Fundamental Astronomy

98
5. Radiation Mechanisms
From (5.2) and (5.5) it follows that
vn = e2
1
4πɛ0 n , rn =
4πɛ0 2
me 2 n2 .
The total energy of an electron in the orbit n is now
En = T + V = 1
2 mv2 1 e
n −
4πɛ0
2
rn
=− me4
32π 2 ɛ 2 0 2
1
n
1
≡−C
2 n
2 ,
(5.6)
where C is a constant. For the ground state (n = 1), we
get from (5.6)
E1 =−2.18 × 10 −18 J =−13.6eV.
Fig. 5.4. Transitions of a hydrogen
atom. The lower
picture shows a part of
the spectrum of the star
HD193182. On both sides
of the stellar spectrum we
see an emission spectrum
of iron. The wavelengths
of the reference lines are
known, and they can be
used to calibrate the wavelengths
of the observed
stellar spectrum. The hydrogen
Balmer lines are
seen as dark absorption
lines converging towards
the Balmer ionization limit
(also called the Balmer discontinuity)
at λ = 364.7nm
to the left. The numbers
(15,...,40) refer to the
quantum number n of the
higher energy level. (Photo
by Mt. Wilson Observatory)
From (5.3) and (5.6) we get the energy of the quantum
emitted in the transition En2 → En1 :
1
hν = En2 − En1 = C
n2 −
1
1
n2
. (5.7)
2
In terms of the wavelength λ this can be expressed as
1 ν C 1
= =
λ c hc n2 −
1
1
n2
1
≡ R
2 n2 −
1
1
n2
, (5.8)
2
where R is the Rydberg constant, R = 1.097 × 107 m−1 .
Equation (5.8) was derived experimentally for n1 = 2
by Johann Jakob Balmer as early as 1885. That is why
we call the set of lines produced by transitions En → E2
the Balmer series. These lines are in the visible part of
the spectrum. For historical reasons the Balmer lines are
often denoted by symbols Hα,Hβ,Hγ etc. If the electron
returns to its ground state (En → E1), we get the Lyman