Upload
julian-david-henao-escobar
View
223
Download
0
Embed Size (px)
Citation preview
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
{ / -
(+^)
8/10/2019 Ch39 Example Solutions
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^--"*
8/10/2019 Ch39 Example Solutions
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
( { )
8/10/2019 Ch39 Example Solutions
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
8/10/2019 Ch39 Example Solutions
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,
8/10/2019 Ch39 Example Solutions
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$t:,
, .ntr*r
Pu
-
l*,u*-,
'---'----
fl*Vn
=
\/,^
=
'y.
)
izr{-e.."J1'a
f
6r};fuo'r
P.+,iu*c.f
}*t-lJ4"r'"'i-o
"''',
*''*-n-'X
-'t-b-\a
:
i-l
=
Er-F,
f**.*)
=
lt.-t5e-J
i -
(
t : . ts.v)
-:
rr.,
C
t'---------------
i ,
------u
i
r',
=
4')",f5
i
i {
(r:
ts.v)
( t 'y"';:t; ' 'V.u)
(1. r ,12, , t ' ' 'u2): 'o ' ) t>xof)
8/10/2019 Ch39 Example Solutions
7/8
Problem 10
(a)
Find, usiug the
energy-level
diagram
of the figue
below, the quanturn
numbers
corresponding to
a transition in
which
the
wavelength
of the
emitted
radiation
is 121 6 nrn.
(b\
To whal seriesdoes
his
tmnsmission
belons?
Nonquantized
-2.0
-10 .0
-14.0
// ..- :- '
l[Il lt_
.
4.(l
6
-6.0
;
H
.i
-B,0
i6;;-;il
E
(
- l
( " )
l? 'GeV
3-rl e V
hi=
/\E
=
Etn*-Eu*
lD.t
".v
-
3.'1-V
-13-4-
n\
--
-
13' t"eV
-..-+
n'
E=hf
.=
\-/\
E=
I
At-f
OeV.om
I
) "1.L
orq
E = lO'l.-V
/r/^t
-lrt
Fr^J
"$
f"a*'-"^-'t^
{u;
lt/.d".-^t/-'r"l
f l t=
=tl
8/10/2019 Ch39 Example Solutions
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