Best, -Your Quantum Mechanics Tutor

[FunkyDwarf]

Hey Guys,

Am working through Relativistic Quantum Mechanics: Wave equations by W.Greiner and have a simple question about the Klein-Gordon equation: is it fair to say that bound states only occur between -m<=E<=m? (c=1). There are a few problems where they show that you can get pair production when the energy of the bound state is -m, and go on to comment about 'diving into the lower continuum'. However, in say the finite coulomb case, if one takes the exterior wave equation, which is just the basic Coulomb problem with a -1/r potential, then the asymptotic behaviour suggests that you only get exponential soltions, ie bound states, in the range described above and not for energies more negative than that.

Are they postulating on more negative bound states purely by extrapolation or is there a way to solve for bound states whos energies are <-m? I know this is somewhat unphysical as one must consider pair production, but just as an academic exercise will those states come out of the mathematics, as i can't see how they would :(

Cheers!
-G

 

[eljose79]

reenScientist

Dear GreenScientist,

Thank you for your question regarding the Klein-Gordon equation and bound states. It is correct to say that bound states only occur between -m<=E<=m in this equation, as it is a relativistic wave equation that describes spinless particles with mass m. The range of energies for bound states is limited to this range because, as you mentioned, energies more negative than -m would result in pair production, which is not a phenomenon that can be described by the Klein-Gordon equation alone.

In the case of the finite Coulomb potential, the asymptotic behavior of the exterior wave equation does suggest that only bound states with energies in the range of -m<=E<=m exist. This is due to the fact that the potential becomes more attractive at larger distances, making it difficult for the particle to escape and form a bound state with a more negative energy.

While it may be possible to extrapolate and postulate on more negative bound states, as an academic exercise, it is not physically meaningful as it does not take into account the phenomenon of pair production. This is why the range of energies for bound states is limited to -m<=E<=m in the Klein-Gordon equation.

I hope this helps clarify your question. Keep up your studies in Relativistic Quantum Mechanics!