Vector Spherical harmonics/spherical coordinates question

In summary, Y110 is a vector of length zero in Cartesian coordinates, but it still serves a purpose in describing the direction of a vector in spherical coordinates.
  • #1
Faust1
1
0
Arfken and Weber lists some of the Vector spherical harmonics in spherical coordinates, and I'm puzzled that one has no radial component. Specifically, $Y(j =1, l=1, m = 0) = i \sqrt[3/(8 \Pi)] sin(\theta) \^{phi} $

To Cartesian components of the vector, it seems you need an r component of the vector in spherical coordinates. Since Y110 only has a phi component, does this mean that the Y110 spherical harmonic is a vector of length zero in cartesian coordinates? If so, why is there even a magnitude in the phi component of the vector? And if so, is Y110 only useful if combined with other vector spherical harmonics?

Thanks in advance.
 
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  • #2
The answer to your question is yes, Y110 in spherical coordinates is a vector of length zero in Cartesian coordinates. This is because the radial component of Y110 is zero, and the other two components are related to each other by the spherical coordinates system. Therefore, Y110 does not have a meaningful magnitude in spherical coordinates, but it is still useful as it helps us to describe the direction of a vector in spherical coordinates. For example, if you have a vector with components (r, θ, φ), then Y110 will tell you the direction of the vector in terms of θ and φ.
 
  • #3


Thank you for bringing up this interesting question about vector spherical harmonics in spherical coordinates. It is important to note that the vector spherical harmonics are not physical vectors, but rather mathematical functions that represent the angular dependence of a vector field in spherical coordinates. Therefore, the lack of a radial component in the Y110 spherical harmonic does not mean that the vector has no magnitude in cartesian coordinates.

To understand this better, we need to look at the definition of vector spherical harmonics. They are defined as the products of spherical harmonics and unit vectors in spherical coordinates. In the case of Y110, the unit vector in the phi direction is multiplied by the spherical harmonic $Y(l=1, m=0)$, which represents a pure cosine function in the phi direction. This results in the imaginary unit $i$ in front of the expression, giving us a purely imaginary vector component in the phi direction.

So, in cartesian coordinates, the vector represented by Y110 does have a magnitude, but it is purely imaginary and therefore does not contribute to the magnitude of the overall vector field. However, it is still a valuable component in representing the angular dependence of the vector field.

Furthermore, the vector spherical harmonics are usually not used alone, but rather in combination with other vector spherical harmonics to represent the full vector field in spherical coordinates. Therefore, Y110 can still be useful in combination with other vector spherical harmonics to fully describe the vector field.

I hope this clarifies your confusion about the lack of a radial component in Y110 and its usefulness in representing vector fields in spherical coordinates.
 

1. What are vector spherical harmonics?

Vector spherical harmonics are a set of functions that are used to describe the behavior of vector fields in spherical coordinates. They are similar to spherical harmonics, but instead of representing scalar functions, they represent vector functions.

2. How are vector spherical harmonics used in science?

Vector spherical harmonics are used in various fields of science, such as physics, mathematics, and engineering. They are particularly useful in studying the behavior of electromagnetic and acoustic fields, and in solving problems that involve spherical symmetry.

3. What is the relationship between vector spherical harmonics and spherical coordinates?

Vector spherical harmonics are closely related to spherical coordinates, as they are used to describe the behavior of vector fields in this coordinate system. The vector spherical harmonics are expressed in terms of the spherical coordinates, and their values depend on the direction and magnitude of the vector field at different points in space.

4. How are vector spherical harmonics calculated?

The calculation of vector spherical harmonics involves solving a set of differential equations known as the vector Laplace equation. This is a complex mathematical process that requires advanced mathematical techniques, such as complex analysis and special functions.

5. Can vector spherical harmonics be visualized?

Yes, vector spherical harmonics can be visualized using computer software or mathematical plotting tools. They are often represented as a series of spherical surfaces with different colors and patterns, which correspond to the different values of the vector field at different points in space.

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