# Derivative of Spherical Harmonic for negative m

• VVS
In summary, the conversation discusses evaluating the derivative of spherical harmonics with respect to the azimuthal angle and expressing it in terms of spherical harmonics. The individual has calculated the derivative and knows how to express it for positive m, but is unsure about negative m. They also mention that the azimuthal angle is the argument of the Exp[] and clarify that they want the derivative with respect to the polar angle. The conversation concludes with the individual stating they have found the correct solution and asking for input on potential mistakes.
VVS
Hello!

## Homework Statement

I want to evaluate the derivative of spherical harmonics with respect to the azimuthal angle and express it in terms of spherical harmonics.2. Homework Equations and 3. The Attempt at a Solution
I have calculated the derivative of the spherical harmonic with respect to the azimuthal angle.
I know how to express the derivative of the spherical harmonic in terms of spherical harmonics for positive m. But I don't know how to do this for negative m.

View attachment Spherical_Harmonics.pdfThanks for your help.

Isn't the azimuthal angle the argument of the Exp[], in this case θ?

Yes you're right. I meant that I want the derivative wrt the polar angle.

Hope this is what you were looking for.

It isn't the final answer because you can still express Cot ø and Exp(iθ) in terms of the spherical harmonics but that part is not the worst. May even be able to find them in a table somewhere.

#### Attachments

• SphHarm..pdf
86.2 KB · Views: 373
Hello!

Yes! That is exactly what I was looking for.
Although I recalculated it and I get a slightly different result.
I will upload my calculations tomorrow or so. Right now I am too tired.

thanks

## 1. What is the derivative of a spherical harmonic for negative m?

The derivative of a spherical harmonic for negative m is given by the expression:

d/dθ [Yl, -m(θ,φ)] = -im√[(l+m)(l-m+1)]Plm(cosθ)e-imφ, where θ is the polar angle and φ is the azimuthal angle.

## 2. How is the derivative of a spherical harmonic for negative m derived?

The derivative of a spherical harmonic for negative m can be derived using the standard formula for the derivative of a complex function and the addition theorem for spherical harmonics.

## 3. What is the physical significance of the derivative of a spherical harmonic for negative m?

The derivative of a spherical harmonic for negative m represents the rate of change of the spherical harmonic function with respect to the polar angle θ. It has applications in studying the behavior of electromagnetic fields and quantum mechanics in spherical coordinates.

## 4. Is the derivative of a spherical harmonic for negative m always a complex function?

Yes, the derivative of a spherical harmonic for negative m is always a complex function, as it involves the imaginary number i. This is due to the fact that spherical harmonics are complex-valued functions.

## 5. Can the derivative of a spherical harmonic for negative m be simplified?

Yes, the derivative of a spherical harmonic for negative m can be simplified by using the recursion relation for associated Legendre polynomials and the differential equation satisfied by spherical harmonics. This can lead to simpler expressions in some cases, making calculations easier.

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