Introduction to Quantum Ideas
53
Now, the dynamical condition for circular orbits is
2
2
2
2
0
4
e e
r
Ze
m r
m r
r
ω =
ω = πε
(2.59)
while the quantum condition for the angular momentum is
L = m r r
2
ω = n
(2.60)
Solving for r, Eqs. (2.59) and (2.60) give
2 2
0
2
4
r
n
r
m Ze
πε
=
(2.61)
This is the radius of the nth Bohr orbit. Furthermore, the equilibrium condition
in Eq. (2.59) allows us to write the total energy as
2
0
8
Ze
E
r
= − π ε
(2.62)
Using the value of r given in Eq. (2.61), the allowed energies of the atom
are
2
,
1, 2, ...
n
hcR
E
n
n
= −
=
(2.63)
where
2
2
3
0
4
4
r
m
Ze
R
c


=


πε
π


(2.64)
Substituting the values of the constants, for the hydrogen atom
R H = 1.09678 × 10
7
m
–1
(2.65)
which is very close to Rydberg’s original value in Eq. (2.40).
The atom can undergo transitions only between the orbits with the discrete
energies E n given in Eq. (2.63). The frequency of the radiation emitted or absorbed
when the atom undergoes a transition from a state with energy E n to a state
with energy E m , is given by
2
2
1
1
cR m
n


ν =
−




for emission,
(2.66)
2
2
1
1
cR n
m


ν =
−




for absorption
It is also implied that if the atom is in the ground state, i.e., the state with the
lowest energy, n = 1, it continues to remain in that state unless an external