egreg said:I have to determine the 18 connection coefficients [tex]\Gamma_{ij}^{k}[/tex] for spherical coordinates.
I know how to calculate said coefficients, but I'm not sure what all 18 are. Can anyone clarify what the possible combinations could be?
Spherical connection coefficients are mathematical coefficients used in the theory of spherical harmonics to describe the relationship between different spherical coordinate systems. They are also known as Gaunt coefficients or Clebsch-Gordan coefficients.
Spherical connection coefficients can be calculated using various methods, including the Clebsch-Gordan recurrence relations, the Wigner 3-j symbol, and the Racah formula. These methods involve summing over different combinations of quantum numbers and applying specific mathematical operations.
Spherical connection coefficients play a fundamental role in the study of spherical harmonics and are used in many physical and mathematical applications, such as quantum mechanics, molecular physics, and electromagnetic theory. They also have connections to group theory and representation theory.
Spherical connection coefficients are closely related to spherical harmonics, as they are used to express spherical harmonics in terms of different coordinate systems. They also provide a way to transform between different sets of spherical harmonics, which are solutions to the Laplace equation in spherical coordinates.
Yes, spherical connection coefficients can be negative. This is because they represent a complex relationship between different spherical harmonics and can have both positive and negative values, depending on the specific quantum numbers involved.