What are the Bound States for a Sum of Two Negative Delta-Function Potentials?

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Discussion Overview

The discussion revolves around the nature of bound and scattering states in quantum mechanics, specifically in the context of delta-function potentials. Participants explore how the sign and configuration of these potentials affect the existence of bound states.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants note that for a positive delta-function potential, there are only scattering states and no bound states.
  • Others argue that a negative delta-function potential can support one bound state, but the specifics require solving the problem mathematically.
  • A participant questions whether the sum of two negative delta potentials would lead to two bound states, indicating a need for further analysis of the potential configuration.
  • There is a discussion about the approach to solving the problem, including breaking the space into sections based on the locations of the delta potentials.

Areas of Agreement / Disagreement

Participants generally agree on the behavior of positive and negative delta-function potentials regarding bound states, but there is uncertainty about the implications of summing two negative delta potentials and whether this leads to multiple bound states.

Contextual Notes

Participants express limitations in their understanding of how to properly analyze the potential configuration and the mathematical steps required to determine the number of bound states.

Logan Rudd
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I'm reading through Griffiths Intro to QM 2nd Ed. and when it comes to bound/scattering states (2.5) they say:

##E<0 \implies## bound state

##E>0 \implies## scattering state

Why doesn't this change depending on whether you have a positive or negative delta-function potential?
 
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It does change. If you have a positive delta-function potential, there are no bound states.
 
Why is that, and why is it not the case for a negative delta-function potential?
 
A positive delta function potential only has scattering states since it's just a scattering problem. The reason it has no bound states is the same reason that a positive finite potential well has no bound states. A positive delta-function potential has no way to "trap" a particle into a bound state.

A negative delta-function potential has 1 (and only 1) bound state, but to figure that out, you actually have to go ahead and solve the problem.
 
Ahh, I see! But if the potential I am working with is the sum of two negative delta potentials then would there be two bound states? I'm trying to work it out in a similar fashion as the text works it out for a a single negative delta potential centered at 0 but since both of mine are centered about ##\pm a## how do I break it up into sections. I tried analyzing from ##x<-a##, ##-a<x<0## to get ##\psi(x) = Be^{-\kappa x}## and for ##x>a##, ##0<x<a## to get ##\psi(x) = Be^{-\kappa x}## but it seems like there is more to it than that.
 

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