Relation between potential and bound states in Schrodinger equation.

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SUMMARY

The discussion centers on the relationship between potential and bound states in the Schrödinger equation, specifically in the form -u''(x) + V(x)u(x) = Eu(x). It concludes that bound state solutions with energy E > 0 are possible when the potential V(x) approaches a constant positive value as |x| approaches infinity. However, plane wave solutions cannot exist if the potential V depends on x. This establishes a clear distinction between the conditions for bound states and plane wave solutions in quantum mechanics.

PREREQUISITES
  • Understanding of the Schrödinger equation
  • Knowledge of quantum mechanics principles
  • Familiarity with potential energy functions
  • Concept of bound states in quantum systems
NEXT STEPS
  • Explore the implications of varying potential functions in quantum mechanics
  • Study the conditions for bound states in different potential scenarios
  • Learn about plane wave solutions and their limitations
  • Investigate the mathematical techniques for solving the Schrödinger equation
USEFUL FOR

Quantum physicists, students of quantum mechanics, and researchers interested in the mathematical foundations of quantum states will benefit from this discussion.

arroy_0205
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Suppose I have Schroedinger equation in the form:
<br /> -u&#039;&#039;(x)+V(x)u(x)=Eu(x)<br />
The potential is such that as |x| -> Infinity, V(x) reaches a constant positive value. In this case can we have bound state/plane wave solutions for u(x) with E > 0 ?
 
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You can have bound state solutions with E>0.
Just take any V that-->0 and add any positive constant to it.
You can't have plane wave solutions if V depends on x.
 

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