130_notes.dvi

29.14.3Estimate of Atomic Decay Rate

We have the formula

Γtot=

e^2 (Ei−En)

2 π ̄h^2 m^2 c^3

∫

dΩγ|〈φn|e−i(

~k·~r)

ˆǫ·~pe|φi〉|^2

Lets make some approximations.

ˆǫ·~p ≈ |p|=m|v|≈mαc=αmc

~k·~r ≈ ka 0 = ̄hω

̄hc

a 0 ≈

1

2 α

(^2) mc 2
̄hc
a 0 =
α^2 mc^2
2 ̄hc
̄h
αmc

=

α

2

e−i(

~k·~r)

≈ e

iα 2

≈1 +

iα

2

≈ 1

Γtot =

e^2 (Ei−En)

2 π ̄h^2 m^2 c^3

(4π)|αmc|^2

=

α(Ei−En)

2 π ̄hm^2 c^2

(4π)|αmc|^2

=

α(^12 α^2 mc^2 )

2 π ̄hm^2 c^2

(4π)|αmc|^2

=

α^5 mc^2

̄h

=

α^5 mc^2 c

̄hc

=

(0.51 MeV)3× 1010 cm/sec

(137^5 )(197 MeV F)

(10−^13 F/cm)≈ 2 × 1010 sec−^1

This gives a life time of about 50 psec.

29.15Homework Problems

1. The interaction term for Electric Quadrupole transitions correspond to a linear combination
   of spherical harmonics,Y 2 m, and are parity even. Find the selection rules for E2 transitions.


2. Magnetic dipole transitions are due to an axial vector operator and hence are proportional to
   theY 1 mbut do not change parity (unlike a vector operator). What are theM1 selection rules?


3. Draw the energy level diagram for hydrogen up ton= 3. Show the allowed E1 transitions.
   Use another color to show the allowed E2 and M1 transitions.


4. Calculate the decay rate for the 3p→ 1 stransition.


5. Calculate the decay rate for the 3d→ 2 ptransition in hydrogen.


6. Assume that we prepare Hydrogen atoms in theψnℓm=ψ 211 state. We set up an experiment
   with the atoms at the origin and detectors sensitive to the polariztion along each of the 3
   coordinate axes. What is the probability that a photon with its wave vector pointing along
   the axis will be Left Circularly Polarized?


7. Photons from the 3p → 1 stransition are observed coming from the sun. Quantitatively
   compare the natural line width to the widths from Doppler broadening and collision broadening
   expected for radiation from the sun’s surface.