Discrete state coupled to a continuum

In summary, the conversation discusses a project involving a discrete state and continuous states connected by a complex coupling. The problem at hand is how to normalize the wavefunction with the continuous states, which is not possible in principle. One attempt at a solution involves adding a factor to the wavefunction, but it is uncertain if this is the correct method. The speaker has recently come up with a potential solution, but welcomes any other ideas or corrections.
  • #1
Heimisson
44
0

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 [tex]e^{\epsilon x}[/tex] 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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  • #2
I actually just realized how this might work, but please leave an reply if you have an better idea or I'm wrong.
 

FAQ: Discrete state coupled to a continuum

1. What is the meaning of "discrete state coupled to a continuum" in scientific terms?

"Discrete state coupled to a continuum" refers to a physical system where there are discrete energy levels (or states) that are connected to a continuous energy spectrum. This can occur in various systems, such as atoms or quantum mechanical systems, where there are discrete energy levels within a larger continuous energy range.

2. How does a discrete state become coupled to a continuum?

A discrete state can become coupled to a continuum through various mechanisms, such as through interactions with other particles or through external fields. This coupling allows for the exchange of energy between the discrete state and the continuum, leading to changes in the energy levels of the system.

3. What are some examples of systems that exhibit a discrete state coupled to a continuum?

Some examples include atoms, where discrete energy levels are coupled to a continuum of higher energy states, and quantum dots, where electrons in a confined region are coupled to a continuum of energy levels in the surrounding material.

4. How does the coupling between a discrete state and a continuum affect the behavior of the system?

The coupling between a discrete state and a continuum can lead to interesting phenomena, such as quantum tunneling and energy exchange between the discrete and continuous levels. It can also affect the stability and lifetime of the discrete state, as well as the overall energy distribution of the system.

5. What are the implications of a discrete state coupled to a continuum in scientific research and applications?

The concept of a discrete state coupled to a continuum is important in many areas of scientific research, such as quantum mechanics, atomic and molecular physics, and solid-state physics. It also has applications in technology, such as in quantum computing and energy harvesting devices.

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