Discrete state coupled to a continuum

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SUMMARY

The discussion centers on the challenge of normalizing a wavefunction that represents a discrete state, specifically a half simple harmonic oscillator, coupled to continuous states via a complex coupling. The user has attempted to modify the wavefunction by introducing a factor of e^{\epsilon x} to ensure convergence of the integral at negative infinity. However, they express uncertainty about the effectiveness of this method and seek alternative solutions or confirmations from the community.

PREREQUISITES
  • Understanding of quantum mechanics, particularly wavefunctions and normalization.
  • Familiarity with simple harmonic oscillators and their properties.
  • Knowledge of complex coupling in quantum systems.
  • Experience with mathematical techniques for integral convergence.
NEXT STEPS
  • Research normalization techniques for wavefunctions in quantum mechanics.
  • Explore methods for handling complex coupling in quantum systems.
  • Study the implications of introducing convergence factors like e^{\epsilon x} in wavefunctions.
  • Investigate the mathematical foundations of discrete and continuous state coupling.
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Quantum physicists, graduate students in physics, and researchers working on projects involving wavefunction normalization and coupling between discrete and continuous states.

Heimisson
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Homework Statement



This is not so much a homework problem but a part of a project I'm working on.

So in just a few words; what I have (at time t=0) is a discrete state (half simple harmonic oscillator) connected to a wire with continuous states. These states are coupled by a complex coupling. My problem is that somehow I will need to normalize the wavefunction with the continuous states but in principle it can't be normalized.



Homework Equations





The Attempt at a Solution



I've tried to add to the wavefunction a factor e^{\epsilon x} to make the integral converge at - infinity, because one could argue that if epsilon is small it shouldn't change the coupling and the coupling shouldn't be present very far away. But I'm not sure if this is the right method and it doesn't seem to give me anything good.
 
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I actually just realized how this might work, but please leave an reply if you have an better idea or I'm wrong.
 

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