How to solve for allowed energies with the WKB approximation?

In summary, the conversation is about using the WKB approximation of the Schrodinger equation to solve for allowed energies with the Morse potential. The equation used is equation 35 from the provided link and the goal is to integrate between the two turning points where V(x) = E. However, the issue arises when these turning points have complex coordinates. The person asking the question is looking for a reference to numerical algorithms that can be used in this situation.
  • #1
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Hello,

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

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

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

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

Thank you!
 
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  • #2
Can you maybe explain how you got complex coordinates in the first place? Because I thought that those are usually real in QM.
 
  • #3
Yes, sorry, I was confused.
 

1. What is the WKB approximation method?

The WKB (Wentzel-Kramers-Brillouin) approximation is a method used to approximate the allowed energies of a quantum system, based on the classical mechanics of the system. It is commonly used to solve for allowed energies in systems where the potential energy function is smooth and slowly varying.

2. How does the WKB approximation work?

The WKB approximation method works by breaking down the wavefunction of a quantum system into smaller, simpler components that can be solved using classical mechanics. These components are then combined together to approximate the overall wavefunction and determine the allowed energies of the system.

3. What are the advantages of using the WKB approximation?

One advantage of using the WKB approximation is that it can be applied to a wide range of quantum systems, including those with complex potential energy functions. It also provides a good estimate for the allowed energies, and can be used to analyze the behavior of a system for both low and high energies.

4. What are the limitations of the WKB approximation?

The WKB approximation is based on classical mechanics, so it is not always accurate for highly quantum systems. It also assumes that the potential energy function is smooth and slowly varying, and may not work well for systems with sharp potential energy changes. Additionally, it does not take into account the effects of spin or other quantum phenomena.

5. How is the WKB approximation used in practical applications?

The WKB approximation is commonly used in fields such as quantum mechanics, solid state physics, and atomic and molecular physics. It can be used to calculate the allowed energies of individual particles, as well as the energy levels of atoms and molecules. It is also used in quantum tunneling calculations and in the analysis of wave phenomena in physics.

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