Optimizing Energy of Hydrogen Atom with 3D Oscillator Wavefunction

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Homework Help Overview

The discussion revolves around optimizing the energy of a hydrogen atom using a trial wavefunction based on the 3D oscillator ground state. Participants are exploring the implications of the term "best energy" in the context of the variational method.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are questioning the meaning of "best energy" and its relevance to the variational method. There is discussion about varying the parameter b to find a higher bound for the ground state energy.

Discussion Status

The conversation is ongoing, with participants seeking clarification on the relationship between the trial wavefunction and the variational method. Some guidance has been offered regarding the approach to take with the parameter b.

Contextual Notes

There is mention of a lack of a fixed textbook and uncertainty about the specific chapter context, as well as previous coverage of related topics like the simple harmonic oscillator and perturbation theory.

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Homework Statement


Take as a trial wavefunction for the hydrogen atom the 3D oscillator ground state wavefunction
ψ(r) = N exp (-br^2 / 2). Calculate the value of b that gives the best energy and calculate this energy.

Homework Equations



Radial part of ∇^2 = 1/r2 (∂/∂r) (r^2 ∂/∂r)

The Attempt at a Solution



I am not sure what best energy is supposed to imply.
 
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Hi.
Is this following a chapter about the variational method?
 
There is no fixed textbook, so I'm not sure what chapter this precedes or follows; we've covered the simple harmonic oscillator in 1D and went into the variational method while covering perturbation theory. I thought the 3D SHO is an extension of the 1D system, but this 'best energy' is throwing me off.
 
In a variational method problem it would make sense to take a trial function and vary the parameter b in order to get a higher bound for the ground state energy ("best" energy?), that's why I'm asking. i can't think of another meaning in the context you're giving...
 

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