- #1
bvol
- 3
- 0
m=1 and l=1
x = cos(θ)
What would be the solution to this?
Thanks.
Last edited:
Associated Legendre polynomials
- Thread starter bvol
- Start date
In summary, the conversation is about finding the solution to a formula involving x = cos(θ). One person provides a solution of 4cos(θ)^3 - 4cos(θ), while another person points out that it should actually be sin(θ). The conversation then turns to the derivation of the associated legendre polynomial and the person realizes they made an error in their calculation. They eventually arrive at the correct solution of sin(θ).
m=1 and l=1
x = cos(θ)
What would be the solution to this?
Thanks.
- #2
ZetaOfThree
Gold Member
- 110
- 23
We won't do the calculation for you. Why can't you do it yourself? What part of it doesn't make sense to you?
- #3
bvol
- 3
- 0
I've tried a lot of things, but I don't get it. (NOT a physics/math student)
I get this:
4cos(θ)^3 - 4cos(θ)
But it should be sin(θ) since I'm applying this formula:
and the solution for 1,1 is Sqrt(3/8π) e^((+/-)iφ) * sin(θ)
I get how the part in italics is derived, but not how the part in bold has been derived from the associated legendre polynomial..
I get this:
4cos(θ)^3 - 4cos(θ)
But it should be sin(θ) since I'm applying this formula:
and the solution for 1,1 is Sqrt(3/8π) e^((+/-)iφ) * sin(θ)
I get how the part in italics is derived, but not how the part in bold has been derived from the associated legendre polynomial..
- #4
ZetaOfThree
Gold Member
- 110
- 23
I think you've made an error in your calculation. If we plug in ##\ell = 1## and ##m=1##, we have $$P^{1}_{1}={(-1)^1 \over 2 \cdot 1!} (1-x^2)^{1/2} \cdot {d^2 \over dx^2}(x^2-1)$$
Do you have that much? It should be easy to simplify that, then plug in ## x=\cos{\theta}##.
Do you have that much? It should be easy to simplify that, then plug in ## x=\cos{\theta}##.
- #5
bvol
- 3
- 0
Great I have it now. -(1-cos()^2)^(1/2), which is equal to sin(θ).
Right I have it now, like you said I made a small error in not applying l+m, but instead just l and some errors on other places as well.
Thanks for the quick reply! :)
Right I have it now, like you said I made a small error in not applying l+m, but instead just l and some errors on other places as well.
Thanks for the quick reply! :)
Related to Associated Legendre polynomials
What are Associated Legendre polynomials?
Associated Legendre polynomials are a special type of mathematical function that are used in the study of spherical harmonics. They are defined as a family of polynomials that are solutions to a specific differential equation, and they are commonly used in various branches of physics and engineering.
What is the formula for Associated Legendre polynomials?
The formula for Associated Legendre polynomials is given by:
Plm(x) = (-1)m(1-x2)m/2(d/dx)m[Pl(x)],
where Pl(x) is the lth Legendre polynomial and m is a non-negative integer.
What is the significance of Associated Legendre polynomials?
Associated Legendre polynomials are significant because they are used to represent the angular part of the solution to Laplace's equation in spherical coordinates. This makes them useful in solving problems involving spherical symmetry, such as in quantum mechanics, electromagnetism, and fluid dynamics.
What are the properties of Associated Legendre polynomials?
Some important properties of Associated Legendre polynomials include orthogonality, recurrence relations, and the Rodrigues formula. They also have specific values at the poles and equator, and they can be expressed in terms of the Gamma function.
How are Associated Legendre polynomials related to Legendre polynomials?
Associated Legendre polynomials are a generalization of Legendre polynomials, where the latter can be seen as a special case when m=0. Both types of polynomials share similar properties, but Associated Legendre polynomials are more useful in solving problems involving spherical symmetry.
Similar threads
-
Calculus
-
Classical Physics
-
Calculus
-
Differential Equations
-
Introductory Physics Homework Help
-
Differential Equations
-
Advanced Physics Homework Help