130_notes.dvi

this but allow us to deal with single exponentials instead of real functions.) Our final result, the
decay rate, will be independent of volume so we can let the volume go to infinity.

kxL= 2πnx dnx= 2 Lπdkx

kyL= 2πny dny= 2 Lπdky

kzL= 2πnz dnz= 2 Lπdkz

d^3 n= L

3

(2π)^3 d

(^3) k= V
(2π)^3 d
(^3) k
That was easy. We will usethis phase space formulafor decays of atoms emitting a photon. A
more general phase space formula (See Section 29.14.2) based on our calculation can be used with
more than one free particle in the final state. (In fact, even our simple case, the atom recoils in the
final state, however, its momentum is fixed due to momentum conservation.)

29.4 Total Decay Rate Using Phase Space

Now we are ready to sum over final (photon) states to get the total transition rate. Since both the
momentum of the photon and the electron show up in this equation, we will label the electron’s
momentum to avoid confusion.

Γtot =

∑

~k,pol

Γi→n→

∑

pol.

∫

V d^3 k

(2π)^3

Γi→n=

∑

pol.

∫

V d^3 p

(2π ̄h)^3

Γi→n

=

∑

λ

∫

V d^3 p

(2π ̄h)^3

(2π)^2 e^2

m^2 ωV

|〈φn|e−i

~k·~r

ˆǫ(λ)·~pe|φi〉|^2 δ(En−Ei+ ̄hω)

=

e^2

2 π ̄h^3 m^2

∑

λ

∫

d^3 p

ω

|〈φn|e−i

~k·~r

ˆǫ(λ)·~pe|φi〉|^2 δ(En−Ei+ ̄hω)

=

e^2

2 π ̄h^3 m^2

∑

λ

∫

p^2 d( ̄hω)dΩγ

pc

̄h

c

|〈φn|e−i

~k·~r

ǫˆ(λ)·~pe|φi〉|^2 δ(En−Ei+ ̄hω)

=

e^2

2 π ̄h^2 m^2 c^2

∑

λ

∫

pd( ̄hω)dΩγ|〈φn|e−i

~k·~r

ˆǫ(λ)·~pe|φi〉|^2 δ(En−Ei+ ̄hω)

=

e^2

2 π ̄h^2 m^2 c^2

∑

λ

∫

Ei−En

c

dΩγ|〈φn|e−i

~k·~r

ˆǫ(λ)·~pe|φi〉|^2

Γtot=

e^2 (Ei−En)

2 π ̄h^2 m^2 c^3

∑

λ

∫

dΩγ|〈φn|e−i

~k·~r

ˆǫ(λ)·~pe|φi〉|^2

This is the general formula for the decay rate emitting one photon.Depending on the problem, we
may also need to sum over final states of the atom. The two polarizations are transverse to the
photon direction, so they must vary inside the integral.

A quick estimate of the decay rate of an atom (See Section 29.14.3) gives

τ≈50 psec.