Finite Square Well Analysis: Odd Bound States & Scattering States

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Discussion Overview

The discussion revolves around the analysis of odd bound states in a finite square well potential, specifically examining the conditions under which these states exist and the nature of scattering states. Participants explore the implications of energy levels, particularly questioning whether E=0 can be classified as a bound state.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions whether E=0 is a bound state or a scattering state, seeking clarification on the definitions of these states.
  • Another participant notes that odd states relate to the behavior of wavefunctions under parity, suggesting that the solutions could involve sine or hyperbolic sine functions depending on the energy and potential.
  • It is mentioned that bound states are normalizable and correspond to physical states, while scattering states are described as nonnormalizable.
  • A participant provides an analogy involving waves generated by a rock in water to explain scattering states, emphasizing their behavior at infinity.
  • One participant expresses concern that if E=0 is a bound state, it would imply the existence of at least one odd bound state, which contradicts their understanding based on a textbook reference.
  • Another participant asserts that a sine wave function is not normalizable, thus not qualifying as a bound state.

Areas of Agreement / Disagreement

Participants express differing views on the classification of E=0, with some suggesting it may not correspond to a bound state, while others argue against this notion based on their interpretations of the problem. The discussion remains unresolved regarding the existence of odd bound states and the implications of E=0.

Contextual Notes

Participants reference the need for a general solution and the implications of energy levels on the classification of states, indicating potential limitations in their current understanding or definitions used.

broegger
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Hi,

I have a problem with the finite square well. I have to analyze the odd bound states of the finite square well,

[tex]V(x)=<br /> \begin{cases}<br /> -V_0 & \text{for } -a<x<a\\<br /> 0 & \text{otherwise}<br /> \end{cases}. [/tex]​

Specifically, I have to examine the limiting cases (wide, deep well and narrow, shallow well) and find out, if there is always at least one odd bound state.

When I try to determine the energies of these odd states, I find that E=0 is always a solution. Is E=0 a bound state, a scattering state or what?

Also, what exactly are scattering states?
 
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broegger said:
Hi,

I have a problem with the finite square well. I have to analyze the odd bound states of the finite square well,

[tex]V(x)=<br /> \begin{cases}<br /> -V_0 & \text{for } -a<x<a\\<br /> 0 & \text{otherwise}<br /> \end{cases}. [/tex]​

Specifically, I have to examine the limiting cases (wide, deep well and narrow, shallow well) and find out, if there is always at least one odd bound state.

Okay,did you find the "normal" /general sollution...?Odd states refer to the behavior of wavefunctions under parity...In your case,depending on the E:V,it could be only [itex]\sin[/itex] or [itex]\sinh [/tex].<br /> <br /> Bound states are normalizable states,physical states according to I-st postulate...In your case,which would be those...??<br /> Scattering states would correspond to nonnormalizable states...<br /> <br /> Daniel.[/itex]
 
Scattering states are those that satisfy the Somerfeld Radiation Condition, which is obeyed if a state behaves like a plane wave at infinity. Drop a rock in a water wave, say from a speed boat. The wave pattern will settle down and stabilize-- the original wave will still be going, as will "scattered waves" generated by the rock -- and these waves will behave like free waves, once they've gone out a bit from the rock's splash.
Regards,
Reilly Atkinson
 
dextercioby said:
Okay,did you find the "normal" /general sollution...?Odd states refer to the behavior of wavefunctions under parity...In your case,depending on the E:V,it could be only [itex]\sin[/itex] or [itex]\sinh [/tex].<br /> <br /> Bound states are normalizable states,physical states according to I-st postulate...In your case,which would be those...??<br /> Scattering states would correspond to nonnormalizable states...<br /> <br /> Daniel.[/itex]
[itex] <br /> My question is really, does E=0 correspond to a bound state? I hope not, because this implies that there is always one odd, bound state (and there isn't according to the book).[/itex]
 
No,the wave function is a "sine",which is not a bound state...It's not normalizable...

Daniel.
 
Ok, thanks!
 

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