130_notes.dvi

Higher order terms can be computed but its not recommended.

Some atomic states, such as the2s state of Hydrogen, cannot decay by any of these terms basically
because the 2s to 1s is a 0 to 0 transition and there is no way to conserve angular momentum and
parity. This state can only decay by the emission of two photons.

While E1 transitions in hydrogen have lifetimes as small as 10−^9 seconds, theE2 and M1 transi-
tions have lifetimes of the order of 10 −^3 seconds, and the2s state has a lifetime of about
1
7 of a second.

33.13Black Body Radiation Spectrum

We are in a position to fairly easily calculate the spectrum of Black Bodyradiation. Assume there
is acavity with a radiation field on the insideand that the field interacts with the atoms of
the cavity. Assumethermal equilibriumis reached.

Let’stake two atomic states that can make transitions to each other: A→B+γand
B+γ→A.From statistical mechanics, we have

NB

NA

=

e−Eb/kT

e−EA/kT

=e ̄hω/kT

andfor equilibriumwe must have

NBΓabsorb = NAΓemit

NB

NA

=

Γemit

Γabsorb

We havepreviously calculated the emission and absorption rates. We can calculate the ratio
between the emission and absorption rates per atom:

NB

NA

=

Γemit

Γabsorb

=

(n~k,α+ 1)

∣

∣

∣

∣

∑

i

〈B|e−i

~k·~ri

ˆǫ(α)·~pi|A〉

∣

∣

∣

∣

2

n~k,α

∣

∣

∣

∣

∑

i

〈A|ei~k·~riˆǫ(α)·~pi|B〉

∣

∣

∣

∣

2

where the sum is over atomic electrons. The matrix elements are closely related.

〈B|e−i

~k·~ri

ˆǫ(α)·~pi|A〉=〈A|~pi·ˆǫ(α)ei

~k·~ri

|B〉∗=〈A|ei

~k·~ri

ˆǫ(α)·~pi|B〉∗

We have used the fact that~k·ǫ= 0.The two matrix elements are simple complex conjugates
of each otherso that when we take the absolute square, they are the same. Therefore, we may
cancel them.

NB

NA

=

(n~k,α+ 1)

n~k,α

=e ̄hω/kT

1 =n~k,α(e ̄hω/kT−1)

n~k,α=

1

e ̄hω/kT− 1