solution set chapter 3 particle nature of light...
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SOLUTION SET
Chapter 3
PARTICLE NATURE OF LIGHTDISCRETE ENERGY LEVELS
"LASER FUNDAMENTALS"
Second Edition
By William T. Silfvast
/ - --- ----- ------ ------····--- -----------------·--····--- ------ -····-
Cll 3
1. Using the Bohr theory, carry out the derivation of the electron orbit radius r ( eqn. 3 .4. and the orbital energy En (eqn. 3.8) for the hydrogen atom.
Fr b ~ ( 3, 2.) ( "3, 3)
h' ~ l ..... Jii - r .~l!i
~er~
n i ~ '- Y TT C:o p f'"tt M W
Y"he. e ~
1. '"l.
- n 1. (]. rr -i1 ) tc - r:~ h n ': ~ u n .. - V\'te rr el. - (j' W'ee ..
-::·--
2. Compute the wavelengths for the followin . . . CH 3 the Bohr theory: g trans1t1ons in the hydrogen atom using ;
!
(a) n == 2 to n == I· '
(b) n == 3 ton== I· '
( c) n == 4 to n == 2 · ' ( d) n == 5 to n == 3 · '
( e) n == 6 to n == 4.
F='ro...._ ( 3. II) * ~ R It ( {2 - ~~):: ~. 0% 77 H'xto ~~· -~) M'
l Oi) Yh :. 1. it\' Vii .:- I J_ -:: ( ~ - l ,_ \ ~ H -J A4.f I 2 }
"Ll ::: (. 2 I )7 x I 0 - '7 M =. 12 I I s 7 V\ l.N\
(b) l ~ ( J. - J_ ) Ru 'A ,.,_ ~...._ fl
)\
)\ 53 = L 2 ~ 2t_ x 10-(pWt = ~, 2.::. .:~~ -. • rr .ti• -uwa ·- n t
_I -::. ( f._ - ~.) f< H
"~
CH 3 3. Compute the ionization potential for five-times ionized carbon (C5+) and also com
pute the wavelength of the transition from n = 3 ton = 2 for that species.
E ::: -~'-Ea - - ?! '- Eo
I '2.. -I
b~ Et> .:= ts. >'fse V a. "1.. cl. ~ ::. ~
b s-+ ( ) l ) ).,_ n -{-o"' j..J - .); lcR C. / s f r o ..._ 5. I '-I :
, r I
D 'f"' .
-v.. -:: c - ~ '"2 2-o ( J_ - J_~) - -jl
~~'- h 2). 3
ie ( \ ~J \ "i? 0 - --- (. h 2.. l ....
r\~'--
\ I (< H ~~ l +'- - ~~)
I.~ 2 .3 s-x '° -8' ~ :: I ~.. 2 3 s- rt t11 -_:.:_ ______________ __
CH3
4~ In the isoelectronic scaling of hydrogen-like species , what is the lowest- atomic number element that would have a possible X-ray laser transition at a wavelength shorter than 4.4 nm? (Here, consider the transition from n = 3 ton = 2 as the laser transition.)
F_...o w... 1·0~o b f.e ~ 3 : I· \ \ - - \. ~ ~ ~H ( ~ - ~) - ~-- /.oei'7x1D
1 1>t-'(i·- ~-)
N () w ~} (.. <. L/. '1 fl IM.
J_ ( lo ~b. 3 V\ IA\_ J ~ '-7, '7- ~ ~ ~ ' .'---=
, r I
-i:- , ·~ a. q \ ~ trv e ,.. ) (4. yt.4.>(, r~ ""'; e ¥"'
l ~ \ 3 -
ll+ 3
5. Solve the Schrodinger equation for the energy eigenvalue of the ground state of th~ hydrogen atom, assuming a solution of the form 'lj;1 (r) = ce-ar (eqn. 3.38).
, r I
, I I
lH 3 6. Write out the wave function for the hydrogen atom for n = 3, l = 2, and m = +1
using (3.65) and (3.66).
Y\ = > _f_ ::.. '- VV\ -:::: + I Pl-: t. i+ t
f,,.b ...._ C 5. ,,') 'f. -= c f,, e.- .. L ti)<') \6l 'R.} vYI.
L~ Lvi - 1 .. t • LS"'=-rio '1 - .,., • • • s--
- -- q,./I.,_
Cll 3
7. Obtain the possible term designations for a hydrogen atom in which the electron is in the 3d level.
S-:::. )/ L. )= z. :::; '.j: Si',,. 'I I l..
T 1. w;.. l., 5"'/i.
$ ~ 1/1- r-~ 3/l..
Cll 3
8. Give the electron configuration for a lithium atom in its ground state, and write ~ut the term designation for that state.
\ s 't 2S L; ~ I"-~ IA .,._e)_ srd:4
1~~ 2 s I s -:.;:. r"L ) ;: 0 =:; s '>~
"2 $_,. I
L :r =)
9. List the electron configurations for the following elements in their ground states: aluminum, fluorine, calcium, germanium, and krypton.
L.#3
10. Determine the number of states and write out the term designations (eqn. 3.76) for those states that result from an outer electron configuration of 2s4p.
2S '-IF
'2.S 9 .-2 =- () s =- v'L Yp ==? 1 :: I S = I/?..
L :. I s--' ~~~~~--.ltllw,i::U
I + \ - L 7 .J:.L+~ - --
I +-- l) - ~ -
I - I - () =? -
L ~ I S =D
T::. L-+ s ::- I + o = I ~
J 1:> '-.
3 [:> (
'3 (=> 0
I /-;;;;
I
11. You are told that a short-wavelength laser is measured to have a wavelength of 30.38 nm and occurs on an H-like transition fro.m n = 4 ton =~-In what element does the laser action occur? Justify your answer. 2..
v=
:t: 'L::
, t- --, '
I'-/ :z_l. G./)X/{)
1, ~? )( l'O /';::;-
&,, f S°"" X ID 1 'I - /~, {)L/(/ -
L-/ I OD ~ ~
lH 3
12. The element magnesium (Mg) is vaporized to produce a gas of magnesium atoms. An electrical current is passed through the gas in order to excite the atoms to levels . above the ground state. Give the complete ground-state electron configuration. Make an energy-level diagram and provide the labels of the ground state and all of the excited energy levels associated with the same n quantum number as that of the ground state. Allow only one of the outer electrons to be excited .
\I
's 0
.R=-o
1. ::. I
£=6 .,R :: 2
____ :-o~ _.. ____ > Di
l -t-__........._._,...,, DI
CH3
13. Obtain the spectroscopic designation and diagram the energy levels for the following three electronic configurations:
' - ~ \)2.
f p~
'
's ()
ls2 2s2, ls2 2s2p, ls2 2s3d.
' sl. 2 s sci --=y
~D 3
lD~
~D,
>pio
lp,o
~p~
s -= l/i.. - I/ z_ ) = o) o
CS] s=V2. ~=-D 5 ~ 'Ii .t. == I
S - l .+ J. :: I L -:: o +I = I J"::. 2 > I> D - 2 2 ) --
s = vi. l::.. o s ~ 'I, )_ = 2.
S=
J:: I
14. Determine the number of energy levels and· the term designations for those levels of an atom in a state in which the two outer electrons are 2s and 2p.
$ ~ 'tz. .P:... i)
5 ': 'I-,_ .). =- '
CH3
('-1. ..j..l=1 :) - "2 "'2..
L -:. D + I = I J;: 2, 'j ()
s ?!) I "2.,1)(.J
s ::. l _.L::D L:: D 4 l =- l .J: J "l- "2.
~ I I ':lo ' I I
'-I ~ ra.te$