Biomedical Engineering Reference
In-Depth Information
13.2 Correspondence Principle
Bohr made much use of the correspondence principle discussed in Chapter 11, which says
that quantum mechanical results must tend to those obtained from classical physics, in the
limit of large quantum numbers. For example, if we consider the transition from level
n
to
level
n
−
1, the emitted frequency is
8ε
0
h
3
n
2
μ
e
4
Z
2
1
1
ν
=
1)
2
−
(
n
−
For large
n
this approximates to
μ
e
4
Z
2
8ε
0
h
3
2
n
3
ν
=
(13.9)
According to Maxwell's electromagnetic theory, an electron moving in a circular orbit
should emit radiation with a frequency equal to its frequency of revolution
v
/2π
r
. Using
the Bohr expressions for
v
and
r
we deduce an expression in exact agreement with the
frequency given by Equation (13.9).
Despite its early successes, Bohr's theory had many failings. For example, it could not
explain the structure and properties of a helium atom. Many ingenious attempts were made
to improve the model, for example by permitting elliptic orbits rather than circular ones, but
the theory has gradually faded into history. It is sometimes referred to as the
old quantum
theory
.
13.3
Infinite Nucleus Approximation
I started the chapter by correctly considering the motion of a one-electron atom about the
centre of mass, and pointed out that this was equivalent to the motion of a single particle
of reduced mass
1
m
e
Because the mass of the nucleus is so much greater than that of the electron, the reduced
mass of the atom is almost equal to (and slightly less than) the mass of the electron. For
that reason, workers in the field often treat the nucleus as the centre of coordinates and
the electron as rotating round the nucleus, which is taken to have infinite mass; this is
called the
infinite nucleus approximation
. It is just a small correction, the two masses μ
and
m
e
are equal to 1 part in 10
4
which is usually good enough, but as we have seen, atomic
spectroscopic data are known to incredible accuracy and it is sometimes necessary to take
account of this difference. We write, for an infinite mass nucleus,
1
μ
=
1
M
+
m
e
e
4
8
h
2
ε
0
1
n
2
ε
n
=−
(13.10)
m
e
e
4
8ε
0
h
3
c
0
ε
0
h
2
π
m
e
e
2
R
∞
=
and
a
0
=
where
a
0
is called the
first Bohr radius
and
R
∞
is 'the' Rydberg constant, with value
R
∞
=
10 973 731.5685458m
−
1
.
