Unraveling the Mystery of Multiple Scattering in Double Well Systems

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SUMMARY

The discussion focuses on the behavior of incident waves in double well systems, specifically how waves are transmitted and reflected between two wells. The author highlights the simplicity of calculating resulting amplitudes through a system of equations, despite the complexity of infinite reflections. The use of periodic boundary conditions in the Schrödinger equation is emphasized as a key mathematical tool in this analysis. The conversation touches on the underlying physical principles that make this process seemingly straightforward.

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  • Understanding of quantum mechanics principles, particularly wave functions.
  • Familiarity with the Schrödinger equation and its applications.
  • Knowledge of periodic boundary conditions in quantum systems.
  • Basic grasp of infinite series and their convergence properties.
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  • Explore the application of periodic boundary conditions in quantum mechanics.
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  • Investigate the implications of wave function behavior in double well potentials.
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Students and researchers in quantum mechanics, physicists studying wave behavior in potential wells, and anyone interested in the mathematical foundations of quantum theory.

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For a double well consider an incident wave from the left.
The wave gets transmitted and reflected at first well. The transmitted wave then hits the second well where it also gets both reflected and transmitted. The reflected part then hits the first well where it reflects and hits the second well and this process continues ad infinum. What I found was that it was not very hard to find the resulting amplitudes traveling in various directions if we just write up a system of equations and solve it (I think you have all seen how one applies periodic boundary conditions to the Schrödinger equation etc.) But why is it fundamentally so easy? I can't really see why it should physically be so easy since I have just accounted for how complicated the infinite process of reflections is. What is the miracle here?
I guess it is a stupid question since I should just accept the maths..
 
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This is like the well-known anecdote about John von Neumann...

"What trick?" asked von Neumann; "all I did was sum the infinite series."
 
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Thank you :)))
 

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