Solve DE for theta component (hydrogen WF)

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
  • #1
Andrew Deleonardis
5
0
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
 
Physics news on Phys.org
  • #2
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:
  • #3
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
 
  • #4
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
 

Related to Solve DE for theta component (hydrogen WF)

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.

Similar threads

  • Differential Equations
Replies
21
Views
2K
  • Differential Equations
Replies
1
Views
2K
  • Differential Equations
Replies
4
Views
1K
Replies
8
Views
451
  • Introductory Physics Homework Help
Replies
2
Views
787
  • Calculus and Beyond Homework Help
Replies
3
Views
737
Replies
2
Views
1K
  • Calculus
Replies
29
Views
1K
  • Introductory Physics Homework Help
Replies
3
Views
362
Back
Top