Show that a complex PE yields a time dependant probability of finding a particle

[jb646]

Homework Statement

Starting from Schrödinger's Equation show that a complex potential energy V=α+iβ yields a time dependent probability P of finding the particle in (-inf,inf), i.e. the particle is unstable and normalization cannot be insured over time. Compute P(+)

Homework Equations

SE: ih/2m=-h^2/2m(∫∂ψ/∂x )+Vψ

The Attempt at a Solution

I tried inserting my value of potential energy into the SE as follows
ih/2m=-h^2/2m(∫∂ψ/∂x )+(α+iβ)ψ
and then i got stuck, am i supposed to integrate next or stick an an A and try to normalize and then prove that because it is not normalizeable it is unstable. Also what is P(+)

Thanks for any advice you can offer me.

 

[dextercioby]

I don't know what they mean by P(+). Look it up in the book / ask your instructor/adviser.

Well, ensuring the conservation of probability (density) over time relies heavily on the fact that this follows from a unitary evolution of states. What you can do is to show that the Hamiltonian will no longer be self-adjoint, since that wouldn't allow us to use Stone's theorem to obtain a unitary evolution.