vanhees71 said:
vanhees71 said:
Real and complex spherical harmonics are different mathematical representations of the same physical phenomenon, namely the spatial distribution of a wave on a sphere. Real spherical harmonics are expressed as real-valued functions, while complex spherical harmonics are expressed as complex-valued functions. For hexagonal symmetry, the main difference between the two representations lies in the number of coefficients needed to describe the wave, with complex spherical harmonics requiring fewer coefficients.
In general, complex spherical harmonics are more commonly used in the study of hexagonal symmetry. This is because they require fewer coefficients to describe the wave, making calculations and analysis easier and more efficient. Real spherical harmonics, on the other hand, are often used in specific cases where the wave is purely real.
While real and complex spherical harmonics are mathematically equivalent, they cannot be used interchangeably in the study of hexagonal symmetry. This is because they represent different physical quantities and have different properties, which can lead to different results and interpretations. It is important to use the appropriate representation depending on the specific problem being studied.
Real and complex spherical harmonics are related through a mathematical transformation known as the complex conjugate. This transformation converts a complex-valued function into its real-valued counterpart, and vice versa. This relationship allows for the conversion between the two representations, but it is important to note that the physical interpretation of the wave may be different in each representation.
The main advantage of using complex spherical harmonics is their efficiency in representing waves with hexagonal symmetry. They require fewer coefficients and can simplify calculations and analysis. However, the use of complex numbers may be more challenging for some researchers, and the physical interpretation of the wave may be less intuitive. Real spherical harmonics, on the other hand, may be easier to understand and interpret, but they require more coefficients and can be less efficient in certain cases.