) þ= gtgj'-...rl ìn thrs ç.oble t-ér-e g oìng kÞ àat

rl ìn thrs ç.oble* t-ér-e g oìng kÞ àat

@ * " * " , " ^ - - . - Þ .

+ô úÒr* gur l€. lt^i: i n ï-e

Ð

qf \ '

P1*1 .^ ( t ) = J t ( , . ,+ )nV i " , t1 åx =-q

dr Æ +.a

a ^ å e u - { ( v ) = É J J - J t ê* fhts ts l1ê '6"^.1l"* u* à

&o u^k'o-gr*'ì f çot't"t . \*t",.g**ss,. e*r"j " f t,s 4^ é}sL Fu^ho'¡

a- g \o l oê e, -Ç joruL.yøu q ffix*98- 6g(ooks { i ke

, qt rI

/ùn c( t*, J g- a

-{7q*¡= rd x

ït

whal-q r^jî'

v(. .'t-È *

l ¡\f\€-€ d k('

X(/ = c4 h:!

, ?("lr"en P¡,rce = ññ- J e-61*'

= * er"Ç

srrrce e.Ç fu) îs *àå .

¿C3,f¡*€.r"þ Ow ( ,*L.Bn.t\

¡ r- 9n ...#L J e-" J"

o

e"-{ to) ; ¿. 8r

"-f ( ' /¡ø¡) = ô. t l

V\ ow \¡/ g

l e b

€o.

. J xðrl = ",úGt-- t t à u = :

{b¿ k-"li,î.k r ¡gq tÈ"is t ool-ca

\il.t eøt

t*.Ço ryh.q er"ç'r *.otc ì'1neü fiîs wq"y

-;a/o - ''l

I - t r ' . I- J e , d u - ¡o

V/q)t

L

L = O : o ( C t ) = "

t= *3 .{(Ð=^F;à P :

s P- -

e'.f, (z]

−8 −6 −4 −2 0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x (a)

|ψ(t)

|2

t = 0t = 3/ωt = 6/ω

0 0.5 1 1.5 2 2.5 3

2

4

6

8

10

α (t

/ω)

0 0.5 1 1.5 2 2.5 3

0.3

0.4

0.5

0.6

0.7

t/ω

A (t/ω

)

ij

--"i-------*'--

I

3 . åoubie eqt t ørqo\

i ) b = onofrce C*vaÞ tgls s\a.hvg Þ î^clreftt¿

* opp.oo.l.ol oe"p"*"ø&h\!'1 'bvf neva.* h"ìlv,oåe :

weL\

ti".\fiq"*ryE t E Y Ç t

w i+t1 trro CSt¿ì .J {

ì ì )

qh *¡

o < b6 -----1

in c\assrcal\.l el\ou*è . .r'Ç3ion V is Stnvsotùot\

a,rLsi &e- oÇ c\ass, c

b) Sctl^ne eS"É_,''', G)

Lhe p \oLs \nq t ;+i-lþ [arg eø-

Þ** hen< ìs the ç'lt p\olù

6K4 )

çL\o)

E(-

:I

,

Ç)

-."' E,(o]

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tl.e elec!,.o^ t^¡ o.r\è { e ".å k, 5.** 'ktne aþç*s . {ageïLe'r . {ro".¿ o vr

ei-p\e r*oåç[ ¡ i^ r *he ex"i-leå .9Ìalg *he e\e,cb.CIt :wov,\à +-e¡¿lD

' F,.r.L, .tUrp j t'vct q,hÐ ,", a.øo,rt - 1 P : t'!."tç r¡o.rr\{ "geesrE i, uun*\ovc\á ,' Þ""L, {'1'rg ' tytc., ctho ,"' o"q^rt " i {- fl{s tçu\{ tge^s' çrurebrygll.e! l* l , t - ¡ a - - Þi L ; v"nqy 'Ize i**S Õç)¡ u"o&e\ òoes^{"{ c.àqq-"&*1y reç**ryrt - . . . - j

I

t

¿¡^1. -iä 1¡,ç¡{s¿'\eg.

I

t 1

I'r ì

i

.:i

i l

I

l - . i

i.

h - 3 -Çq; 7¿ççf: ilel"t tvug¡¡'goqti ç f tR SQøi ng zo lf

o) àe-Ë1,'uq w{..Ç*

check iç

in -L{,rrS pnobLe*r , . ,Je * i \ l q(t Sorq€ pr.u.eheq r¡ .stnq opqrca-\n"J:;\n qvo.rrhr ' r- t y,"çç.?r, f t ' ; { f \ ù*o "6,{ tue ø'" .os\ rn^çor\a.n&- *ç€. . r . i : . {a,(e the posr\ oo crnr ò brn evqgpl f r ; r"*r C'çF" *r l -* rs. aS 4üe þroLtg-w\eç^,\ror.g a^ ópe,/Äto/ ts qr\ ¿*9F tch'o^ bo, *o So\t \4elç-,r*Q,ws-.'\\ fitå

-ort lä*e- lna{, ct c¿nhr^r.- ctois

-.*i oc*'*L.s

rChcr,¡^f tiæ.)veprese"'r\ phyçr.rl ly 6þgs,rvaL,\es we vint.ih{- \dkC fr n4er\s\r/¿ tn

{.E T qb" o. ' i ren I'p., eà øV CIve 4*" *oX w¿\T ["ok a\ {cu".t*l ¡q&",'a"nr

oI fhe pÕorr'i 't *r*r, g.^* Í"ro r'*gyrl'v *i OpqrqÍars-.

çosìÌ i on s¡otce. rePreseb'- ' t-a+ì€)!nL{ gosrl¡ ü t o,,nå u¡oov--.enlvs't oçeir*uÞ*or s

o^à o\ \^ravg &,rreh'o^ tl'= D ;V*-^ / n

Õ { 6 * ;

å 9c') e-^-/*- = it, D (#) ef/*)É rens¡nbe,- Ç(") iso\n Oiqe.-v', $r*ch a3- ìfô+C"i= À {c,'} ú.r he'¡e\ ì= o. corusLa"nt

þ tc, : ," , ^_1frí ¿x7-')

eigav-fuv.cho"., oÇ Ê

l = x Iê r - i r . ,* J

= î

P

â 9c*)t 5

å) '(rs we. see F.o* 4

w\o.e gene.cr\\y \Ê f(X) \s

lfr,îJ Êr") --_ ^ r tx]èÊ6e -Êr )cc . )

Y t") is sgf, av\ ei geu. fir*ct.on

b) àe{t"E q y.rew o?€-ner bcr ê

fa¡ = ìhft x D"-*/^' i+, D lu-*/*'olso YroX- G¡n= i r ( t - #) t ( r )

c) tt. ' ì.à ôperabcr A - ê o.,å checþ< ô ttn'¡ t

êVr,) = -W*fr.) - ìh (\- #)*"¡Yt¡)

ê9cx) = ÀV&) - i t1^

crì øtge* fu"cho^ ç&

ìh&x €G) -* ìh [ " *r o À = ì f t

^ Aof c{

Î qn å ¿heck

4n 9ùgc.vr (vnçhorl of t,

t î 'Êl lhen;

- f - . * ll= ir ro)

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I

' t \

ö ) , he..e v'e \^.tq,,-E+ +. qet- q b¿tle"- ,^&.C&*n'å i^q o{ þe giqewþ@

of {he h.o., ln} ," ucy sa\v¡^g Çø, ("), (x;}, (p) ""å iÉ)

i f we i.e preser'rk Î o',å Ë l+ tere,,l o{ fhe fcq{strrq e* åtow¿,r r'nt /

rr" i o hL eype*h Èo r o"

is '^ -ltr boifo* oÇ

p

t , ri Þ , ì n \ ln tc çøer tLør* , w¿.¿ot . r f id¿r a

a. \ Þ hctr¡*o¡i( p o\-e'",t i al, |ì¡ep"tr k,6\4 is ?ve* by t

r lC*, t )=As-#ç* ' ' -

*it-rn A= (#ä)'tn

*r"g ç"ob\¿*- r*ø^.i-uo.'ç 1L.ìs i: a\476,

6.U-sr,\d t\.ct{ * U,v) is a çr\,h o. \- t1ê sch.oJ¡ etpe.* equ&þ ';dr.:

iv.åY = f##' * **.*="{ f

,

ih *g = [- å ¡^ az *' e-¿i*t + lzh* {- rnq r,^Êx t\'* J Y

^^à ** = (-Ëft) [t. - uo'.-'-uJ f

# - -+,.- (x - 'ê'*+)åI- - ryv- * f f ) " ( * - q e i * t f y - ä v

=ffi)" (*" - zax €'*" + ar, e)tt*t) * - ËV-## + f v-.,.r-x ry= f+>r"# * w,ax Lr'dì*** åk*_ à*f-.t'*

+ Ly'ã'x'J V- t- Lw,du¡uøtz*+. *k- {- n¿a-=x e*ì-I = ì}.'È'y "/

çc^., ti Crg o{ v.4o.çS lrvj lnw ave Çv..ch s¡. øÇ *rg

o *t t+ e-r"urt) 'Lqxel ld = * t l

Ønù trvs s Lç.'çe $ E" ìoeð k scÇ,'n $r r c-e r-G"tI c t

q\

b )

,l,U ,^,o) [" =

I - r rf t^ r5 tS J V5f

I -d,\'/z\ r rn I

t r "g Srov* t å cheÞe o{ lrê h.o. chif+eJ bv q

ù3r''¡¿'à eTÞtg

#nø{-et we'Tt Q^b o*& T^t-e- 4'\.*åtr";s co(.espo^à; to qCohrr e*l s\s{e , w hi ch, is rfevl

i ^ n g o ' l a ^ ì ! n q . ? - . d ,

[9{' ' ,0) lz



.l îY (", t)\¿ = Y* Y = ¡" *-n#-(*'.S)e-trq'-$ j*ï

.- ñ ø-åof*' * * * $ cnt1*u)- {atx cos{-ç}]

&r"* fhe ç"ob\e*,* we kr^.o-, that $tr".o, (r*t$ =

--'La,Y.os(*d) + at .os(.t)]

¿.risx (d*'* ¿ îc 't / -Z.os (* t )

{ cosz (* *)

(x')/ -w Y¿

= t f i l t i

(Å^= $ V \ w /

û| (^ uhr¡r

L J , z { * u Y . À y-oo ---:r-,FTzv'n'ú

r *

* J zvacos('t)u#t-àt-æ * f*cos"Vù;9-;'J;- @ 4

ffiq?*ft*ù/ un i l s o f x '

Àx is r^ à¿pç^$s*t eÊ 'þr'*e J

i€ Tó, I , ¡ þu , ral tu !øïi o"

*t'(ú)

(ffi)'

*-À x a p = Q

V O Uq nð

'Vfre

p 6 " l òlhal"

co her* t

E€e tt"aF Àp 15

[*)'-

c . h . Ò t s a ,slqf€ J

heq-e wg har$¿ q ¡,u, n iffiVrï v rnce*)^ihl y .kk (ul"i ch uve,\ qrunþnn v,¿e chau"'ica\) -rL'6{F osci\faþ5 e-x.r.ch{y . og

q c\osd i ca\ objecf ,,,,t6u\ä .th tt e 9a'¡o"e E"t'e*Fàø t- tt"i.;> c\ v¿øy tc^por\av'T eFar,P\0.

t ¡ ^ lsþ;te o* lûeW,rf n i r'tu:11 urrcev"kint)

−4 −3 −2 −1 0 1 2 3 40

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

x (a)

|ψ(t)

|2

Coherent state of the quantum S.H.O

t = 0t = π/ωt = 2π/ω

0 1 2 3 4 5 6−4

−3

−2

−1

0

1

2

3

4

t/ω

Coherent state motion

*) þ= gtgj'-...rl ìn thrs ç.oble* t-ér-e g oìng kÞ àat

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) þ= gtgj'-...rl ìn thrs ç.oble t-ér-e g oìng kÞ àat