How to solve for allowed energies with the WKB approximation?

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SUMMARY

The discussion focuses on solving for allowed energies using the WKB approximation of the Schrödinger equation with the Morse potential. The integral equation provided, \int_a^b \sqrt{2m(E-V(x))}dx=\left(n+\frac{1}{2}\right)\pi\hbar, is essential for determining these energies. The challenge arises when integrating between complex turning points where V(x)=E. Participants seek numerical algorithms to handle these complex coordinates effectively.

PREREQUISITES
  • Understanding of the Schrödinger equation
  • Familiarity with the WKB approximation
  • Knowledge of Morse potential in quantum mechanics
  • Basic skills in numerical integration techniques
NEXT STEPS
  • Research numerical algorithms for complex integration
  • Study advanced applications of the WKB approximation
  • Explore the properties of the Morse potential in quantum systems
  • Learn about complex analysis in the context of quantum mechanics
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Quantum physicists, students studying quantum mechanics, and researchers working on numerical methods in quantum systems will benefit from this discussion.

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Hello,

I'm trying to solve for the allowed energies with the WKB approximation of the Schrödinger equation, using the Morse potential.

So I have (as per equation 35 at http://hitoshi.berkeley.edu/221a/WKB.pdf),

\int_a^b \sqrt{2m(E-V(x))}dx=\left(n+\frac{1}{2}\right)\pi\hbar

However, how do I integrate this between the two turning points (those x such that V(x)=E) when they are complex? I would appreciate a reference to any numerical algorithms I could use.

Thank you!
 
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Can you maybe explain how you got complex coordinates in the first place? Because I thought that those are usually real in QM.
 
Yes, sorry, I was confused.
 

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