DATE:
September 30, 2011
DUE:
October 7, 2011
ENG45, Solution to homework set #1.
1.
The Bohr Atom. This is the most primitive model for why the atom is stable and for how
atomic energies are quantized. It is built on a planetary type motion of an electron cir
cularly orbiting a (Hydrogen) nucleus. Based on realizations from the photoelectric
effect and special relativity, where photon energy
E
ph
=
hν
and photon momentum
is
p
=
h/λ
[
h
is Planck’s constant,
ν
is the frequency (in Hz) and
λ
is the wave
length. The speed of light is
c
=
νλ
], the electron momentum is associated with a
wave length, and the quantization of orbits is accomplished. The results are:
r
n
=
r
B
n
2
, r
B
=
¯
h
2
4
πε
0
m
e
e
2
(1)
v
n
=
v
B
1
n
, v
B
=
¯
h
m
e
r
B
(2)
E
n
=

E
B
1
n
2
, E
B
=
1
2
e
2
4
πε
0
r
B
(3)
where
n
= 1
,
2
,
···
is the (primary) quantum number,
¯
h
=
h/
2
π
,
ε
0
is the vacuum
permittivity,
m
e
is the mass of the electron
1
, and
e
is the proton charge. The three cal
culated constants are the Bohr radius
r
B
, the Bohr velocity
v
B
, and the Bohr Energy
E
B
.
a. Calculate
r
B
,
v
B
, and
E
B
as given by Eqs. (13) as well as with the revision
given by the footnote. Find the differences between the values. Express
E
B
in
both Joules and eV.
Looking up the natural constants we get:
r
B
= 0
.
529178
˚
A
(4)
v
B
= 2
.
18769
×
10
6
m/s
(5)
E
B
= 2
.
179905
×
10

18
J
= 13
.
606
eV
(6)
If we use the reduced mass, which is
μ
= (1 + 5
.
45
×
10

6
)

1
m
e
, we get:
˜
r
B
= 0
.
529466
˚
A
(7)
˜
v
B
=
v
B
(8)
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 Fall '11
 jensen
 Periodic Table, Photon, Ode, Atomic orbital, Fundamental physics concepts

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