A question about Dirac Delta Potential Well solution

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Discussion Overview

The discussion revolves around the interpretation of solutions to the Schrödinger Equation for a Dirac delta potential well, specifically comparing scattering states (E > 0) and bound states (E < 0). Participants explore the behavior of the wave function solutions in different regions and the implications of complex exponentials in these contexts.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant notes a contradiction in Griffith's text regarding the behavior of wave function terms for scattering states compared to bound states.
  • Another participant suggests that the difference lies in the nature of the terms: scattering states involve oscillatory complex exponentials, while bound states involve regular exponentials that can diverge.
  • A participant expresses understanding that complex exponentials do not blow up as x approaches infinity, which clarifies the confusion regarding the wave function behavior.
  • Another participant shares a similar experience of confusion and highlights the lack of clarity in their instructor's explanation, indicating a common challenge in understanding this topic.

Areas of Agreement / Disagreement

Participants express varying levels of confusion and understanding regarding the behavior of the wave function terms, indicating that multiple interpretations exist. There is no clear consensus on the explanation of the observed phenomena.

Contextual Notes

Participants reference specific terms and behaviors of wave functions without resolving the underlying mathematical details or assumptions that may affect their interpretations.

Who May Find This Useful

Students and educators in quantum mechanics, particularly those grappling with the concepts of scattering and bound states in potential wells.

Positron137
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In Griffith's Introduction to Quantum Mechanics, on page 56, he says that for scattering states
(E > 0), the general solution for the Dirac delta potential function V(x) = -aδ(x) (once plugged into the Schrödinger Equation), is the following: ψ(x) = Ae^(ikx) + Be^(-ikx), where k = (√2mE)/h. After that, he states that in the general solution for ψ(x) (stated above), both terms do NOT blow up in the section of the well where x < 0. But this doesn't make sense, because earlier, when he was demonstrating bound states (E < 0) , he stated that the second term, Be^(-ikx), blows up at infinity when x < 0. But here, for scattering states, he states that NEITHER term blows up as x < 0, which seems contradictory. Could anyone explain why this is true (why neither term blows up for a scattering state, when x < 0)? Thanks!
 
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The difference, I believe is in the "i" (of the beholder BWAAHAHAHA). But seriously. The scattering states have an i, thus are oscillatory, and the bound states don't have an i, and hence are 'regular' exponentials, which blow up at one of the infinities (+ or -).
 
Ah ok. Thanks! LOL I was getting confused. So the reason why it doesn't "blow up" as we would expect it to is because for complex exponentials, as x -> infinity, e^(ikx) and e^(-ikx) don't blow up? Actually, that kinda makes sense because e^ix is like going in a circle in the complex plane. Thanks for the clarification!
 
It's funny you ask this, I asked the exact same question and didn't get a good explanation; I don't think my instructor understood my question. I actually still have the equation circled in my textbook with a "why" written next to it. This does clarify it though, it's pretty obvious now that I think of it... I didn't notice the distinction... thanks for posting.
 
No problem! LOL yeah, I was also confused - why for bound states, one of the terms blew up, and why for the scattering states, both e^(ikx) AND e^(-ikx) terms were kept, even though x tended to negative infinity.
 

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