Introduction to Quantum Ideas
55
2
(He )
(He )
(H)
(H)
+
+
=
r
p
p
r
m
E
E
m
(2.67)
The ratio of the reduced masses is approximately
(He )
1
1
1
(H)
(H)
(He )
+
+
≈ +
−
r
e
r
m
m
m
m
m
(2.68)
i.e., about 1.000408. Therefore, the transitions between these He
+
levels
correspond to frequencies which are slightly higher than those of the hydrogen
atom. Indeed, measurements of these small differences give a fairly accurate
determination of the ration of m e /m(H).
Bohr’s ideas can be extended to noncircular orbits also. This leads to the
conclusion (see Example 7) that the angular momentum does not uniquely
determine the energy of the atom, and that there are several angular momentum
states which correspond to the same energy. This is an example of what is
known as the degeneracy of an energy level. However, inclusion of the relativistic
corrections shows that these different angular momentum states have slightly
different energies. This results in the multiplicity of the corresponding spectral
lines. Such a fine structure of the lines (the structure is narrower for larger n
values), is indeed observed experimentally, but the quantitative predictions of
the simple model are not in agreement with the experimental observations.
The Bohr theory of the atom is essentially a theory of single-electron atom.
It does not allow a simple generalization to many-electron atoms, not even to
helium, and being ad hoc, it is not logically consistent. But its picture of an atom
with quantized orbits for the electrons, has retained its utility till today, especially
for qualitative arguments.
The existence of discrete atomic energy levels can be observed
experimentally from an analysis of collisions between atoms and electrons with
known energy. In these collisions, since the mass of the atom is much larger
than that of an electron, very little energy is carried away as kinetic energy of
the atom. However, if the energy of the electron is sufficient to raise a bound
electron to a higher energy orbit, the electron may transfer most of its energy to
the atom. This phenomenon was demonstrated by Franck and Hertz (1914).
Electrons from a filament are gradually accelerated through a vapour in a tube
[Fig. 2.8 (a)], towards a grid G and are subjected to a small retarding potential
V 0 between the grid and the plate P. When the accelerating potential is sufficiently
large to excite an atom, the electron may undergo a collision near G and transfer
most of its energy to the atom.
55
2
(He )
(He )
(H)
(H)
+
+
=
r
p
p
r
m
E
E
m
(2.67)
The ratio of the reduced masses is approximately
(He )
1
1
1
(H)
(H)
(He )
+
+
≈ +
−
r
e
r
m
m
m
m
m
(2.68)
i.e., about 1.000408. Therefore, the transitions between these He
+
levels
correspond to frequencies which are slightly higher than those of the hydrogen
atom. Indeed, measurements of these small differences give a fairly accurate
determination of the ration of m e /m(H).
Bohr’s ideas can be extended to noncircular orbits also. This leads to the
conclusion (see Example 7) that the angular momentum does not uniquely
determine the energy of the atom, and that there are several angular momentum
states which correspond to the same energy. This is an example of what is
known as the degeneracy of an energy level. However, inclusion of the relativistic
corrections shows that these different angular momentum states have slightly
different energies. This results in the multiplicity of the corresponding spectral
lines. Such a fine structure of the lines (the structure is narrower for larger n
values), is indeed observed experimentally, but the quantitative predictions of
the simple model are not in agreement with the experimental observations.
The Bohr theory of the atom is essentially a theory of single-electron atom.
It does not allow a simple generalization to many-electron atoms, not even to
helium, and being ad hoc, it is not logically consistent. But its picture of an atom
with quantized orbits for the electrons, has retained its utility till today, especially
for qualitative arguments.
The existence of discrete atomic energy levels can be observed
experimentally from an analysis of collisions between atoms and electrons with
known energy. In these collisions, since the mass of the atom is much larger
than that of an electron, very little energy is carried away as kinetic energy of
the atom. However, if the energy of the electron is sufficient to raise a bound
electron to a higher energy orbit, the electron may transfer most of its energy to
the atom. This phenomenon was demonstrated by Franck and Hertz (1914).
Electrons from a filament are gradually accelerated through a vapour in a tube
[Fig. 2.8 (a)], towards a grid G and are subjected to a small retarding potential
V 0 between the grid and the plate P. When the accelerating potential is sufficiently
large to excite an atom, the electron may undergo a collision near G and transfer
most of its energy to the atom.