0Introduction

IB Quantum Mechanics


0.2 Bohr model of the atom
When we heat atoms up to make them emit light; or shine light at atoms so that
they absorb light, we will find that light is emitted and absorbed at very specific
frequencies, known as the emission and absorption spectra. This suggests that
the inner structure of atoms is discrete.
However, this is not the case in the classical model. In the classical model,
the simplest atom, the hydrogen, consists of an electron with charge
−e
and
mass m, orbiting a proton of charge +e and mass m
p
m fixed at the origin.
The potential energy is
V (r) = −
e
2
4πε
0
1
r
,
and the dynamics of the electron is governed by Newton’s laws of motions, just as
we derived the orbits of planets under the gravitational potential in IA Dynamics
and Relativity. This model implies that the angular momentum
L
is constant,
and so is the energy E =
1
2
mv
2
+ V (r).
This is not a very satisfactory model for the atom. First of all, it cannot
explain the discrete emission and absorption spectra. More importantly, while
this model seems like a mini solar system, electromagnetism behaves differently
from gravitation. To maintain a circular orbit, an acceleration has to be applied
onto the electron. Indeed, the force is given by
F =
mv
2
r
=
e
2
4πε
0
1
r
2
.
Accelerating particles emit radiation and lose energy. So according to classical
electrodynamics, the electron will just decay into the proton and atoms will
implode.
The solution to this problem is to simply declare that this cannot happen.
Bohr proposed the Bohr quantization conditions that restricts the classical orbits
by saying that the angular momentum can only take values
L = mrv = n~
for
n
= 1
,
2
, ···
. Using these, together with the force equation, we can solve
r
and v completely for each n and obtain
r
n
=
4πε
0
me
2
~
2
n
2
v
n
=
e
2
4πε
0
1
~n
E
n
= −
1
2
m
e
2
4πε
0
~
2
1
n
2
.
Now we assume that the electron can make transitions between different energy
levels
n
and
m > n
, accompanied by emission or absorption of a photon of
frequency ω given by
E = ~ω = E
n
− E
m
=
1
2
m
e
2
4πε
0
~
2
1
n
2
−
1
m
2
.
0
E
m
E
n
γ
This model explains a vast amount of experimental data. This also gives an
estimate of the size of a hydrogen atom:
r
1
=
4πε
0
me
2
~
2
≈ 5.29 × 10
−11
m,
known as the Bohr radius.
While the model fits experiments very well, it does not provide a good
explanation for why the radius/angular momentum should be quantized. It
simply asserts this fact and then magically produces the desired results. Thus,
we would like a better understanding of why angular momentum should be
quantized.