QM M97 1
From QED
Ionizing the delta function. It is not difficult to show (and you can take as a given) that an attractive one-dimensional delta function potential V(x) = − λδ(x), with λ a positive, real constant, has one bound state with energy . The normalized wave function for this state is:
with α a real, positive constant and .
Consider placing electrons in this state. We can ionize the system by applying an oscillating electric field along the x direction a constant, the 2 present for later convenience, and is small enough that first-order perturbation theory can be applied. For ease of calculation, you may ignore the effect of the original potential on the ionized (positive energy) states and treat them as one-dimensional plane waves. For a given (fixed) value of ω, and an oscillating electric field that turns on at t=0, what is the ionization rate for the system after a time t that is large?
We can write the perturbing potential as:
Using Fermi's golden rule for periodic perturbations:
Where ρ is the density of final states, and the F's are defined:
So that:
Which boils down to calculating the matrix element \langle n|x|m\rangle . The free particle has wavefunction:
(where we have normalized for convenience later), so that:
Integrating by parts:
So that:
In our approximation, the density of final states is given by:
where . Accordingly:
since , and En > 0 for a free particle, only the second delta function will be nonzero in our range of interest. All that is left is to integrate over the states k:
where . Solving also for α using the equivelence of energies:
So that finally:
This page was recovered in October 2009 from the Plasmagicians page on Prelim_M97_QM1 dated 01:35, 20 December 2005.