path
as
indicated
in
the
figure.
Note
that
topiece
the detector.
involve
multiple
reflections
at
the
bottom
surface
of the
and another
is located
o f The
glassprobability
ofphotodetcctor
detecting
a photon
in either
one-half.
Mthe
Half-si
lvered
acetoneglass
surfaces
0.1.
r splitter
I within
includes
travel
from
beam
M,
to
the
film
can
be
neglected
in
yo
ur
calculat
ion.
Express
your
the glass.PM
AtI/ normal
reflects
photomultiplier
orS mirror
PM2 incidence,
varies w iththe
theglass
position
of
d
A
r
answer
in
terms
and
as
well
as
the
thickness
and
movable
mirror
MI
and that
back
tofirst
thebeam
beam
splitter,
M2
Fithe
xed
mirror
1.26.
Assume
theincident
at A while
inasthe
of the
light.
A photon
is
onsplitter
glass,
4%movable
B. Determ
the
mirror,
say mirror
ine
the
Path
21/ of
theMach-Ze
indextrave
ofhnder
refraction
the acetone.
What
is the
path
2
includes
l
from
the
beam
splitter
to
M,
interferometcr
(Fig.
1.23)
is
a
(a) the
indicated
in Fig. 1.42.and
W hat
is the magnitude
ofthe
the
maxi
mum
prObability
minimum
probability
of
minimum
thickness
coating
necessary
toume
produce
the
1---1of
,---1
"thi
rd-sil
vered
mirror,"
is, asplitter.
mirror
that
reflects
mirror
and
back
to
th
ethat
beam
Ass
fixed
M2
a/llp/ill/de
for
refl
ection
of
the
photon?
probability
obtainingzero
a one-thi
cou
in,thesay,
PM
What
is two-th
the visibility
I .transmits
NOle:
reflnt
ection?
the air-acetone
and
rd
light
and
irds.
two
the
introduces
aFor
phase
change
iT The
for
(b)beam
Whaacetonetsplitter
is the glass
mag
nitude
of
the
probabi
lityofampl
itude
surfaces
0.1.
r
mirrors at Band
Cmax
reflect
100%
of
the
light.
and
the
TIlin
=I photon?
P
- Psource
light
that follows path
from
the
to the detector
fo r transmission
ofvthe
second beam
splitter at D is a traditional half-silvered
+ ?lIlin
Pthe
relative to
light
that fothat
ll ows
path
2beam
fromsplitter
the source
the
rn ax
1.26.
Assume
first
A intothe
mirror
that reflects
one-half
the light andattransmits
detector.
Also
assume
mirrors
and
lv/Pmin
reflect
Mach-Ze
hnder
interferometcr
(Fig.
isare
a in
fringes,
whereofPMl
and1.23)
theeither
of
the interference
one-half.
The the
probability
detecting
a 1photon
max
Figure
1.43vered
The
Michelson
interferometer.
"thi
rd-sil
mirror,"
that
is,
a
mirror
that
reflects
100%
of
th
e
light
incident
upon
them
and
the
photomultiplier
PM
or
PM2
varies
w
ith
the
position
I
maximum and minimum probabilities,
respectively, that of
one-thi rd the light and transmits two-th irds. The two
P = 0. 16 sin (2iTd I):)
I"," ..
I
where )c ' is th e wavelength of light in glass, i.e.,
A' = )./n, where II is the index of refraction of glass.
M assume
Half-si
1.42calculation
Part ial reflecti
on oflvered
light
single surface
by amagnitude
NoteFigure
: In this
that the
of
/ S mirror
of
glass.
the ampli tude for reflection from the M2
top Fi
orxed
themirror
bottom
and2that there is an add itional
surface of the glass is 0.2Path
1.24. (a) Show that1---the
probab
il ity from
ofa photon
1,---1
phase change of iT in the reflection
the top of
surfacc.
wavelength
being
refl ected that
fromarise
a th in
layer
of glass of
Also
assume),that
amplitudes
from
multiple
thickness dbetween
at normal
is givensurfaces
by
reflections
theincidence
top and bottom
of the
glass can be neglected in your 2ca lculation. Given the
P = 0. 16 sin (2iTd I):)
resu lt of Probl em 1.23, it is okay to approximate Ihe
Figure the
1.43amplitude
The Michelson
interferometer.
magnitude
forlight
transmission
as one.
where )c ' isof
th e wavelength
of
in glass, i.e.,
l-liw:
extra distance
does light
travel in of
being
)./n, where
the index
of refraction
glass.
A' = What
photodctector
PM II(aisphotomultiplier)
is 100% efficient
reftected
from
the
bottom
surface
relat
ive
to
the
top
Note
:
In
this
calculation
assume
that
the
magnitude
as we ll. (a) Use the principles of quantum mec hanicsof
surface?
(b)
In
QED:
The
Strange
TheO/}'
oj
Lig/II
and
ampli tudetheforprobability
reflection from
top or
the bottom
tothedetermine
that athe
photon
entering
the
Feynman
states
thatand
as the
thickness
ofadd
a thin
Maller
of
the
glass
is
0.2
that
there
is
an
itional
surface
interferometer is detected by the photodetector. Exp ress
layer
ofchange
glass increases
from
zero thickness,
the surfacc.
phaseanswer
the
iT in
your
A. (b) Find
, I" the
in of
term
s of
threflection
e lengths IIfrom
andtop
probabi
lity
of
refl
ection
fi
rst
reaches
a
value
of 0.16
Also
assume that
that
arise), from
12 and
an
expression
for IIamplitudes
in terms of
such multiple
that there is
ness of thetoplayer
glass surfaces
isProblems
5 mi lli of
onths of
when
the thick
reflections
between
andof
bottom
100%
probabi
li ty th aithethe
photon
is detected
by the the47
an inch. What index of refraction is being assumed? Take
glass can be neglected
in your
lculation.
photodeteetor.
(e) Suppose
that ca
the
movableGiven
mirrortheis
air
the
wavelength
o
rthe
light
in
to
be
the
same
asIhe
you
resu lt of
Probl em
it is okay
to approximate
Movable
Idistance
byM
a1.23,
shifted
upward
1. /mirror
6 from
the position(s)
determ
ined of
in the
Problem
1.22. for
What is the minimum
(c) transmission
magnitude
amplitude
as one. that
that
you de lermined
in pa rt (b).
Find th e probability
valuc of d necessary to produce zero reflec tion?
l-liw: What extra distance does light travel in being
the photon is detected at the photodetector in this case.
reftected from the bottom surface relat ive to the top
1.25. Suppose that a thin film of acetone (index of
A beam
monM
ochromati
c light
fromojaLig/II and
1.28.
surface?
(b) InofQED:
The
Strange
TheO/}'
Half-si
lvered
d is coating a thick
refractionn = 1.25) of
thickness
mirror
heliumneon laserstates
(i. / = Sthat
633
) illuminates
a do
uble
asnmthe
thickness of
athe
thin
Maller Feynman
Take
plate of glass (index of refractio n = 1.50).
M2
Fi
xed
mirror
sli
=
t.
From
there
the
light
travels
a
di
stance
D
10.0
m
layer of glass increases from zero thickness, the
Pathfor
2 reflection ofa photon
magnitude of the amplitude
If the
toprobabi
a screen.
distance
between
interference
lity (a)
of refl
ection
fi rst
reaches
a value
of 0.16
1,---1
from the top or the 1---bottom surface
of the acetone at
= glass
maxima
the ness
screen
to layer
be 8 of
10.0 mm,
what
sho uld
of isthe
is 5 mi
lli onths
of
when theonthick
normal incidence to be r and assume that
there is47an
Problems
d between
(b)assumed?
be
e distance
the two isslits?
What wou
ld
of refraction
an thinch.
What index
being
Take
additional phasc change of IT in the reflection from the
you
sec on the screen
sheet
ofccllophallc
were
a thin
iflight
air
the wavelength
o rthe
in
to
be
the
same
as
you
Movable
top and the bottomMIsurface
ofmirror
the acetone, since at each
placed
over
one
of the slits
so(c)
thatWhat
there were
2.5 more
determ
ined
in
Problem
1.22.
the minimum
of these surfaces light is pass ing
from aismedium
with a
wavelengths
wit hinThe
thetoMichelson
cellophane
than
within
valuc ofFigure
produce zero
tion?a layer of
d necessary
interferometer.
lower
index
of 1.43
refraction
to one with
areflec
higher
index of
Problems
47
air of the same thickness? (Assume the interference
1.25. Suppose
that(aaare
thin
of
acetone
(index
of
M
Half-si
lvered
maxima
in quePM
stion
at film
only
a small
with
photodctector
photomultiplier)
isangle
100%
efficient
/ S mirror
M
Movable
mirror
I
=
d
refractionn
1.25)
of
thickness
is
coating
a
thick
respect
laser
d irection.)
as
we ll. to
Use
the beam
principles
of quantum
mec hanics
(a)the
M2 Fixed mirror
2
of glassthe
(index
ofPath
refractio
1.50). entering
Take thethe
toplate
determine
probability
that na =
photon
1.29.
Suppose
thatamplitude
the two
very
narrow slits
(widths
1---1,---1
magnitude
of
the
for
reflection
ofa
interferometer is detected by the photodetector.photon
Exp ress
A) the
in the
double-slit
experim
ent
are
not
the
same
«from
topinorterms
the bottom
of
the
acetone
at Find
your answer
A.
II
,
I"
(b)
ofMthHalf-si
e surface
lengths
and
lvered
ili
ty
amplitude
fo
r
a
pho
ton
width
and
that
the
probab
normal
incidence
r andofassume
there
an ofis
/ S mirror
12 and that
an
expression
for IItoinbeterms
), such
thatisthere
A
wavelength
to
strike
a
photomultiplier
centered
at
a
IT in the reflection
additional
phasc
from
M2 Fixed by
mirror
100%
probabi
li ty change
th ai theof
photon
is detected
the the
Path
2
particular
point
P in
thMichelson
e detection
plane thatsince
makes
an
Figure
1.43
The
interferometer.
top and the
bottom
surface
of the
at each
photodeteetor.
(e)
Suppose
that
theacetone,
movable mirror
is
1---1,---1
angle
with
the
hori
zontal
from
onc
slits
is
larger
e
of
the
of
these
surfaces
light
is
pass
ing
from
a
medium
with
shifted
upward PM
by (a
a distance
1. / 6 from
the efficient
position(s) a
photodctector
photomultiplier)
is 100%
by
a fa index
ctor ofof.Ji
Ihan
fortothe
olher
Determ
ine the
lower
refraction
one
withs lit.
a higher
index
of
in pa rt (b).
Find thmec
e probability
that
that
as you
we ll. de
Use the principles
of quantum
hanics
(a)lermined
visi bility
to
determine
the
probability
that
a
photon
entering
the
the photon is detected at the photodetector in this case.
v = byPm3x
Pmin
interferometer is detected
the -photodetector.
Exp ress
POlin
max +c II
yourAanswer
A. (b)
, I" and
inof
terms
of
thMichelson
ePlengths
beam
monThe
ochromati
light
from
a Find
1.28.
Figure
1.43
interferometer.
II
12
an
expression
for
in
terms
of
and
),
such
that
is
helium- neon laser (i. = 633 nm ) illuminates there
a do uble
photodctector
PM
(athe
photomultiplier)
is 100%
efficient
100%
probabi
li
ty
th
ai
photon
is
detected
by
the
sli t. From there the light travels a di stance D = 10.0 m
the principles
ofmovable
quantummirror
mec hanics
as we ll. (a) Use
photodeteetor.
(e) Suppose
that the
is
to a screen. (a) If the distance between interference
shifted
upward the
by aprobability
distance 1. / that
6 from
the position(s)
to determine
a photon
entering the
maxima
screen
is rtto(b).
be
8 = th10.0
mm, what
uld
that
youon
dethe
lermined
in pa
e probability
thatsho
interferometer
is detected
by Find
the photodetector.
Exp
ress
d
(b)
be the
th
e
distance
between
the
two
slits?
What
wou
photon
is
detected
at
the
photodetector
in
this
case.
your answer in terms of th e lengths II , I" and A. (b) Findld
youansec
on the screen
sheet
ofccllophallc
were
a thinof
12 and
expression
for II inif terms
), such that there
is
1.28. A beam of mon ochromati c light from a
placed
over
one liof
the
slits
so that isthere
wereby2.5
more
100%
probabi
ty
th
ai
the
photon
detected
the
helium- neon laser (i. = 633 nm ) illuminates a do uble
(e) light
Suppose
movable
mirror
is of
wavelengths
wit the
hin
the cellophane
than
amlayer
sliphotodeteetor.
= 10.0
t. From there
travelsthat
a dithe
stance
Dwithin
by
a
distance
upward
1.
/
6
from
the
position(s)
airtoshifted
ofa the
same
thickness?
(Assume
the
interference
screen. (a) If the distance between interference
movable
mirror,
say mirror
Determ
ine the
a photon is the
cou
nted
by
the detecto
r, asisB.the
position
of
photodctector
PM
photomultiplier)
efficient
mirrors
at (a
Band
C reflect
of100%
the light.
and theof
maxi mum
prObability
and100%
the minimum
probability
mirror
varies? Note:
In the experiment
of
Use
the principles
of
mec hanics
asthewemovable
ll. (a)
second
beam
D quantum
isPM
a traditional
half-silvered
obtaining
a splitter
cou nt in,atsay,
visibility
I . What is the
Aspect
et
al.
described
in
Section
1.5
the
visib
ility
of
to determine
thethat
probability
that a photon
the the
mirror
reflects one-half
the lightentering
and transmits
v
= Pmax - PTIlin
fringes
is
0.987
0.005.
±
interferometer
is detected
by the of
photodetector.
Exp ress
one-half.
The probability
detecting a photon
in either
Prn ax + ?lIlin
photomultiplier
or PM2IIvaries
w ithA.the(b)position
your answer
, I" and
in term s of PM
th e Ilengths
Find of
shows a Michelso
n in terferometer
1.27. Figure
Pmax and Pmin are the
fringes, where
of 1.43
the interference
the movable
mirror
Determ
inethere
the is
an expression
for II inmirror,
termssay
of 12
and B.
), such
maximum
andAminimum
probabilities,
respectively,
that
with a movable
mirror
a
fixed
mirror
and a
lII,
Nhthat
.probability
maxi
mum
prObability
and
the
minimum
of
100% probabi
li ty th is
ai cou
thented
photon
isdetecto
detected
by
the
a
photon
by
the
r,
as
the
position
of
beam splitter
which
is
a
half-sil
vcred
mirror
that
M"
obtaining
a cou nt in,that
say,the
PMmovable
ismirror
the visibility
I . What
photodeteetor.
Suppose
is of
Note:
the (e)
movable
mirror varies?
In the
experiment
transmits one- half the light and reflects one-half the light
Aspect
al. described
Section
1.5position(s)
the visibility of the
shifted upward
by et
a distance
from
max
v 1.=of/ 6inPthe
PTIlin
- the
incident upon
it independent
direction
of the lig ht.
pabe
rt 8(b).
Find
th eangle
probability
that you
lermined
fringes is 0.987
0.005.Find
± (b).
= a10.0
maxima
the
screenare
isintoat
mm,
what with
sho uld that
maxima
inonde
que
stion
only
small
Prn ax +
that you de lermined
in pa rt
th e?lIlin
probability that
The source emi ts monochromatic light of wavelength A.
the
photon
is
detected
at
the
photodetector
in
this
case.
d between
be th e to
distance
two slits? (b) What wou ld
the laser
beam the
d irection.)
the photonofisthe
detected
at1.43
the
photodetector
this
case.are the respect
interference
fringes,
Figure
shows
Michelso
nininand
terferometer
1.27.
There are two
paths
that
light
cana where
fo llowPmax
from
th Pmin
e source
you sec on the screen if a thin sheet ofccllophallc were
photo n is detected. ilf 3
together and (b) using geometry to "add the arrows"
abi lity that a
= 2 = e / by (0) adding the real and imaginary pieces
1.34. Starting from first principles, show that the
representing each of
these
numbers.
Check
probabi
litycomplex
that a photon
of wavclcngth
A hitsthat
a
(b) using geometry to "add the arrows"
together
and
1.30. Suppose that a thin piece of glass were placed in
yo ur results fo r thephotomultiplier
mag nitude and
phase
theP in
comp
lex
centered
on a of
point
the detection
representing
each of these
complex
numbers.
Check that
front of the lower
slit in a double-slit
apparatus
so that
the
plane that makes an angle f) with the horizontal for a
nu mber::: I + 12 agree.
ere placed in
yo
ur
results
fo
r
the
mag
nitude
and
phase
of
the
comp
lex
grating
composed of three very narrow slits each
amp litude for a photon o f wave length ).. to reach that sli t
atus so that the
separated by a distance d is given by
nu mber:::
0 I + 12 agree.
differs in phase by 180 with the amplitude to reach the
1.33. A photon with wave length A is incident on a single
reach that sli t
= ,.'(probability
Prob the
1 + 4 cos</> + amplitude
4 cos' </»
slit.
in detail the interference pattern on
slit of finite width a. Calculate
e to top
reach
the(a) Describe
1.33. A photon with wave length A is incident on a single
1.34. Starting from first principles, show that the
where r'
is the probability
thatatthea photon
would strike
the screen. A t what angles will there be bright frin ges?
fo r the photon to strike
a detector
located
di stant
Problems
49
nce pattern
on
slit of finite width a. Calculate the probability amplitude
probabi
lity that a photon
of wavclcngth
A hits a
thethe
photomultiplier
with
a single
sl it open and
by
point P at angle e photomultiplier
in
detection
pla
ne
integrat
ing
(b) What is the minimum thickness of glass requi red,
centered
on
a
point
P
in
the
detection
ight frin ges?
fo r the photon to strike a detector located at a di stant
</> = kdsinl} = 2ndsin l}/ A.
fromisfirst
principles, show
that the
that makes
this lresprobability
ult angle
reduces
to amplitude
thethe
double-slit
result
across
the slit the Verify
infinitesima
assuming
the index
of refraction1.34.
for Starting
the glass
,,"
f) with
plane that
an
horizontal
for a
by
point P at angle eprobabi
in thelitydetection
pla
ne
integrat
ing
s requi
red,
that a photon of wavclcngth A hits a
for
N
2.
=
(1.60)
grating
composedtheofprobability
three verythat
narrow
slits each
a photon
is detected
1.35. Determine
across the slit thephotomultiplier
infinitesima l probability
amplitude
ass is1.31
,," . (a) A monochromatic
on a point
P in the detection
at the location
the firs
of a three-sli t grating
separated
by a of
distance
is given by
·'(1
.tdminimum
0)
light source S ofcentered
wavelength
tI,
XSIn
=The
(li r+
d:::
plane that makes an angle f) with the horizontal for a
1.39.
ght from
sod ium
ists of two
thecons
magnitude
ifpthe
thirdyellow
slit is closed.
Assume
of the
a Prob
A is located to th e left or an opaque
screen
with
two very
= nm
cos</>
cos'
</»as the
,.'(due
1 +to4589.6
4known
+
grating
composed
of three
very
narrow
slits
each
wavelengths,
589.0
and
nm
,
probability
amplitude
each
sli
t
is
r.
Suggestion:
·'(1
.
0)
of wavelength
tI,
XSIn
=
(
r+
d:::
prigh tby
separated
a distance d is given by
sodium
doublet.
general,
when incident
uponfrom
a each
Inhow
narrow sl its of equal width. To the
of
Start by
showing
the complex
tudes
a the screen in
with two very
r'the
where
is point
the probability
that theampli
photon
would strike
for the photon to reach
P by
passing
through
diffraction
two
slightly
diffe
sli t add up grating,
to zero atlight
the of
fi rst
minimum.
Whatrent
is the
the
detection
plane
atProb
is
a
photomultiplier
point
P.
The
the a
photomultiplier
with a single
sl it of
open and
= ,.'( 1 + 4 cos</> + 4 cos' </»
the screen in
distance
below
theamp
top
elx, which is located
x and
wavelengths,
say A
generates
A +third
l:iA,
resulting
am plitUde
if the
litude the
ismaxima
eli minated?
for Sthe
photon
to reach
theIpoint
Pand
by passing through
</> = kdsinl} = 2ndsin l}/ A.
distance
between
and
P
along
the
path
is
elt
sati
sfying
sin
l}
iliA
and
sin(1}
+
d
d
t-I})
=
=
point P. The
where r' is the probability that the photon
would
strike in Fig. 1.44. The distance d) is the distance
slit,
as
shown
apath
distance
the top of the
elx, which is located
. below
2 is d x2with
1.36.+ Determine
the probability
thatinteger
a photon
is detected
where the
the
t-A), respectively,
111(1-.
III labels
the photomultiplier
a single sl it open and
is eltthe
anddi stance between Sand P along
Determine
the the
probability
a photon
is detected
1.35. in
trave led by the photon
reaching
point that
P Show
from
the
at theof
first
ofamaximulll.
five-slit grating
ifthat
the the
bottom
slit, as shown
in Fig.
1.44.
The
distance
order
interference
theminimum
</> = kdsinl}
2ndsin
l}/ P
A. d) is the distance
=photon
Show
that
the
probability
of
detecting
the
at
at the
locationprobability
of the firs t minimum
of a three-sli
t grating
is d 2 .
top ofthe
the slit. Show
that
the
three
sl itsthe
arethe
closed.
Assume
thedetecting
of/ dthecosl}.
dispersion
of
grating
t-8 /of
t-A
ismagn
givenitude
by III
trave led by the photon in reaching the point P from
is
given
by
if the third slit is closed. Assume the magnitude of the
probability
slit is
r. be
Suggestion
:
photon at P
photon is is
detected
1.35. Determine the probability that a photon
t-I} amplitude
« 1. Thusdue
thetodieach
spersion
can
increased
given byAssume
top of the slit. Show
theofprobability
of detecting
the
probability
amplitude
due to each proba
sli t is r. Suggestion:
Start
by show
how the
at thethat
location
the firs t minimum
of a three-sli
t grating
by
reducing
theingseparat
ion complex
d between the bility
slits in the
.each
, slit
Start
byandlor
showing
how
the
complex
ampliattudes
from each
Prob =
2,.2 (I by
costhird
+
4» slit is closed. Assume the magnitude of the
photon
is given
amplitudes
from
add
up order.
to zero
the first
if the
grating
working
at higher
Verify that this res ult r
(1.60) for N = 2.
1.39. The yellow li ght
wavelengths, 589.0 nm
sodium doublet. In gen
diffraction grating, ligh
wavelengths, say A and
u
satiVerify
sfying that
l} =res
iliA
d sinthis
for respective
N = 2.
(1.60)
+ t-A),
111(1-.
order of the interferenc
1.39. The
li
dispersion
of yellow
the gratin
wavelengths,
t-I} « 1.589.0
Assume
Thu
In
by sodium
reducingdoublet.
the separa
grating
andlor grating,
working
diffraction
wavelengths, say A
1.40.
sati Light
sfyingofd wavelen
sin l} =
the111(1-.
normal
on a respec
transm
+ t-A),
between
t, as sh
order each
of thesliinterfe
to dispersion
the normal of
willthe
diffr
gr
Assume t-I} « 1. T
by reducing the sep
grating andlor wor
1.40. Light of wav
•
25 111. a
sli
t
add
up
to
zero
at
the
fi
rst
minimum.
What
is
the
the normal on a tra
minImum
.
probability amplitude
Z I'Zp
r - -, . due to each sli t is r. Suggestion:
ct-if the third amp litude is eli minated?
resulting
amofplitUde
between each sli t, a
by
how
the
a complex ampli tudes from each
• showing
25
111.the
where,.2 is the probability that aStart
photon
strikes
I-. is incident at an angle", to
Light
wavelength
1.40.
1.37. Determine the probabi li ty that a photon is detected
Z I'Zp
r -at the
-, to the normal will
t add
upObta
to zero
What is the
the
normal on a transmission grating with spacing d
w ith a single slisli
t open.
in ctan fi rst minimum.where
1.36.
Determine
theofa
probability
that a if
photon
is detected
at
the
first
min
imum
six-slit
grating
two
the bottom
rikesphotomultiplier
the
resulting am plitUde if the third amp litude
is eli minated?
between each sli t, as shown in Fig. 1.45. At what angles ()
expression for 4> in terms of eI" el2 , and A. (6) Now
at theare
first
minimum
ofathefive-slit
gratingthe
if the
bottom
closed
. Assume
magnitude
probability
toslits
the normal
diffraction
maxima bcoflocated?
ta in an
sineach
kawill
where
three
its are
closed.
Assume
the magn itude
amp lisltude
due
to
slit is r. Suggestion:
Startofbythe
1.36. si
Determine
the probability
that a photon is detected
Figure 1.45 Lig ht inc
suppose
that
thin
pieces
of
partially
lvered
glass
are
ct=
(6) Now
showing how
th2
e complex
fro m :
probability
amplitude
due probabi
to eachlity
slitamplitudes
is r. Suggestion
at the first minimum
ofa
five-slit
grating
if
the
bottom
ka sin
placed
in this double-slit
ed glass
are in front of each of the slits
each slit
add up
zerothe
at the
fi rst minimum.
Start
by show
ingtohow
complex
proba bility
three slct=
its are closed. Assume the magn itude of the
1.41. Figure 1.46 dep i
experi
ment.
AsslIme
the
top
the glass covering
2 slit
amplitudes
from each slit
add upattothe
zeroangles
at the first
probability amplitude
due to each slit isVerify
r. Suggestion
:
that minima
in the probability
occur
ouble-slit
observation
of interfer
1.38. For a grating with N equally spaced narrow
while
1/2 the light incident upon
Start byitshow
ing the
how glass
the complex proba
bility in (1.17). Evaluate
minImum .
monochromatic light o
given
the
probability
in
the
limit
that
e toptransmits
slit
slits,
the
amplitude
for
detecting
a
photon
with
a
Verify that minima
in
the probability
occur
theatangles
amplitudes
each
slit add
up toatzero
the first
covering the bottom slit transmits
light
incident
1/4 thefrom
narrow slit S. The slit
al poliint
photomu
lt iplier centered
inthe
the
detecti is
ondetected
plane
I, that is, a 1.37.
Figure
1.9
shows
how
ka
A.Determine
ile the
glass
the probabi
ty Pthat
a photon
minImum . the probability in the limit that
given
in
(1.17).
Evaluate
mirror. Derive an expr
upon it. D etermine the probability that a photon of
is
given
by
minaimum
a six-slit
f) for
= A,ofa
va ries atwithe
thfirst
5>", grating
and if the bottom two angles I} at which brig
ight incident
that is, a1.37. A.
Figure
1.9 shows how probability
the
ka the I,photomultiplier
Determine
the probabi li ty that a photon is detected
wave
length
A
hits
in
thi
s
case.
ikdl
itPthe
slits
are
closed
.
Assume
magnitude
the I}4I]
probability to constrllctive interfer
r
e
e
e2it/l
Zp
+
... +ofe i (N=
+
+
hoton of
aif=thelOA.
a = ofa
f) min
probability va riesat wi
forimum
A, asix-slit5>",
and
theth
first
grating
bottom two
amp li tude due to each slit is r. Suggestion: Start by
of
rand
Express
your
answer
in
terms
4>.
Figure
directly
from1.45
the Lig
sli thta
hi s case.
Not ice that
each
in thisprobabi
series of
can be fro m
slits are closed . Assume the magnitude of the probability
a = lOA.
showing
how
th eterm
complex
lityterms
amplitudes
from the mirror. Simp
obtained
fromLig
by
by
amp li tude due to each slit is r. Suggestion: Start by
Figure
incident
on multiplying
a grating.
each
slit 1.45
add
uptheht
toone
zeropreceeding
at attheangle",
fi rstit minimum.
possible.
Note: Experi
1.41. Figure
1.46 d
e i ¢. Thus it is a geometric
showing how th e complex probabi lity amplitudes fro m
.p series that can be summed.
band
(destructive
inter
observation
of
inte
Show
that
the
probability
of
detecting
a
photon
is
given
each slit add up to zero at the fi rst minimum.
1.38. For a grating with N equally spaced narrow
this?
Eval
Suggestion:
Figure
dep
icts
Lloyd's
mirror
for
the
1.41.
1.46
.p
lig
monochromatic
by
slits, the amplitude
for detecting
photonscreen
with of
a
paths shown in the figu
observation
of interference
upon aadistant
1.38. For a grating with N equally spaced narrow
narrow slit S. The
2 P in the detecti on plane
al po
photomu
lt
iplier
centered
int
sin
the point P is sufficie n
monochromatic light,."'of
7 wavelength
- ,.2 _ _'_), from a single
slits, the amplitude for detecting a photon with a
mirror. Derive an e
-p-p sina 2distance
1!.
is givenslit
byS. The slit
point P can be taken to
d
narrow
is
located
above
the
photomu lt iplier centered al po int P in the detecti on plane
angles I} at which b
mirror.ZpDerive
an expression
terms
of' A+and
the
d for
e i (N+ ...
I}4I]
= r eikdl
+ eitP +ine2it/l
is given by
to constrllctive inte
angles I} at which bright bands appear on the screen due
directly from the s
Zp = r e ikdl
+ eitP + e2it/l + ... + e i (N- I}4I]
to
constrllctive
interference
from
lightofreaching
P be
Not
ice that each
term in this
series
terms can
from the mirror. Si
directly
t andpreceeding
light reachiitngbyPmultiplying
by reflectionby
obtainedfrom
fromthetheslione
Not ice that each term in this series of terms can be
possible. Note: Exp
i
from
the mirror.
li fy your
ex pression
as
geometric
it is a Simp
series
that can as
bemuch
summed.
e ¢. Thus
obtained from the one preceeding it by multiplying by
band (destructive i
possible.
Experimentaoflly,detecting
it is observed
that is
a dark
Note:
Show
that
the
probability
a
photon
given
i
e ¢. Thus it is a geometric series that can be summed.
this? Suggestion: E
band
(destructive interference) occurs as I} -+ O. Why is
by
Show that the probability of detecting a photon is given
paths shown in the
this? Suggestion: Eval uate the path2 difference fo r the two
by
sin
the point P is suffic
to '_arrive at P. Assume
paths shown in the figure
__
,."' 7 -for,.2li ght
-p-p sin 2 1!.
point P can be take
sin2
the
point
P
is
sufficie
ntly
far
away
that
the
two
rays
to
the
_ _'_
,."' 7 - ,.2 Single-slit
Figure
diffraction.
-p-p1.44
sin 2 1!.
point P can be taken to be paralic\'
=
,
=
e
e
«
«
«
«
[I
=
[I
=
,
[I
,
Figure 1.44 Single-slit diffraction.
,