Understanding Scattering and Bound State Solutions in Quantum Mechanics

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SUMMARY

In quantum mechanics, scattering states exist when the energy (E) is greater than zero, resulting in an imaginary wave function (ψ). Bound states can also exhibit an imaginary ψ, but their characteristics differ from scattering states. Griffiths' "Introduction to Quantum Mechanics" (2nd ed.) discusses the representation of ψ using sine and cosine functions for symmetric potentials, such as the finite square well, to exploit even and odd solutions. The delta potential, however, is treated differently due to its lack of symmetry, which affects the approach to finding solutions.

PREREQUISITES
  • Understanding of quantum mechanics principles, particularly scattering and bound states.
  • Familiarity with wave functions and their representations in quantum systems.
  • Knowledge of potential wells, including finite and infinite square wells.
  • Basic grasp of symmetry in quantum mechanics and its implications on solutions.
NEXT STEPS
  • Study the differences between scattering and bound states in quantum mechanics.
  • Explore the mathematical representation of wave functions in quantum systems.
  • Learn about the implications of symmetry in quantum mechanics, particularly in potential wells.
  • Investigate the treatment of delta potentials in quantum mechanics and their unique characteristics.
USEFUL FOR

Students and professionals in physics, particularly those focusing on quantum mechanics, wave functions, and potential theory. This discussion is beneficial for anyone seeking to deepen their understanding of scattering and bound state solutions.

Logan Rudd
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1)So from my understanding, as long as ##E>0## you will have scattering states and these scattering states will always result in an imaginary ##\psi##, but bound states can also have an imaginary ##\psi##? Is this correct and or is there a better way of looking at this maybe more conceptually?

2)I noticed in a footnote in griffiths intro to qm 2nd ed. in the finite square well section he mentions that he wrote ##\psi## in terms of sin/cos instead of exponentials because we know the solutions will be even or odd because the potential is symetric. But he also does it for the infinite square well when it's not really symmetric (at least with respect to the y-axis) and doesn't do it for the delta potential. How come he doesn't do it for the delta potential, is it not really symmetric? and why does he do it for the finite square well?
What is the benefit of "exploiting the even/odd solutions" and what does he mean by that?

Thanks
 
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