Calculating Spherical Harmonics Cuadratic Dispersion

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SUMMARY

The discussion focuses on calculating quadratic dispersion in quantum systems by expanding x² in terms of spherical harmonics using Clebsch-Gordan coefficients. The user initially expands x as a linear combination of spherical harmonics and attempts to compute x² by multiplying x with itself. However, the resulting sum of spherical harmonics, each multiplied by a Clebsch-Gordan coefficient, does not yield the correct expression. The user seeks assistance in resolving this issue.

PREREQUISITES
  • Understanding of spherical harmonics and their properties
  • Familiarity with Clebsch-Gordan coefficients
  • Knowledge of Gaunt integrals and their applications
  • Basic concepts of quantum mechanics and state vector couplings
NEXT STEPS
  • Study the properties and applications of spherical harmonics in quantum mechanics
  • Learn about the calculation and application of Clebsch-Gordan coefficients
  • Research Gaunt integrals and their role in quantum mechanical calculations
  • Explore advanced topics in quantum state vector couplings and their implications
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Quantum physicists, researchers in quantum mechanics, and students studying advanced topics in spherical harmonics and their applications in quantum systems.

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Greetings,

I´m calculating cuadratic dispersion of some quantum systems. I need to expand x^2 in terms of spherical harmonics (using Clebsch-Gordan coefficients, or threeJ as well) in order to be able to use Gaunt espression in the integral solving.

I start from the expansion of x as linnear combination of Spherical Harmonics (with Clebsch-Gordan coef.). Then, in order to get x^2, I calculate x*x (studying the couplings of the state vectors term by term) and I get what I need: a sum of Spherical Harmonics (each multiplied by a C-G coef.) but it´s not the correct one!

Could anyone help me? Thanks
 
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wow K-12 got really hard really fast!
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