QuantumPhysics.dvi

13.4 Switching on an interaction

We consider perhaps the simples of time-dependent potentials,

V(t) =Vθ(t) (13.36)

whereθ(t) is the step function, andVis a time-independent operator. This potential represents
an interaction that is turned on abruptly at timet= 0 and then stays on. This circumstance is a
reasonable approximation to what happens in physical situations. For example, an atom prepared
in a certain excited energy eigenstate may decay under the influence of an external perturbation,
such as an incoming wave or a vacuum fluctuation. Or a probe mayhit a target in a scattering
experiment.

We use (13.34) to first order to compute the time evolution. For simplicity, we shall assume
that the system has been prepared at timet= 0 in one of the eigenstatesn=i, so thatcn(0) =δn,i,
and we obtain,

cn(t) = δn,i−

i

̄h

Vni

∫t

0

dt′eiωnit

=

Vni

En−Ei

(

1 −eiωnit

)

(13.37)

where we have used,

Vni = 〈n|V|i〉

ωni = (En−Ei)/ ̄h (13.38)

The probability for measuring the system in staten 6 =iafter timetis then given by

Pni(t) =|cn(t)|^2 =

4 |Vni|^2

(En−Ei)^2

sin^2

(En−Ei)t

2 ̄h

(13.39)

Clearly, this probability distribution is peaked towards the smallest values of|En−Ei|. To study
this behavior more closely, it is convenient to introduce the density of statesρ(E), so that we can
treat discrete and continuous spectra on the same footing. The density of states is defined so that
ρ(E)dEequals the number of states between energiesEandE+dE. The probability for the decay
of the initial state at energyEiinto a stateEnbetweenEandE+dEis then given by

dP(E,t) =ρ(E)dE|cn(t)|^2 =ρ(E)dE

4 |VE,Ei|^2

(E−Ei)^2

sin^2

(E−Ei)t

2 ̄h

(13.40)

This gives us the probability rate, per unit energy,

dP(E,t)

dE

=ρ(E)

4 |VE,Ei|^2

(E−Ei)^2

sin^2

(E−Ei)t

2 ̄h

(13.41)

Now, we are primarily interested in this probability for long time scales, where we mean long
compared to the transition energiesE−Ei.