Δ
k
=
Δ
p
⇒
Δ
k
=
1
Δ
p
Δ
k
·
Δ
x
=
1
Δ
p
·
Δ
x >
2
π
Δ
x
·
Δ
p >
×
2
π
=
h
2
π
×
2
π
=
h
Δ
x
·
Δ
p > h
The Bohr model
In 1897, J.J. Thompson discovered the electron.
It was then clear that matter, whch is
macroscopically neutral, is made of charged parts, the electron being a negatively charged
particle. There must be therefore a positively charged part, and Thompson suggested the
“plum pudding” model, in which here is a continuum of positive charge distribution, in which
negatively charged electrons are embedded.
In 1909, Geiger and Marsden found that alpha particles (that are positively charged) are
sometimes (about 1 in 10,000) reflected by very large angles when are fired at a thin gold
leaf. It is very hard to reconcile this experimental result with the “plum pudding” model.
Rutherford described this expriment as follows: “It was almost as if you fired a fifteen–inch
shell at a piece of tissue paper and it bounced back and hit you.”
Rutherford was motivated by the Geiger–Marsden results to propose the “solar system”
model, in which most of the atom’s mass is in a small nucleus
, and the electrons orbit the
nucleus. There is an obvious problem with this model: an electron orbiting the nucleus must
radiate, becasue it is constantly accelerated.
Losing energy to radiation, the orbit must
decay. We next estimate the lifetime of the hydrogen atom based on the Rutherford model.
The radiated power is given by Larmor’s formula,
P
=
2
3
ke
2
a
2
c
3
where
k
= 1
/
(4
π
0
),
e
is the electron charge, and
a
is the acceleration. Take now
a
=
v
2
/r
,
and find
P
=
2
3
ke
2
v
4
r
2
c
3
=
2
3
ke
2
(
v/c
)
4
c
r
2
.
As typical parameters, to be jusified below, we take
v/c
= 1
/
137 and
r
= 0
.
0529 nm, and
use
ke
2
= 1
.
44 eV
·
nm. We therefore find,
P
=
2
3
×
1
.
44 eV
·
nm
×
1
137
4
×
3
×
10
17
nm
/
s
(0
.
0529 nm)
2
= 2
.
92
×
10
11
eV
s
124
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 Spring '09
 LIORBURKO
 Physics, Atom, Charge

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