Atomic Orbitals & Circular Membranes: Is There a Connection?

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SUMMARY

The discussion centers on the relationship between the probability density distribution function of atomic orbitals and the modes of a circular membrane. It concludes that while there are similarities in the mathematical functions involved, specifically the spherical Bessel function in 3D and the circular Bessel function in 2D, they are fundamentally different due to the dimensionality and the presence of spherical harmonics in the 3D solutions. The wave-like behavior of electrons leads to stationary states that correspond to normal modes of the Hamiltonian, but this does not imply a direct equivalence between the two systems.

PREREQUISITES
  • Understanding of quantum mechanics and atomic orbitals
  • Familiarity with wave functions and their properties
  • Knowledge of Bessel functions, specifically spherical and circular Bessel functions
  • Basic concepts of Hamiltonian mechanics
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  • Study the properties of spherical Bessel functions and their applications in quantum mechanics
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  • Explore the relationship between wave functions and normal modes in various physical systems
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dimensionless
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Is there a particular reason why the probability density distribution function of atomic orbitals bears a similarity to the modes on a circular membrane? Is this just a coincidence? Is it not actually that similar?
 
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This is a direct consequence of the wave-like behaviour of electrons.
Because of this, the stationary states are normal modes (of the hamiltonian).
 
dimensionless said:
Is there a particular reason why the probability density distribution function of atomic orbitals bears a similarity to the modes on a circular membrane? Is this just a coincidence? Is it not actually that similar?

It is not "similar" simply because a circular membrane is 2D. You have to solve a 3D equation for atomic wavefunction. Now, the spherical Bessel function that one obtains for the Radial part may have the similar profile with the circular Bessel function that one gets in 2D. But that is as far as it goes. It doesn't have the equivalent of the Spherical Harmonics of the 3D solution.

Zz.
 

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