# Legendre equation and angular momentum

• Angsaar
In summary, the conversation discusses the significance of the second type of solution, ln((1+x)/(1-x)), for the Legendre differential equation in the context of quantum mechanics. The equation (1-x^2) (d^2P/dx^2) - 2x (dP/dx) + l(l+1) P = 0 is mentioned, with l being the total angular momentum quantum number and P being the solution to the theta part of the eigenfunction from the Schrodinger equation. It is noted that the Legendre functions of the second kind, Q_n(x), are irregular at the poles and cannot be included in a normalizable wave function, making them not commonly used in EM theory. They
Angsaar
Hi all,

I've been doing a math problem about the Legendre differential equation, and finding there are two linearly independent solutions. When I was taught about quantum mechanics the polynomial solutions were introduced to me as the basis for spherical harmonics and consequently the eigenfunctions of the hydrogen atom, etc etc.

But if there's a second type of solution (involving ln((1+x)/(1-x)) ), what significance to QM do they have? I've not been able to find any mention of them in the context of physics, only in math textbooks.

Thanks for any help!

So what is your equation...?Post it,so i can have a clear picture of you're trying to say.

Daniel.

(1-x^2) (d^2P/dx^2) - 2x (dP/dx) + l(l+1) P = 0

l being the total angular momentum quantum number, and P being the solution to the theta part of the eigenfunction from the schrodinger equation after using separation of variables, with x=cos theta

I found the first solution type, say y1, using a series method, and the second, say y2, by doing y2(x)=y1(x)v(x) and finding out what v(x) had to be.

If you're going to claim that the second independent solution to Laplace equation is also elligible to describe physically possible quantum states,then u should check it out:do these new wavefunctions obey the orthonormality condition that a complete system of eigenfunctions of the H atom's Hamiltonian must fulfil...?My guess is NO,this thing being similar to the "radial wavefunctions issue" and those Whittaker functions...

Daniel.

The Legendre functions of the second kind, usually designated as Q_n(x), are irregular at the poles. Because of this, they can't be included in a normalizable wave function.
The same argument holds for non-integral l for the P_l(x).
That's also why they don't usually come up in EM theory.
They are used sometimes in scattering theory (along with non-integral l)
when the scattering amplitude is continued into the complex plane.

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## 1. What is the Legendre equation?

The Legendre equation is a second-order ordinary differential equation that arises in mathematical physics, specifically in the study of spherical symmetry and angular momentum. It is named after the French mathematician Adrien-Marie Legendre.

## 2. What is the significance of the Legendre equation?

The Legendre equation is used to solve problems related to the behavior of physical systems with spherical symmetry, such as the motion of planets or the behavior of atoms. It is also important in the study of quantum mechanics, where it is used to describe the behavior of particles with spin.

## 3. How is the Legendre equation related to angular momentum?

The Legendre equation is closely related to angular momentum because it describes the behavior of particles moving in a spherically symmetric potential. The solutions to the equation represent the possible angular momentum values that a particle can have in a given system. This makes it a valuable tool in the study of angular momentum in physics.

## 4. What are the applications of the Legendre equation?

The Legendre equation has many applications in physics and mathematics. In addition to its use in solving problems related to spherical symmetry and angular momentum, it is also used in the study of heat transfer, electrostatics, and fluid mechanics. It has also been applied in other fields such as statistics and signal processing.

## 5. Are there any special techniques for solving the Legendre equation?

There are several techniques that can be used to solve the Legendre equation, depending on the specific problem at hand. These include the power series method, the Frobenius method, and the separation of variables method. In some cases, the use of special functions, such as Legendre polynomials, can also simplify the solution process.

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