EXAM #1
SC346, Spring 2016 Prof. M. Elert
NAME ____~X_t_-1~
___________ 1. (6 points) A laser emits photons that have a wavelength of 550 nm. The power rating ofthe laser is 1.0 watt.
(1 W = 1 lis) How many photons does the laser emit in one second?
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S"S-O
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tL
t()
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s
2. a. (6 points) Nonnalize the wavefunction 'I'(x) = Nx(l-x) over the range O:s x:S 1.
I"
(V (~) 4-y. ~
N '-
~
X ( , - ,,)
3]1
X"
--+3
o
~
~ "-"< ~
N
2....
'2. ( ;
I'J ~ ( ( ,~ -
2 ,<'
z: t- '3
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5 -
(
')
~
&
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I 2- (
,
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h. (6 points) Calculate <x2> for a particle that has the wavefunction of part (a), again assuming the particle is trapped in the range 0 :s x:s 1. <>':>:::
[,+"0)/' 'f 4
'"
o
"" 1'1'-
r",l.'/I~
0
J.'k¥('-x~4
'" 3 0
[:;;
~ ~
lo!'~
L
-=(
['C)('-2.. +-\C")JA<
-irir:: 3c(f-t+-f)~ ~C{r-3"+-d
- . ,-3'"
-
30
k s'
3. (4 points) Find the value of the commutator
(f - x ) ( ~
4-
~) f ~ (~- ~) ( f
'.;- x f)
;:
1; (f (
f-
-X
f) -
(f f- ~ +J
X
r
f" 4- x .f ( 4- f - X += I - x2 f ::: f ({ +- f - X -+=
f ~ (~+- x) (fl_ x.?) .~
(f '- X t=) t- ~ ( f~ ~ t)
-=
(~+~)( f
- X)
cL 2.
II
f
-:::
f
f
-)(
I
(1...r
-.f-t-?<-P-K1':::
[~-\(/ ~f''1l]+ ~ (r"f.f:-)(L+-)~(f('_f-,,1.fJ ~ 2-f
r
I,
T
f'
--r-X
SIS
'2.
r
1'""
[~-~/-t+)(]:::
2­
4. (6 points) Calculate the zero-point energy for an electron in a three-dimensional cubic box of length 0.1 0 nm,
which is about the size of an atom. Express your answer in joules.
E" I I I
(I
~
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:t
-+
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_
? (<t, iG' )((()-'J(~) (1.0>«0 -(()~t-
_1'1
f.fl \1/6
:r
5. (6 points) Calculate the wavelength of a photon needed to excite a transition between neighboring energy
levels of a harmonic oscillator of effective mass equal to that of an oxygen atom (15.9949 mu) and force
constant kf 455 N/m.
il£
~
1;w
-
=-
1i
(~)''2. -.: :-
(f.oS-S--
)(,/()-3i1.
~
J'5)[-(IS'. 99'fcr.)((,f;b(k!(j'-t.
NIl'¥..
J"L
I../-S'!>-
>-
,
~)
( (•OS'S" 'If 10- 1 If J. ~ ) ( I. '50 f 7 ~ I () f If S - ()
1.38'{
-~(\
'c
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T
c=,. bZ(.~/(J-1'1- J .5) (G. 9977 -,.(C/rr-t/s)
,~
.
-5
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3 K{ >/6-2.'"
J
~
It.(-. tf~
6. (4 points) Locate the nodes of the harmonic oscillator wavefunction with v=3. (You can express the positions
in terms ofthe dimensionless length variable y.)
WAV£.,....·f.J/VCTI<J'V
1-13 (~)::
g-1 3 -
(J ~€".k.O
12..;/ = 0
~ (&,,~1._(1.) ~O
WNel!.E {.(=-kMtle !'OLYNOf..fIAL If l{'lfO.
7. A molecule is found to have a £=1 rotational energy level at 7.50 x 10-23 J.
a. (4 points) Calculate the moment of inertia ofthis molecule.
{J/r
E -=- ~ (£+-1) tiXr
I:; t
2
--;:
-= JJ. Q;-~- )( 1(; -'3
F:()~
J. 5;) 7.
7. r() '" 10-'1.'3 J
j
/E
If-
_
b. (4 points) What is the value of the angular momentum of the molecule in its £=1 state?
J
=-1/~(lyt)
t
=
-:::
h (LO:':S" y.(()-~t/- J-:J
I. 1(..7 2...l< 10- ~ if J. J
A.!Z- 1:;
?-
c. (4 points) What wavelength of light will be required to excite the molecule from the £=1 to the £=2 level?
'tit
11'2. E -::. (I) Ct 1-' () if -::- -:;::: -: :
-.L­
I
I~l.
-::
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~ e.. - to,
(")(V·')
-::
h~
:0
7.
sa It>
-t3
>A
T
(t'L)
~ 3 T
3 (}) -
(f) -: "- Lt'}
2-( v'a ~/6- D r)
-:t.<.. ­
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J
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