Bound State Condition: Definition Explained

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Discussion Overview

The discussion centers around the precise definition of the bound state condition in quantum mechanics, exploring various interpretations and examples, including harmonic oscillators and potential wells. It involves theoretical considerations and conceptual clarifications regarding wavefunctions and energy levels.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants propose that a bound state condition is defined by energy being less than zero (E < 0) and that the wavefunction decays to zero as r approaches infinity.
  • Others argue that the harmonic oscillator, which has all energies greater than zero, still represents bound states, suggesting that the decay of the wavefunction must be further qualified.
  • A later reply points out that while a harmonic oscillator is bound by definition, if it were not in an infinite potential well, tunneling could occur, which would affect its bound state status.
  • Another participant questions whether the decay of wavefunctions to zero at infinity is sufficient for defining bound states, noting that bound states can exist in finite wells or attractive delta functions, indicating that the bound state condition may require more than just the decay property.

Areas of Agreement / Disagreement

Participants express differing views on the definition of bound states, with no consensus reached on whether the decay of wavefunctions or energy conditions alone suffice to define bound states.

Contextual Notes

There are limitations in the discussion regarding the assumptions made about potential wells and the nature of wavefunctions, as well as the implications of tunneling on bound state definitions.

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What is the precise definition of the bound state condition? Thanks in advance.
 
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E < 0 and that the wavefunction "decays" to 0 as r-> infty
 
malawi_glenn said:
E < 0 and that the wavefunction "decays" to 0 as r-> infty
Disagree with you, though I'm not insisting. Consider a harmonic oscillator: all energies > 0, still bound states. Wave function should "decay" at infinity, but something must be said about when it decays: e.g. a moving Gaussian wave packet decays faster than an exponent, but does not correspond to a bound state.
 
Well, a harmonic oscillator is bound by definition though. It's in an infinite potential well.

If it wasn't an infinite potential well, the particle could tunnel out sooner or later. So it wouldn't be bound then.
 
Yes, "If it wasn't an infinite potential well, the particle could tunnel out sooner or later. So it wouldn't be bound then".
 
Doesn't any allowed wave function decay to zero at infinity as part of the L2 condition? But not all wave functions are bound states, and we do have bound states in finite wells or attractive delta functions, so the bound state condition can't be just that.
Or am I missing the point, maybe?
 

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