quantum mechanics - Can Schrödinger Equation be derived from Huygens' Principle?

Notes of Enrico Fermi start from an analogy between mechanics and optics and with 4 pages he derives the Schrödinger equation. In all my courses, I have seen as an axiom - this is how wave-particles behave. Here it is being derived from Fermat's least action principle.

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  Maupertuis Principle -- $$S = \int mv\,\mathrm ds = \int p \,\mathrm dq$$

  
  
  
  
  • This is integral over phase space so this is not the same as Hamilton principle $$S = \int L \, \mathrm dt$$
  
  
  • Maupertuis felt space and time should be put on equal footing
  
  
  
  
  


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  Fermat principle -- $$S = \int n \, \mathrm ds$$ principle of least time, Snell's law of refraction, etc

  
  
  
  
  • this is principle of least time rather than principle of least action
  
  
  • Similar to Huygens principle.
  
  
  
  

Can Quantum Mechanics be understood in terms of wave-fronts and their singularities? I would wonder what how the free particle or scattering might look in this setting.

The only other source I could find is Sir Michael Berry, who has written quite a lot.