Delta well + infinite barrier -> bound state

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SUMMARY

The discussion focuses on the quantum mechanical analysis of a delta potential well represented by V(x) = -λδ(x-d) for x > 0, with an infinite barrier at x = 0. Participants explore the conditions for the existence of bound states, emphasizing that the wavefunction must decay exponentially for x > d. The derived condition for at least one bound state is λd > ħ²/(2m), indicating a direct relationship between the strength of the delta potential (λ) and its position (d). The conversation also touches on the nature of bound states and the implications of energy levels in this context.

PREREQUISITES
  • Quantum mechanics fundamentals, including Schrödinger's equation
  • Understanding of delta potentials and their implications in quantum systems
  • Familiarity with wavefunction behavior in bound and unbound states
  • Knowledge of mathematical techniques for solving transcendental equations
NEXT STEPS
  • Study the properties of delta potentials in quantum mechanics
  • Learn about the mathematical derivation of bound state conditions in quantum systems
  • Explore the implications of energy levels in quantum mechanics, particularly in relation to bound states
  • Investigate the use of transcendental equations in quantum mechanics and their solutions
USEFUL FOR

Quantum physicists, graduate students in physics, and researchers focusing on quantum mechanics and potential theory will benefit from this discussion.

  • #31
reilly said:
Think of the delta function, with positive lambda, as a very narrow, deep potential well. Use your common sense and experience with the potential well to get the appropriate general form of the solution.
Regards,
Reilly Atkinson

Reilly, thanks for your reply, but which part are you referring to?

As I stated in an earlier post, I was able to get the bound state solution. But in post #18, I posed the second part of the problem, which deals with the asymptotic nature of the wavefunction in case of a scattering problem for the same V(x). I solved that and got an equation for the phase shift.

But as far as I can tell, the phase shift can be determined exactly from that equation and unless I have made a mistake in the computation, there is no need to assume that \phi \approx ak and determine a as the question demands.
 

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