Modern Physics Solution

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Widener University

Summer COOP 2004

PHYS 261 Modern Physics Name _______________________

Prof. Augensen

Worksheet for Chap. 3

Quantum Theory of Light

Exercise #1 (Similar to Problems 2 & 4)

An iron ball of radius 0.10 m is heated to 500 . !al"ulate#

a) the $a%elength at $hi"h the energ distribution is ma'imum.

b) the total o$er radiated per unit area b the iron ball.

c) the total o$er emitted b the entire surfa"e of the iron ball.

Exercise #2 (Similar to Problems * & +)

A hoton has a fre,uen" of 1 -/. hat is its#a) energ in both and e3 Note 1 e 1.02 1061+ .

b) $a%elength in nm3



Exercise #3 (Problem 12)A sodium %aor lam has a o$er outut of 10 . 7sing 5*+.8 nm as the a%erage $a%elength of the

sour"e9 "al"ulate the number of hotons emitted er se"ond.

Exercise #4 (Problem 1*)

:ight of $a%elength 500 nm is in"ident on a metalli" surfa"e. ;f the stoing otential for the hotele"tri"

effe"t is 0.45 9 "al"ulate#

a) the ma'imum <ineti" energ of the e=e"ted hotoele"trons.

b) the $or< fun"tion.

c) the "utoff $a%elength

Exercise #5 (Problem 85)

;n the original !omton e'eriment9 !omton used hotons of $a%elength 0.0>11 nm. !al"ulate#

a) the energ of these hotons before "ollision. b) the $a%elength of the hotons s"attered at an angle 1*0 (ba"<s"attering).

") the energ of ba"<s"attered hotons.

d) the re"oil energ of the ele"trons.



Widener University

Spring 2004


Prof. Augensen

Worksheet for Chap. 4The Particle Nature of Matter


a) hat are the %alues of n for the transition "orresonding to the first t$o lines of the Pas"hen series3 b) !al"ulate the $a%elength for the first t$o lines of the Pas"hen series.

") !al"ulate the radii for the orbits "orresonding to the three different %alues of n in art a).

Exercise #2 (Similar to Problems 1 & 1>)

!onsider the hdrogeni" ion :i2?9 $here @ 8. !al"ulate#

a) the energies of the first three le%els of :i2?

.b) the $a%elengths of the first t$o lines of :i2?.

c) the orbital radius of the first ohr orbit of :i2?.

d) the ioni/ation energ for :i2?.



Exercise #3 (similar to Problem 20)

a) !al"ulate the energ of the hoton that "auses an ele"troni" transition bet$een the n4 to n5 states of

hdrogen

b) !al"ulate the energ of the hoton that "auses an ele"troni" transition bet$een the n5 to n states of

hdrogen.

c) !al"ulate the $a%elengths of the hotons in arts a) and b).


A hdrogen atom is in the first e'"ited state (n2). !al"ulate from ohr theor#a) the radius of the orbit.

b) the linear momentum of the ele"tron.

") the angular momentum of the ele"tron.d) the <ineti" energ.

e) the otential energ

f) the total energ



Widener University

Spring 2004


Prof. Augensen

Worksheet for Chap. 5 Matter Waves


!al"ulate the de roglie $a%elength for ea"h of the follo$ing#a) an ele"tron (me +.1 10681 <g) mo%ing at 2.0 10 mBs

b) a roton $ith <ineti" energ 100 e.

") a 500 <g "ar mo%ing at a seed 20 mBs.

Exercise #2

A roton (mass 1.> 1062> <g) has <ineti" energ 1.0 Ce. ;f its momentum is measured $ith an

un"ertaint of 5.0D9 $hat is the minimum un"ertaint in its osition3



Exercise #3 (Problem 25)An e'"ited nu"leus $ith lifetime 0.100 ns emits a 6ra of energ 2.00 Ce.

a) !al"ulate the energ $idth (un"ertaint) of the 2.00 Ce 6ra emission line.

b) !an this energ $idth be dire"tl measured if the best gamma dete"tors "an measure no smaller than 5e3

Exercise #4 (Similar to Problem 2>)

A monoenergeti" beam of ele"trons is in"ident on a single slit of $idth 1.50 nm9 and a diffra"tion attern isformed on a s"reen 20 "m from the slit. Ehe distan"e bet$een su""essi%e minima of the diffra"tion attern

is 2.1 "m. !al"ulate the energ of the in"ident ele"trons in e.



Widener University

Spring 2004


Prof. Augensen

Worksheet for Chap. 6Quantum Mechanics in One Dimension

Exercise #1 (Similar to Problem 8)

A free ele"tron has $a%e fun"tion gi%en b (') A "os (2 1010 ')9 $here is in m. Find the ele"tronGs#a) de roglie $a%elength in nm

b) momentum in <g mBs

") energ in e

Exercise #2

Ehe $a%e fun"tion des"ribing a state of an ele"tron "onfined to mo%e along the '6a'is is gi%en at an time/ero b

('90) A e'(6'2B42)

Find the robabilit of finding the ele"tron in a region d' "entered on#

a) ' 0 3

b) ' 3") ' 2 3

d) here is the ele"tron most li<el to be found3

e) hat is the normali/ation "onstant A3



Exercise #3 (similar to Problem 10)An ele"tron is "ontained in a one6dimensional bo' of $idth 0.100 nm.

a) !al"ulate the energ le%els n for the ele"tron for le%els n 19 29 89 and 4.

b) rite all ossible transitions of the ele"tron from the n 4 state to the n 1 state") !al"ulate both shortest and the longest $a%elengths of the emitted hotons "orresonding to the

ossible transitions found in art b).

Exercise #4 (Problem 11a)

!onsider a arti"le mo%ing in a one6dimensional bo' $ith $alls at ' 6:B2 and ' :B2.

a) rite the $a%efun"tions n(') for the states n 19 n 29 and n 8

b) rite the "orresonding robabilit densities Pn(') for states n 19 n 29 and n 8




An ele"tron is traed in an infinitel dee otential $ell 0.800 nm in $idth.

a) ;f the ele"tron is in the ground state9 $hat is the robabilit of finding it $ithin 0.100 nm of the left hand

$allH i.e.9 ' in the region (0.009 0.100) nm b) Ieeat art a) for an ele"tron in the ++th energ state abo%e ground state (i.e.9 n 100)

") Are the abo%e ans$ers "onsistent $ith the "orresonden"e rin"ile3

Exercise #

A arti"le is in the ground state of an infinite s,uare6$ell otential gi%en b

7(') ' J 0

0 0 J ' J :

' K :

Find the robabilit of finding the arti"le in the inter%al ' 0.002 : at the follo$ing ositions#

a) ' :B2

b) ' 2:B8") ' :

LNote# sin"e ' is %er tin9 ou do not need to do an integrationM



Exercise #! (Problem 12)

A rub laser emits light of $a%elength +4.8 nm. ;f this light is due to transitions from the n 2 state to the

n 1 state of an ele"tron in a bo'9 find the $idth : of the bo'.

Exercise #"

A mass of 106+ <g is mo%ing $ith a seed of about 1068 mBs in a bo' of length 0.01 m. Ereating this as a

one6dimensional infinite s,uare6$ell otential9 "al"ulate the aro'imate %alue of the ,uantum number n.

Exercise #

A arti"le is in the first e'"ited state (n 2) of an infinite s,uare otential. !al"ulate#

a) J'K b) J'2K



Widener University

Spring 2004


Prof. Augensen

Worksheet for Chap. 7$unneling Phenomena


hen arti"le energies are $ell belo$ the to of the barrier ( JJ 7)9 the robabilit for transmission isgi%en aro'imatel b#

P 1 (B7) e62: $here 2m(76)O1B2 B

7sing 1.055 10684 .s9 "al"ulate the e'onential fa"tor e62: for ea"h of the follo$ing "ases#

a) a roton (m 1.> 10

62>

<g) $ith 7 6 10 e and : 10

612

m b) a baseball (m 0.10 <g) $ith 7 6 5 and : 100 m



Exercise #2 (similar to Problem 1)An alha arti"le is "onfined to the nu"leus of an unse"ified radioa"ti%e element. Assume a semiinfinite

s,uare $ell $ith infinitel high $all at r 0 and $all of height 22 Ce at the nu"lear radius I >.0 fm.

Ea<e 1+>.8 Ce.fmB"

a) 7se the interati%e method of 'amle 5.+ to estimate lo$est (ground state) energ ermitted for the

alha arti"le. b) !al"ulate the smallest %alue of %elo"it of the alha arti"le from the result of art a).



Widener University

Spring 2004


Prof. Augensen

Worksheet for Chap. 8Quantum Mechanics in $hree Dimensions


Assume that the nu"leus of an atom "an be regarded as a three6dimensional bo' of $idth 2 10614 m. ;f a roton (m 1.> 1062> <g) mo%es as a arti"le in this bo'9 "al"ulate (using h .8 10684 .s)#

a) the ground state energ of the roton in Ce

b) the energies of the first and se"ond e'"ited states

") hat are the degenera"ies of these states3

Exercise #2 (Problem +)

;f an ele"tron has orbital angular momentum of L 4.>14 10684 .s9 $hat is the orbital ,uantum number for

this state of the ele"tron3



Exercise #3 (Problem 1)!al"ulate the ossible %alues of :/9 the /6"omonent of angular momentum for an ele"tron in d subshell.

Exercise #4 (Problem 1>)

!al"ulate the magnitude of angular momentum : for an ele"tron in#a) the 4d state of hdrogen

b) the % state of hdrogen


Suose that a hdrogen atom is in the 2 s state. Ea<ing r a09 "al"ulate %alues for#a) 2 s (a0)

b) 2 s (a0)2

") P2 s(a0)



Widener University

Spring 2004


Prof. Augensen

Worksheet for Chap. 9 &tomic 'tructure


:ist the ossible sets of ,uantum numbers for an ele"tron in the#a) 4 p subshell

b) 4 % subshell


An ele"tron is in the 4F5B2 state.

a) Find the %alues of the ,uantum numbers n9 9 and =. b) hat is J9 the magnitude of the ele"tronQs total angular momentum3

") hat are the ossible %alues for /9 the /6"omonent of the ele"tronQs total angularmomentum3



Exercise #3 (similar to Problem 1>)Si' identi"al9 non6intera"ting arti"les are la"ed in a "ubi"al bo' of sides : 0.500 nm. Find the lo$est

energ of the sstem (in e) and list the ,uantum numbers of all o""uied states if#

a) the arti"les are ele"trons

b) the arti"les are identi"al to ele"trons9 e'"et do not obe the Pauli e'"lusion rin"ile


rite out#

a) the ele"troni" "onfiguration for nitrogen (@>) b) the %alues for the set of ,uantum numbers n9 9 m and ms for ea"h ele"tron in

nitrogen.



Widener University

Spring 2004


Prof. Augensen

Worksheet for Chap. 10'tatistical Physics


A sstem of si' indistinguishable arti"les has total energ *. !al"ulate the robabilit of finding the arti"le $ith energ (a) 1 and (b) *.

Exercise #2 (similar to Problem )

An unse"ified element ossesses an energ differen"e of 1.+2 e bet$een n 1 (ground) state and n 2(first e'"ited) state9 assumed to ha%e e,ual statisti"al $eights. ;f a samle of 1020 atoms of this substan"e is

in an en"losed "hamber in thermal e,uilibrium at E 1000 9 "al"ulate the aro'imate number of atoms

in the ground state (n1) and in the first e'"ited state (n2). Assume n2 JJ n1.



Exercise #3 (Problem 15)Ehe Fermi energ of aluminum (atomi" mass 2.+*9 densit 2.>0 gB"m8) is 11.8 e.

a) Assuming that the free ele"tron model alies to aluminum9 "al"ulate n NB9 the number of free

ele"trons er unit %olume at lo$ temeratures.

b) Retermine the %alen"e of aluminum b di%iding the ans$er found in art (a) b the number of

aluminum atoms er unit %olume9 as "al"ulated from the densit & atomi" mass.


Find the robabilit that a "ondu"tion ele"tron in a metal has an energ e,ual to the Fermi energ F at the

temerature 800 .


Ehe Fermi energ for sil%er is 5.4* e at E *00 . Also9 the robabilit of finding the ele"tron in that

state is 0.+5. Note the olt/mann "onstant is < *.1> 1065 eB.

a) !al"ulate the energ of a "ondu"tion ele"tron in sil%er at E *00 .

b) !al"ulate the Fermi seed %F for ele"trons in sil%er.



Widener University

Spring 2004


Prof. Augensen

Worksheet for Chap. 12$he 'olid 'tate

Exercise #1 (Problem >)

Ehe "rstal stru"ture of !l is the same as Na!l. !al"ulate the#a) ioni" "ohesi%e energ for !l. Ea<e r 0 0.814 nm and m +.

b) atomi" "ohesi%e energ of !l b using the fa"t that ioni/ation energ of otassium is 4.84 e

( ? 4.84 e ? ? e) and that ele"tron affinit of "hlorine is 8.1 e (!l6 ? 8.1 e !l ? e)


Sodium (Na) is a mono%alent metal (ea"h Na atom "ontributes one e6) ha%ing a densit 0.+>1 gB"m89 an

atomi" $eight 28.0 gBmol9 and resisti%it 4.20 106* .m at 800 . Assume the "lassi"al free6ele"tron model

(C distribution). !al"ulate the#

a) a%erage densit n NB of free ele"trons b) a%erage time bet$een "ollisions of the ele"trons ") thermal %elo"it %rms of an ele"tron

d) mean free ath :

e) ti"al latti"e sa"ing d in nm9 and "omare $ith :



Exercise #3 (Problem 14)Sodium (Na) is a mono%alent metal (ea"h Na atom "ontributes one e6) ha%ing a densit 0.+>1 gB"m89 an

atomi" $eight 28.0 gBmol9 and resisti%it 4.20 106* .m at 800 . Assuming the Fermi6Rira" gas9 use this

information to "al"ulate#

a) the densit of free ele"trons n NB b) the Fermi energ F at 0

") the Fermi %elo"it %F of an ele"trond) the a%erage time bet$een ele"troni" "ollisions

e) the mean free ath : of the ele"trons (assuming F at 800 is the same as at 0 ) and "omare $iththe ti"al latti"e sa"ing d (from 'er"ise 2)

f) the thermal "ondu"ti%it of Na


From the oti"al absortion se"trum of a "ertain semi"ondu"tor9 one finds that the longest $a%elength of

radiation absorbed is 2.0+ m. !al"ulate the energ ga for this semi"ondu"tor.



Widener University

Spring 2004


Prof. Augensen

Worksheet for Chap. 12 Ae!"#'uperconducti(ity

Exercise #1

An iron6"ore toroid is $raed $ith 250 turns of $ire er meter of its length. Ehe "urrent in the $inding is*.00 A. Ea<ing the magneti" ermeabilit of iron to be m 500009 "al"ulate the#

a) magneti" field strength H (in ABm)

b) magneti" flu' densit $ (in E)

Exercise #2

A $ire made of Nb8 Al model has radius 2.0 mm and is maintained at 4.2 . 7sing data for E" and "2(0) ro%ided in Eable 12.59 "al"ulate#

a) the uer "riti"al field "(E) for this $ire at this temerature

b) the ma'imum "urrent ;ma' that "an ass through the $ire before its suer"ondu"ti%itis destroed ") the magneti" field at .0 mm from the $ire surfa"e (*.0 mm from "enter)

$hen the "urrent has its ma'imum %alue.

Exercise #3

Ehe enetration deth for lead at 0.0 is 8+ nm. 7sing the "riti"al temerature E" >.1+8 for lead9 findthe enetration deth in lead at# a) 1.0 b) 4.2 ") >.0



Exercise #4 Persistence currents. ;n an e'eriment "arried out b S.!. !ollins bet$een 1+55 and 1+5*9 a "urrent $as

maintained in a suer"ondu"ting lead ring for 2.5 ears $ith no obser%ed loss. ;f the indu"tan"e in the ring

$as 8.14 106* 9 and the sensiti%it of the e'eriment $as 1 art in 10+9 determine the ma'imum resistan"eof the ring. )int # Ereat this as a de"aing "urrent in an I: "ir"uit9 and use the fa"t that

e6' 1 6 ' for small '

Exercise #5

!al"ulate energ gas g as redi"ted b !S theor for (a) te ; suer"ondu"tor @n and (b) te ;;

suer"ondu"tor Nb8Sn. E" is listed in Eable 12.+. !omare the %alues for tes ; and ;; suer"ondu"tors.

Exercise #

Ehe radius of a ring that $ould fit on a finger is T *.0 mm. !al"ulate#

a) the magneti" flu' through the ring due to arthQs magneti" field ( 5.* 1065 E)

b) the number of flu'ons that the ring $ould en"lose.



Widener University

Spring 2004


Prof. Augensen

Worksheet for Chap. 13 *uclear 'tructure


Ehe :armor re"essional fre,uen" is

f Bh 2Bh

!al"ulate the radio6$a%e fre,uen" at $hi"h resonan"e absortion o""urs for#a) free neutrons in a magneti" field 1.0 E

b) free rotons in a magneti" field 1.0 E

") free rotons in the arthGs magneti" field at a lo"ation $here 5.0 1065 E

Exercise #2 (Problem >)

!onsider a atom $ith the ele"tron in the 2 p state. Ehe magneti" field at the nu"leus rodu"ed b the

orbiting ele"tron has a %alue 12.5 E. Ehe roton "an ha%e its magneti" moment aligned in either of t$o

dire"tions erendi"ular to the lane of the ele"tronGs orbit. e"ause of the intera"tion of the rotonGs

magneti" moment $ith the ele"tronGs magneti" field9 there $ill be a differen"e in energ bet$een the states$ith the t$o different orientations of the rotonGs magneti" moment. Find that energ differen"e in e.



Exercise #3 (Problem 11);n 'amle 18.89 the binding energ of the deuteron $as "al"ulated to be 2.224 Ce. Ehis "orresonds to

a %alue of 1.112 CeBnu"leon. hat is the binding energ er nu"leon for the hea%iest isotoe of

hdrogen9 8 ("alled tritium)9 for $hi"h m(8) 8.0105 u.

Exercise #4 (Problem 1>)

7sing the grah in Fig. 18.109 read off the %alues of binding energ er mass number bBA at A1 100 andA2 200 to estimate ho$ mu"h energ is released $hen a nu"leus of mass number 200 is slit into t$o

nu"lei9 ea"h of mass number 100.



Exercise #5 (Problem 24)Eritium has a half6life of 12.88 r. hat er"entage of the 8 nu"lei in a tritium samle $ill de"a during a

eriod of 5 r3

Exercise # (Problem 2)

o$ man radioa"ti%e atoms are resent in a samle that has an a"ti%it of 0.2 !i and a half6life of *.1

das3



Exercise #! (Problem 82)

A b6rodu"t of some fission rea"tors is the isotoe 28+Pu+49 $hi"h is an alha emitter $ith a half6life of

249000 r#

28+Pu+4 2857+2 ? 4e2

!onsider a samle of 1 <g of ure 28+Pu+4 at t 0. !al"ulate#

a) the number N0 of 28+Pu+4 nu"lei resent at t 0 b) the initial a"ti%it I 0 in the samle

") the time t re,uired for the a"ti%it to de"rease to 1 de"aBs.

Exercise #" (Problem 41)

Find the energ released in the alha de"a 28*7+2 sho$n belo$. 7se the mass %alues in Eable 18..

28*7+2 284Eh+0 ? 4e2



Widener University

Spring 2004


Prof. Augensen

Worksheet for Chap. 14 *uclear Physics &pplications


Ehe follo$ing rea"tion9 first obser%ed in 1+809 led to the dis"o%er of the neutron b !had$i"<#

+e4 (9 n) 12!

!al"ulate the U %alue of this rea"tion.


Ehe densit of li,uid hdrogen target in a bubble "hamber is >0 <gBm8. ;f 20D of a beam of slo$ neutrons

in"ident on the bubble "hamber has rea"ted $ith the hdrogen b the time the beam has tra%eled 2 m

through the hdrogen9 $hat is the "ross se"tion9 in barns9 for the rea"tion of these slo$ neutrons $ithhdrogen atoms3



Exercise #3 (Problem 20)A arti"le "annot generall be lo"ali/ed to distan"es mu"h smaller than its de roglie $a%elength. Ehis

means that a slo$ neutron aears to be larger to a target arti"le than does a fast neutron9 in the sense that

the slo$ neutron $ill robabl be found o%er a large %olume of sa"e. For a thermal neutron at roomtemerature (800 )9 find#

a) the linear momentum

b) the de roglie $a%elength

") !omare this effe"ti%e neutron si/e in b) $ith both nu"lear & atomi" dimensions.


Ehe Sun radiates energ at the rate 4 102 . Assuming that the rea"tion

4(11)4e2 ? 2e? ? 2 ?

a""ounts for all the energ released9 "al"ulate#a) the energ U released e%er time this rea"tion o""urs

b) the number of rotons fused er se"ond

") the mass transformed into energ er se"ond.

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