Guessing trial wave function with variational method

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Discussion Overview

The discussion revolves around the process of guessing trial wave functions in the context of the variational method of approximation, particularly for quantum mechanical systems like the harmonic oscillator and hydrogen atom. Participants explore general strategies for selecting appropriate trial wave functions when faced with various potentials.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant inquires about general strategies for guessing trial wave functions when applying the variational method.
  • Another participant suggests that experience and familiarity with solved problems can inform the choice of trial wave functions.
  • It is proposed that the selection process involves considering the desired properties of the wave function, such as asymptotic behavior and behavior near specific points.
  • A suggestion is made to use classes of functions that exhibit the required properties, with references to paradigmatic cases that can guide the selection.
  • Participants discuss the form of trial functions, suggesting an ansatz involving exponential decay and polynomial contributions, or using Gaussian functions for ease of calculation.
  • It is noted that variational methods are generally not very sensitive to the choice of trial functions, provided they are not drastically incorrect.
  • There is an emphasis on the necessity for trial functions to have qualitatively correct forms, as the accuracy of the approximation is limited by the chosen ansatz.

Areas of Agreement / Disagreement

Participants express a range of views on the process of guessing trial wave functions, with no clear consensus on a single method or approach. The discussion reflects multiple perspectives on the role of experience and the properties that trial functions should exhibit.

Contextual Notes

Participants mention the importance of asymptotic behavior and the influence of known solvable cases on the selection of trial functions, but do not resolve the specifics of how to apply these considerations universally.

physicist 12345
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i want to ask about guessing the trial wave function at variational method of approximation

usually for example at solving harmonic oscillator or hydrogen atom we have conditions for trial wave function
but in fact i want to ask generally how could i make the guessing .. some problems give a particle of mass m moving at certain potential and want to use variational method to find gs energy now how could i guess the trial wavefunction
 
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physicist 12345 said:
how could i guess the trial wavefunction
By having solved a number of problems, or seen how others make the choice.
 
A. Neumaier said:
By having solved a number of problems, or seen how others make the choice.
then it some thing come with experience ?
 
physicist 12345 said:
then it some thing come with experience ?
With experience, or with trial and error. Generally one first thinks about the properties the function wanted should have (asymptotic behavior or behavior near distinguished points). Then one selects a class of functions having this property. Often there are paradigmatic exactly solvable cases that show what kind of solution is reasonable, and one can choose similar functions. Normalized wave functions typically decay exponentially. Generic variability is created by polynomial contributions. This suggests an ansatz ##e^{-a|x-x_k|} p(x)## with a polynomial ##p(x)##, or (suggested by the linearity of the Schroedinger equation) linear combinations of these. If one wants to have an easy time in calculating inner products, one uses instead Gaussians times polynomials, etc..Unnormalized ones have an asymptotic form reflecting knowledge about scattering, and again one can make up trial functions with this behavior.
 
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Variational methods are often not very sensitive to the choice of trial functions, so long as they're not way off.
 
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marcusl said:
Variational methods are often not very sensitive to the choice of trial functions, so long as they're not way off.
They must have qualitatively the correct form, and the approximation cannot be better than what the ansatz allows.
 

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