I II III-

.

let77

For' '

even

"

solutions,4€ = cos Knx

Tv match exponential and be barelybound

,slope at a f -a mast be zero

,

90

Rna =0,IT,

21T,31T

and because E -0 at threshold,

v. = tiki,

so Vo > In'

me!?2M

For ground state,

Vo 70

For a 2nd even state , Vo > ITFor 3rd - -

-

,v.> In 'II

For"odd " states

,III = sin knx

kn a = Ah,

3 Th,

5th - - -.

For l't odd state

,Vo ) #II ÷

For 2nd old state , Vos EITI. ÷- - 3

- I-.

- -- -

. 2514

Eoxpyresseedas formula

,

for > n bound state

" C IIT.

an :÷÷÷÷÷4 ,

n=5

osnvlx )

Ht'↳I I

⑨4, = sinhqx ,

4⇐= Be- H - a )

qfsq

qsij.IE?IIgB*rsBtanh ga

=

€ (

tanh qaso, so p%8 ↳④intersects once only ,

and I/only if slope e's higher at q

--o

for tanh⇒ a > f un p > at

② never more than one

bound stale

•

My X 0.99,1. ul,1.5

I IItap.-Bscx -a)

HI = As in kx ,4¥ = e- it's Life .

- kx

A sin ka = e-'

' ha- ji Seika = - z .

- Lids in@ ats)

k Aws ka = - i k e-'' ha

-ikeiteihazikeitws ( Kats )

+ PA sink asin ka - is in (Kats )k.ws#sin-ka)=-cFkat )tan @ ats) = ksinka

Roos ka -Tsin ka= &

ksinkaS = - ka + tan

'

Ca ) x = -> has ka -Bsinka

I I

*:

F- O

② 4g = eik "

+ A e-ikx

,¥= Be

ik "

I +A = B

ik ( I - A ) = ik B - BB

zik = 2 ik B - P B -

iftar

is = tie ,A -

- B - I =Ii

fraction reflected= IAI

'

= 974k£It 1745⑤

-

=him - Intl

'

= t.TN#pYh4k 'I left

Jnisht = First =- Im yuh

.

✓

hxy

M×Lnxg) = E I = ( nxyti )Ny O

M×yfN)=n!E=o(hast ') = tenth

I,N = O

=(NHlzNI ) = {3 ,

NH

6,N = 2

10,N =3

-

N = 0,( nx , ng , nz ) = l 0,44

N = I,(nx ,

ng , Ms ) = ( o , o,l ),10,1 , u ) , 40,0 ) ③V

N = 2,(O, 0,2)

E 0,70 ) , (2,0-0) , ( I , 1,01 ,( 1,0 , l ) ,

41,1 ) ⑥✓

N =3 , (O,0,3 ) , 143,

o ),( 9,40 ) , ( 2, 1,0 )

,( 2,41 )

,

(o, 2,1 ) , 11,30)

( o, 1,2 ) , 11,0, 2) ,( I,

I,

I ) ①✓

( o l a a at a at a" at tu) = ( ul aa N atat lo)

=52 I N 127 . 2 = 4

Flat at at a 1ms = fn,m + zEn I atatattm -Dim

= fn.mx . ( n I mtD m Tmt

at = TEX - i REW f

4.=

"ne

- + "Y z = IT't b

,

b = FTW

at lo> = ÷zf{TEX - i C- ima} e-""b'

= If {TEX + RE E } e-"' b '

= 't {MEt÷Em÷} e- t""

= Fee (FF ) e-"" "Exe"""

= N,¥ e

-

Mz b-

=TImyX - ir¥EP

a= TEX tinter P

X = VII ( at at )

⑧P = inner (at- a )

( n I X' In> = Ew Cnt @+ attire)

= Fmw Cnl aattataln)

= Img - (2h ti )

( n l P' In> = tzur ( 2hti )

Lux -Bsnl H' In> = @ nti )' h÷

For G. s.

= tyg r

⑦ (Pfm) = hq ( anti )

⇐mix) = hj hate)

xxx xx

-

-

⑨ Lpl Ad 14> =/Cpl x> x 6147 dxdx

= it 2p fsplx> 5×14> dx

= it ap split>

⑥ H= II + IKK

= I tikaii +Izm

H -4 ( p ) = E tcp )

I II NI

Iwr→X= 0

Consider Solution-t÷mx4C× ) tuk ) Xcx ) = E4Cx )

with 4.(×< a) = e

ik ×

+ Be-

ikx

4,4×5 a) =Ceikx

Because Sch. eq

is real, youcan

take

complex conj . of solution , of =-4*4,=e-ik "

t pg*

e

ikx

%,= C

*

e

- ikx

Now,consider X = B

- 4 - Q

XECB-

B - t ) e-ik "

,X, " = @

* ceik " - c.*

e-ikx)

transmission ( Right → left =XB§Bt=lf=k4①

d.

Iact) = . Iz ein

t Ita e-

i H t th

= -ie ite th f H , if e-itt ta

ti

H =t w fatat E )

(H ,a] = ti w ( at a a - a

at a)= t w ( at a a - at a a -

Ck )

= -

tr w a

dat aft ) =-im at

,

Similar }Azt at ft ) =

tier at Lt )

- i w t

a ft ) = ea

at (t ' ) = e i

wt

'

at

fact ) , at c t' )) = e

i wet

'- t)

From previous problem⑤ act ) = e-

i w ta,at # e

iwt at

xct) =y€[e- in ta + ein tat )

Lol xxx# to-

- Ew fake" :

e.a:*,

=÷w e iw ft ' - t )

⑤ He tew Cata tf ) - FAX

= him Cata + I ) eaten( at at ) Fct)

dat x# TEW dat ( a# tata )

= ME ( - i ac.tl t iate) etizctleih.tt/EgxJe"¥µ¥(iwata + c'waxy

"

%,

pye-itE-e-mw-xitt-FC.cl#--w'

xx ) t EMEI X = TEI Cata't

m dI + mw-x# = F

P = inner (at- a 1

G- I# IF > =/self> soil # IT' >soil e 's

disqdqi⇐It I

6

= getit ' - T

'

-''

T.it#ETIYd3qd3g '

-

-

'

= y¥hJq÷so e

i qtr -F' loose

=z¥[dg= .¥i÷i

① Xx ) = sinlttxlaa ),Let g =

Tl?

Let x' = x- ah

4 ( x ) --Fausta'

= us g. x'

,

- Is x' ⇐

Let x' > O

fck,x ) = Zag 'dTy4singfxtIz ) single-E) dy e'

' k 't

= f%gTuq(x 't't ) asqcx'- 7) e'

' ht

* Imax ← fan x' 70

only need coscqy ) part because it is even

f- ( Bx ) = If'

gas g.( x' '' E) cos qlx

'- 7) cos qty

-Snax y - ay a- 2x '=2 Ca-x)

iAM [as Lqx

'

+ cos of g) cos Kyfck,xl= -5 dya sinlqtklgwoyt sing - thorax

- ymnx t--

zwszgxt-kudmayoftk.la (q- Ha

= ka

gum,= {"a- x )

,x > §

2 X,X C

-

y'

= X - ale