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AriAstronomer
2010-Nov-02, 08:09 PM
Hey guys,
So I know that I've been asking alot of quantum questions lately, but I'm taking the course this year, and questions keep looming:
With respect to the schrodinger equation, I've been told that when E>0 you encounter a scattering state, and when E<0 you encounter a bound state. The question I have is qualitatively determining whether the energy is greater or less than 0.
For instance, if we have a 1-D finite square well, where V(x), the potential, equals -Vo between -a and a (centered around x = 0), and otherwise the potential equals 0 (as x-> +/- infinity), how can one instantly tell if bound states or scattering states are going to occur. The real point is being able to look at a situation, and know if the energy will be less than or greater than the potential (which is general 0 at +/- infinity)

Thanks a bunch.

Ken G
2010-Nov-02, 10:54 PM
You would need to know the "wave function", because the expectation value of the energy is <psi|H|psi> where psi(x) is the wave function, H = KE + V(x) where KE is a second order in x differential operator for the kinetic energy (look it up ), and < | > means take a dot product of the psi*(x) and psi(x) with the H operator in between (in the x basis, a dot product means integrate over all x from -infinity to infinity). Then you have the expected energy of the state.

In practice, invariably a scattering problem involves a "matrix element" between an incoming and outgoing plane wave (see "Fermi's Golden rule"). A matrix element looks just like the above dot product, but it has two different psi functions, not the same one, and they are generally plane waves (eigenstates of the momentum operator). So you can usually tell if you are doing a scattering problem-- the particle comes in from infinity and goes out to infinity-- versus a bound state problem, where the wave function is concentrated in the well and you seek energy eigenstates of the H operator. The two problems are connected-- what couples incoming and outgoing wave states is related to the bound states inside the well, bringing up the possibility of "resonance".

Andrew D
2010-Nov-03, 03:59 PM
Where H is the hamiltonian?

Ken G
2010-Nov-03, 04:29 PM
Yes.

astromark
2010-Nov-03, 06:59 PM
Stop it gentleman... Surly the answer you seek is in the question. If E > 0 then its potentiating kinetic energy, and thus open... or unbound...

+ / - infinity, is still infinity. where's the question ?