130_notes.dvi

Hereαrepresents all the (other) quantum numbers of the state, not the angular momentum quantum
numbers.jmrepresent the usual angular momentum quantum numbers of the states.〈α′j′||V||αj〉
is a reduced matrix element. Its the same for all values ofmandq. (Its easy to understand that if
we take a matrix element of 10rit will be 10 times the matrix element ofr. Nevertheless, all the
angular part is the same. This theorem states that all vectors have essentially the same angular
behavior. This theorem again allows us to deduce that ∆ℓ=− 1 , 0 .+ 1.

The theorem can be generalized for spherical tensors of higher (or even lower) rank than a vector.

29.10Exponential Decay

We have computed transition rates using our theory of radiation. In doing this, we have assumed
that our calculations need only be valid neart= 0. More specifically, we have assumed that we start
out in some initial stateiand that the amplitude to be in that initial state is one. The probability
to be in the initial state will become depleted for times on the order ofthe lifetime of the state. We
can account for this in terms of the probability to remain in the initial state.

Assume we have computed the total transition rate.

Γtot=

∑

n

Γi→n

This transition rate is the probability per unit time to make a transitionaway from the initial state
evaluated att= 0. Writing this as an equation we have.

dPi

dt

∣

∣

∣

∣

t=0

=−Γtot

For larger times we can assume that the probability to make a transition away from the initial state
is proportional to the probability to be in the initial state.

dPi(t)

dt

=−ΓtotPi(t)

The solution to this simple first order differential equation is

Pi(t) =Pi(t= 0)e−Γtott

If you are having any trouble buying this calculation, think of a large ensemble of hydrogen atoms
prepared to be in the 2p state att= 0. Clearly the number of atoms remaining in the 2p state will
obey the equation
dN 2 p(t)
dt

=−ΓtotN 2 p(t)

and we will have our exponential time distribution.

We may define the lifetime of a state to the the time after which only^1 eof the decaying state remains.

τ=

1

Γtot