# Solve DE for theta component (hydrogen WF)

• A
• Andrew Deleonardis
In summary: Although never tried it myself, associated Legendre differential equation can be solved by power series expansion. For the ordinary Legendre differential equation see this. I guess you should be able to get the idea from that link when applied to the associated Legendre equation.By the way, no one can solve the equation you have there until you add equal sign and some terms in the other side.Oh oops, it's meant to equal zero, I forgot
Andrew Deleonardis
The particular equation I would like to see solve is:
##\sin\theta\frac{d}{d\theta}(\sin\theta\frac{d\Theta}{d\theta})+\Theta(l(l+1)\sin\theta-m^2)##

The solution for this equation is the following associated laguerre polynomial:
##P^m_l(\cos\theta)=(-1)^m(\sin\theta)^m\frac{d^m}{d\cos\theta^m}\bigg(\frac{1}{2^ll!}\frac{d^l}{d\cos\theta^l}(cos^2\theta-1)^l\bigg)##

This equation is involved in solving the schrodinger equation for the hydrogen atom.
Even though I already know the answer, I would like to know HOW to solve it

Although never tried it myself, associated Legendre differential equation can be solved by power series expansion. For the ordinary Legendre differential equation see this. I guess you should be able to get the idea from that link when applied to the associated Legendre equation.
By the way, no one can solve the equation you have there until you add equal sign and some terms in the other side.

Last edited:
blue_leaf77 said:
Although never tried it myself, associated Legendre differential equation can be solved by power series expansion. For the ordinary Legendre differential equation see this. I guess you should be able to get the idea from that link when applied to the associated Legendre equation.
By the way, no one can solve the equation you have there until you add equal sign and some terms in the other side.
Oh oops, it's meant to equal zero, I forgot

Andrew Deleonardis said:
The particular equation I would like to see solve is:
##\sin\theta\frac{d}{d\theta}(\sin\theta\frac{d\Theta}{d\theta})+\Theta(l(l+1)\sin\theta-m^2)##
Edit: =0

The solution for this equation is the following associated laguerre polynomial:
##P^m_l(\cos\theta)=(-1)^m(\sin\theta)^m\frac{d^m}{d\cos\theta^m}\bigg(\frac{1}{2^ll!}\frac{d^l}{d\cos\theta^l}(cos^2\theta-1)^l\bigg)##

This equation is involved in solving the schrodinger equation for the hydrogen atom.
Even though I already know the answer, I would like to know HOW to solve it

## 1. What is a differential equation (DE)?

A differential equation is a mathematical equation that relates a function to its derivatives. In other words, it describes the relationship between a function and its rate of change. It is used to model various physical systems and phenomena in science and engineering.

## 2. What is the theta component of a hydrogen wave function (WF)?

The theta component of a hydrogen wave function refers to the angular dependence of the electron's position in the atom. It describes the probability of finding the electron at a specific angle from the nucleus.

## 3. Why is it important to solve DE for the theta component of a hydrogen WF?

Solving the DE for the theta component of a hydrogen WF allows us to understand the behavior of electrons in a hydrogen atom. This is crucial for understanding chemical bonding, spectroscopy, and other physical and chemical processes involving hydrogen.

## 4. What is the process for solving DE for the theta component of a hydrogen WF?

The process for solving DE for the theta component of a hydrogen WF involves using the Schrödinger equation, which describes the behavior of quantum particles, to derive a differential equation for the theta component. This differential equation can then be solved using various mathematical techniques, such as separation of variables or numerical methods.

## 5. What are some applications of solving DE for the theta component of a hydrogen WF?

The solutions of the DE for the theta component of a hydrogen WF can be used to calculate the energy levels and orbitals of the hydrogen atom. This information is essential for understanding the electronic structure of atoms and molecules, as well as for predicting and interpreting experimental results in fields such as chemistry, physics, and materials science.

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