Hydrogen atom ground state with zero orbital angular moment question.

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Discussion Overview

The discussion revolves around the ground state of the hydrogen atom and the implications of symmetry on the orbital angular momentum of its wavefunction. Participants explore the relationship between mathematical solutions of the Schrödinger equation and physical concepts of symmetry, particularly in the context of the Coulomb potential.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant notes that the ground state wavefunction of the hydrogen atom is independent of angular variables and concludes that the expectation value of the orbital angular momentum is zero.
  • Another participant emphasizes the symmetry of the Coulomb potential and suggests that the wavefunction must be symmetric or antisymmetric, particularly in the ground state where there is no degeneracy.
  • A later reply questions whether symmetry implies spherical symmetry, seeking clarification on the nature of the symmetry involved.
  • One participant explains that "c.f." means "compare with" and discusses the parity property of wavefunctions, indicating that it does not necessarily imply spherical symmetry.
  • Another participant argues that the wavefunctions need only have positive or negative parity eigenvalues and challenges the notion that the symmetry of the ground state wavefunction must match that of the potential.

Areas of Agreement / Disagreement

Participants express differing views on the implications of symmetry for the ground state wavefunction and its relationship to the potential. There is no consensus on whether symmetry necessarily entails spherical symmetry, and the discussion remains unresolved.

Contextual Notes

Participants acknowledge the complexity of the relationship between symmetry and the properties of wavefunctions, with some limitations in defining the nature of symmetry and its implications for the ground state.

xfshi2000
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Hi all:
As we know, if we solve the Schrödinger equation, the ground state wavefunction is independent of theta and psi. We find the expectation value of ground state orbital angular momentum is zero. But if we don't do any mathematical calculation, can we conlude that?
For example, Due to symmetry of coulomb potential, Hydrogen atom (one proton plus one electron) wavefunction must be symmetric or antisymmetric (for ground state, there is no degeneracy). Then I am stuck. How can I conclude that only by virtue of physical concept or symmetry? Does symmetry means spherical symmetry?
thanks

xf
 
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well symmetric with respect to partiy : spatial inversion. C.f with particle in box solutions.
 
malawi_glenn said:
well symmetric with respect to partiy : spatial inversion. C.f with particle in box solutions.

Thank you. What is C.f? parity property only determines psi(-x)=+/-psi(x). where x is vector. It doesn't mean it is spherical symmetry. Could you explain more? thanks
 
c.f means "compare with"

No, the wavefunctions just need to have positive or negative parity eigenvalue. I don't think there is a theorem which states that the symmetry of the ground state wave function must have the same symmetry as the potential.
 

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