ch39 example solutions

8/10/2019 Ch39 Example Solutions

1/8

Someproblems

deal with

the

allowed

values

of the

energy

of a hydrogen

atom,

with

the

possible

wavelengtls

and

requencies

f the

radiation

emitted

when

a hydrogen

atom

makesa transition

from

one state

to a lower

energy

state, or with

the frequency

and wavelength

ofradiation

that can

be absorbed

by a hydrogen

atom.

The aliowed

values

ofthe energy

are

given

by

4 .

-

t ru '

I

13.6

eV

o =-a+FF=---7-.

where is

a

positive

nteger.

Remember

=

0 means

heelectron

s

ust

free

of the

proton

and

hasno

kineticenergy.

he

energy

f the

photon

s

the magnitude

fthe

difference

n energy

f

the

wo statesnvolved

n

the

kansition:

f:14- E1l.

e-

Elis positive

or

an absorption

vent

andnegative

or an emission

vent.

some

probiems

nvolve

he wave

unctionsor the

electron

n a hydrogen

tom.The

electron

moves n

three-dimensional

pace

nd

y42

v

g|esthe

probability

t can

be ound n

the

infinitesimal

olumedZ.

The

adial

probability

ensity

s

given

by

p(r)

:

{

mzlyy'z rild,

(r)dr

gives heprobabiiity hat theparticlecan be found n thespherical hell with inner radius and

outer

adius

I

dr'.

Questions

and

Example

Problems

from

Chapter

39

Question

1

The igure

below

shows hree

nfinite

potential

wells, each

on an x

axis. without

written

calculation,

determine

he wave

unction ry

for a

ground-state

lectron rapped

n

eachwel1.

Vn

*)

=

X

Y

\l

ff*,e

v

,^ (# -)

(t) =

I

L

\P,*) =

tP, )

13

{ / -

(+^)


2/8

Problem

A

proton s confined o a one-dimensionalnfinite

potential

well 100

pm

wide. What s its

ground-state

nergy?

)

X"

\ r , ^

rQr=

I ' t ,?*,o"?Kr

En

=

(-*; ,)

"^

n=))e'3>

""

L= looprn

=

loo"

lo-'*^

g/t-*J

arta&

--+

n

=

I

L.

E

,

=

--)>

'

8"q

L}

p.=

(G'ut* lo-zq5' ' )a

-

'

8( t . t "?*

tdt"rg;(roo^ d ' \ ) '

Problem2

Aa electron s trapped

n

a

one-dimensionalnfinite

potential

well.

(a)

What

pair

of adjacent

energy

evels

(if

any)will have lree times he energydifference hat exists between evelsn: 3

and

n

:

4?

@)

What

pair

(if

any) will have wice that energydifference?

. / hl- \

, '

, . .a,^, in#o*,,-

|

E-)In*r)^

r l--L-- I nl

____-_>

A,a^r;*.,te.J"g?

Fn*,

=

(

,.r",,

_

Xn=

(8.qL',/

' '

fu"r]_

r^"^"+-"A"

'r+t

\

z

=

A, nt

ja

nr l

'

. E,

(n* \ )

n

(")

^-,

-

2 .

-

F,

(.* l) ' -

E,

D'

)

, t^rt ,uj-^f-

t^*r*

En

=

71,

(a"*r)

=

3

(Er-trt)

A,

(t,.*,)

=

3

f

a,

t"')*-

E,

(3)"1

i , ,., .\:i'

'

o r'- ,-

-'l'

- /\ ' :

-;

f-----

r--

1

d.n* 'J

.- . -

L,

E

rJ

* An-t 1= Al 2l n=-,o n'"n

'

,

P^*^'W

-Xt/v414

b

b)

,.^,*

"u*.t

4*,

-

En=

E,(ln*l)

='I(tr,-tr")

)n:*^

"':

A,

(an*r)

=

t

f

e,

(q')"-

{l

(3)"1

*

''

[

?A']

.

A1

(antt)

=

if

tr,

---"

An-n1

1

---+

|

w,

nAaa

N"I4Aa^'f'^-

\

^

^^

".,San^t,L

r-n^a

W

IA,'x@

E

:

3.2,1*

1d'r5

=

O. o?'o6q

V

: 3l '7F'l * An* 1="|1 -"21*''-.t9^--"*


3/8

Problem

3

Supposehat ar electron rapped

n

a one-dinensional

nfrnite

well

of width 250

pm

is excited

fiom its first excitedstate o

jts

third

exiited srate.

a)

[n

e]eclron-volts,

hat eneigymust

be

transfened o the eiectron or

this quantum

ump?

Ifthe

electron

hen

de-excites

by emitting

light,

(b)

what wavelengths

can it

emit

and

(c)

in which groupings

(and

orders)

can they be

emitted?

d)

Show the several

possible

ways he electron

can

de-excite

on an energy-level

diagram.

(

l

n\

l )QoeV.nrn

=

' - . ' - - - - - - .

.=

13

( t

o1"v) ( '1 ' - l ' )

= :

P"aarJA

fi*r"'lon"

n=q

-+

n=

3

---:=:'

q

'L{

-+

h=

A.

=e

:=:-

y^1 ] - -s n= l

p1

=) - r r6= |

.:--:---

'

\ =

( . )

[ "

&-q'E,

f t= {

n=3

rr=:.

n=\

4/

->

\

?---r

a

l--t I

, l-a3

3

--

|

/ i--r3 3-ra

a+l

E:

(-n

a

8

r.rL*

n\=

( r.a3*ro-']f,..)'

En=

(l.us*

lo-'*r)

nr

(

a.oa

v)

n\

(")

M

z'pfutali,b

*-r

6=

q

A

F= Er-

Ar

p.'rivJ

r,r4}.A,utdo

---+

n='l

=

(6.o3,rv)

( , i t -2-)

D

E

'

??,.a1-V

(u)

hf

=

AE

6/r=oF -: )=

\/ng

=

?.j.tl

.. r

\

:

lJ - Inm

\

=

4l 'Xn$

-=::,

X = )-5.?n.' \

'===:

X

=

68'5n' ,

e--->

6

(q-n"

to

' tcz)(15arto ' ' lo)a

( { )


4/8

Problem 4

A

particlb

is

confined o the one-dimensionalnfinite

potential

well

ofthe figure belor,.

If the

particle

is in its first excited state

n

:

2), what is its

probability

of

detectionbetween

a)

x

:

0

and x

:

0.25L,

(b)

x

:

0 and

x

:

0.50L, and

(c)

x

:

0.25 L altd x

:

L?

_,,-Nonquandzed

-Top

of rvcll

\(r)

=

i{

**.

(li')

-

,P"c.)=f;(

V"*t*) YL,.,\\ ^"L)*

/ ' o ' L 5 L

(4)

(

n,

-

e/

^Anx\

r

-?

u

=

2"rx / t

du

=

L/ t

dr

\

/

\ / , ' t t 'n(t) ly

Jt=LA"dq

-r/t tn/^

x :o

/ ,x t \ t

'n / t

(

/")(Lrn)J

-"'"in

=

*

LAx-/q

'

'r)1

oo

= +

|

Y^ntr-+"'r- (t/s'oY^*4f =

W

L - ' - 1

(u)

r,--*

ttz,,*

y,1 y^

hr

r*{;*

ts

.x=\Z:;r"oL

-)W

1. \

. a

I t t I z t I

probfems

'/

f=o.rr.=l-)"

=

l- ' / ' t=)71

(a;

The igurebelow

gives'the

nergy

evels or an electronrapped

n Tifri6p-otenrialenergy

well 450 eV deep.f the electrons i,n he

n: 3 statgwhat s its kineticenergy?

t)

Theelectron

then absorbs500 eV of energy rom an exlemal source.What

s its kinetic energyafter his

absorption, ssuminghat he electron

moves

o

a

position

or which

x

>

L.

/ c , )

{ ** t - r

=

}3o


5/8

Problem

6

Ar electron

s contained

n

the

rectangular

box

of the

figure

below,

with

widths

L_

= g00

pm,

Lr= 1600pm,

andr-:400

pm.

what

is

the

electron's

lound-state

energy

n

eleciron-voits?

"r-

=

h'

1":

-:---G

.

L

nx)

n)

I

rr

-6;

t

e""

Zr.

-{,,

, I

^

-i vlt.r"i.,p,_t,Tsh s q,*r,


6/8

Problem 8

(a)

What is the wavelength

oflight

for the

east energetic hoton

emitted

n the Paschen

eriesof

the hydrogenaton

spectrum ines?

(b)

What

is the wavelength

ofthe

series

imit for

the Paschen

series?

P^*r^r^)r)

,

t+'3

hf

'at

=

(-t:'a"v)

t,-;k

)

ll::=

a

e

=

(Bl "v) +

-*-i)

(")

X^,*

r^-4'fi2

4

nr"ra

rl

i

A

E,

(t:.{"eV)

+.

*)

=

WV

|

875

"

\ = ltqo"V.t'cq=- lr-=j=;il

/ . j t l -V

l "

" -" - ' I

Problem

9

A hydrogen

alom, nitially at

rest n the n

=

4

quantum

state,undergoes

transition

o the

ground

state,emitting a

photon

n tle

process.

What s the

speed f the recoiling

hydrogenatom?

D*V*

'

l '+/-

=

1

l ,

b

l r ,

l o

'

' tl

?

"+

.).i\",I',,^-ixl,"dr^r

*e*']'u"

;","r*k.'r"{-{l4

r* ,"&i

{

,il

=

a),

)*,r^

-{r+,

As,,r'*ili-tr;ro.i';-a>

",

fl

p,rr-*^rfur.1.,

ldu

..r.'-.c...r,-,rfi+,"1

g

E|*

F$-J*4r"1 "r."J

3"

-.t'.*'s",1.,s"-."{Ir'1

,-i J;ou ,.

w;-"+"- "re,,n

}.,1,-;L.zrvou-1

*b*r'

rw,-,,,-e*

-.$r^,.,+

jo$t:,

, .ntr*r

Pu

-

l*,u*-,

'---'----

fl*Vn

=

\/,^

=

'y.

)

izr{-e.."J1'a

f

6r};fuo'r

P.+,iu*c.f

}*t-lJ4"r'"'i-o

"''',

*''*-n-'X

-'t-b-\a

:

i-l

=

Er-F,

f**.*)

=

lt.-t5e-J

i -

(

t : . ts.v)

-:

rr.,

C

t'---------------

i ,

------u

i

r',

=

4')",f5

i

i {

(r:

ts.v)

( t 'y"';:t; ' 'V.u)

(1. r ,12, , t ' ' 'u2): 'o ' ) t>xof)


7/8

Problem 10

(a)

Find, usiug the

energy-level

diagram

of the figue

below, the quanturn

numbers

corresponding to

a transition in

which

the

wavelength

of the

emitted

radiation

is 121 6 nrn.

(b\

To whal seriesdoes

his

tmnsmission

belons?

Nonquantized

-2.0

-10 .0

-14.0

// ..- :- '

l[Il lt_

.

4.(l

6

-6.0

;

H

.i

-B,0

i6;;-;il

E

(

- l

( " )

l? 'GeV

3-rl e V

hi=

/\E

=

Etn*-Eu*

lD.t

".v

-

3.'1-V

-13-4-

n\

--

-

13' t"eV

-..-+

n'

E=hf

.=

\-/\

E=

I

At-f

OeV.om

I

) "1.L

orq

E = lO'l.-V

/r/^t

-lrt

Fr^J

"$

f"a*'-"^-'t^

{u;

lt/.d".-^t/-'r"l

f l t=

=tl


8/8

0 -

4.0

-6.0.

_-8.0'

-10.0.

-12.0

-140

Problem1I

A hydrogen,atom

n a statehaving a

binding

energy

the

energy equired

o

remove

an electron)

of 0.85 eV

makesa transition o a state

with

an excitation energy

the

difference

between

he

e,nergy

f the stateand hat of the

ground

state)of 10.2

eV.

(a)

Wlat is

the energy

of the

photon

emitted

as a result ofthe transition?

b)

Identifi

this traasition,

usrng

he energyJevel

diagramof

the figure below.

= -D-85-V

Ee

-

(-3.' i

"v)

* L

Er=

-)3.t"eV+

lo.)eV

g,

Azr.o

D=4

(

b

)

bt*r"

F,

+/qll""

)

^4,o^^

r+.r

/040

Jiat

^

pr"b'l

-,r.u'l^,oaa

'"bqv

h

D=?.

-

13.

"

e-V

(-

13,6

u

. - - . . - -

n '

aa

t)

bpl**,

.ttbtet

--->

A

F

=

En

-

Ae

=

-

tzss

-

tt::9

n).

-

13.

"'v

=-..-

f i \

/ \a

/.-\

|,

+

t '1zV

=

=

l '55

e/

1..55-

V

---->

13,G4V

na

=

-o.85ev

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