I have a particle which is initially in a bound state for a given voltage in the form of a delta function at the origin,
V = -αδ(x)
initial state is ψα = (√αm)/h2*exp(-m*α*|x|/(h2)
At t=0, voltage is changed to V = -βδ(x)
both α and β are greater than zero. Right now I'm just struggling to find ψβ
All equations I have are included above.
The Attempt at a Solution
I realize the new bound state is in the same form as it was earlier, as all bound states for attractive delta functions have only one bound state and if I solve schrodinger's equation for this delta function, it comes out to the same equation except replaced with β. However, I realize there's also a probability of the particle radiating away. My attempt at solving this was treating it as a free particle radiating away to the left or right and looking into what I get , thinking that if I added them together then I would get the answer for ψβ, but it blew up and made no sense (and seemed unsolvable, at least for how I tried to do it).
I'm a bit lost as to how to do this problem. I know there are three possibilities: It stays in the bound state, it moves off to the left, and it moves off to the right. However, I have no clue as to find the function for the unbound particle state.