PHY 251 Fall 2009: Final Exam (Thursday, Dec. 17). Some masses in u:
1
1
H
1.007 825
206
82
Pb
205.974 449
210
84
Po
209.982 857
2
1
H
2.014 102
208
83
Bi
207.979 727
211
84
Po
210.986 637
3
2
He
3.016 029
209
83
Bi
208.980 383
212
85
At
211.990 735
4
2
He
4.002 603
209
84
Po
208.982 416
214
86
Rn
213.995 346
1. Calculate the wavelengths associated with a 100 eV photon, a 20 eV electron, and a 200 keV
electron.
Answer:
The wavelength of the 100 eV photon is
λ
=
hc
E
=
1240
eV
·
nm
100
eV
= 12
.
4
nm
.
The 20 eV electron is nonrelativistic, so we have
E
k
=
p
2
/
2
m
and
p
=
√
2
mE
k
giving a
wavelength of
λ
=
h
p
=
h
√
2
mE
k
=
hc
√
2
mc
2
E
k
=
1240
eV
·
nm
r
2
·
(511
×
10
3
eV
)
·
(20
eV
)
= 0
.
274
nm
.
The 200 keV electron energy is close enough to the rest mass that we must use a relativistic
approach. We know
E
k
= (
γ

1)
mc
2
so
γ
= 1 +
E
k
mc
2
= 1 +
200
keV
511
keV
= 1
.
39
and then from
γ
≡
1
/
√
1

β
2
we can find
β
=
r
1

1
/γ
2
=
r
1

1
/
(1
.
39
2
) = 0
.
695
We can then use
λ
=
h/p
and
p
=
γmv
=
γβmc
to find
λ
=
h
γβmc
=
hc
γβmc
2
=
1240
eV
·
nm
1
.
39
·
0
.
695
·
(511
×
10
3
eV
)
= 0
.
0025
nm
.
2. A He
+
ion has its electron undergo a transition from the
2
p
to the
1
s
state. What is the mean
wavelength of the emitted photon? If the
2
p
state were to have a lifetime of
2
×
10

14
seconds
or 20 fsec, what would be the wavelength spread about the mean?
Answer:
We have
E
=

E
0
Z
2
/n
2
and a correction of
m
r
/m
e
with
m
r
m
e
=
(4
·
939
·
0
.
511)
/
(4
·
939 + 0
.
511)
0
.
511
= 0
.
99986
which we will ignore. The photon energy is then
Δ
E
=
E
2

E
1
= (

13
.
6)
2
2
2
2

(

13
.
6)
2
2
1
2
= 13
.
6(4

1) = 40
.
8
eV
and the photon wavelength is
λ
=
hc/E
= 1240
/
40
.
8 = 30
.
4
nm. The energy width is given
by the uncertainty principle
(Δ
E
)(Δ
t
) = ¯
h/
2
or
Δ
E
=
¯
h
2Δ
t
=
6
.
582
×
10

16
eV
·
sec
2
·
(2
×
10

14
sec
)
= 0
.
016
eV
1
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View Full Documentand since
Δ
E/E
= Δ
λ/λ
we have
Δ
λ
=
λ
Δ
E
E
=
hc
E
Δ
E
E
=
(1240
eV
·
nm
)(0
.
016
eV
)
(40
.
8
eV
)
2
= 0
.
012
nm
as the
1
σ
spread in wavelength.
Note: it is incorrect to say
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 Fall '01
 Rijssenbeek
 Physics, Radioactive Decay, Energy, Mass, Photon, ev

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