Do Delta Potentials Allow for Even Solutions in Quantum Mechanics?

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Homework Help Overview

The problem involves analyzing a delta potential in quantum mechanics, specifically focusing on the potential defined by V(x) = αδ(x) for -a < x < a and V(x) = ∞ for |x| > a. The original poster is tasked with examining both even and odd solutions to find the allowed energies.

Discussion Character

  • Exploratory, Conceptual clarification

Approaches and Questions Raised

  • The original poster attempts to analyze even solutions using the wave function form ψ(x) = A cos(kx) for 0 < x < a and questions whether this leads to A = 0 at the delta barrier, suggesting a potential absence of even solutions. Other participants question the understanding of even and odd solutions and clarify what is meant by these terms.

Discussion Status

The discussion is ongoing, with participants exploring the initial approach to the problem. Some guidance has been offered regarding the definitions of even and odd solutions, but there is no explicit consensus on the correctness of the original poster's approach or the existence of even solutions.

Contextual Notes

Participants are navigating potential confusion regarding the definitions of even and odd solutions in the context of quantum mechanics, which may affect their interpretations and approaches to the problem.

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Homework Statement


Consider the potential:

[tex]V(x) = \alpha\delta(x)[/tex] -a<x<a
[tex]V(x) = \infinity[/tex] |x|>a


Analyze the even and odd solutions separately, and find the allowed energies.

Homework Equations





The Attempt at a Solution



So far, I looked at the even solutions:

[tex]\psi(x)=A\cos(kx)[/tex] 0<x<a
[tex]\psi(-x)[/tex] -a<x<0

With this solution, Acoskx must equal zero at the delta barrier correct?
Since the only way this could happen is for A=0, am I to assume the even solutions don't exist?

Am I going about this correctly? Thanks for any help you can offer.
 
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I think I may be confused on what they mean by "the even and odd solutions."
 
Last edited:
Solutions which are odd and even under coordinate inversion...
 
Ok I think I get that much, did I at least start this problem the correct way? Like, do I have the correct form of the even solutions?
 

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