Solving a differential equation similar to Legendre

Remember, the key to solving non-separable differential equations is to make use of symmetries and clever substitutions to reduce the problem to one that is more manageable.In summary, the given differential equation is non-separable and cannot be solved using the properties of spherical harmonics alone. Other techniques, such as separation of variables or taking advantage of symmetries, may be useful in finding a solution.
  • #1
Physicslad78
47
0
I am trying to solve the following differential equation:



where is the angular momentum given by:



goes from 0 to while goes from 0 to . w and k are constants and E is the energy of the system.. This differential equation is non separable. However i have realized that



I plugged that in differential equation above and multiplied the whole equation by and then used the integral properties of three spherical harmonics multiplied together (in terms of wigner 3 j symbols). This is to get a recursion relation between different coefficients found in the solution that I assumed :



I am looking at the case when m=1 hence the substitution by 1 for m in above equation. My two questions are: (1) How will I treat the term above contatining i? I only took the part without i and got solutions but some of the eigenvalues are complex (it is an energy term) so it is not possible! (2) even inserting i would cause more problems as u still get complex values..can anyone tell me what might be going wrong?.Thanks

Thanks
 
Last edited by a moderator:
Physics news on Phys.org
  • #2
for your question. The issue you are facing is due to the fact that the differential equation you are trying to solve is non-separable, meaning that it cannot be written as two separate equations in terms of $\theta$ and $\phi$. This makes it difficult to use the properties of spherical harmonics to derive a recurrence relation between the coefficients in your solution. One way to approach this problem is to use the method of separation of variables. This involves assuming a solution of the form $\psi = \Theta(\theta) \Phi(\phi)$, and then substituting this into the original differential equation. This will lead to two separate equations in terms of $\theta$ and $\phi$, which can then be solved using the properties of spherical harmonics.Another approach that may be useful is to make use of the symmetry of the problem. Since the equation is invariant under rotations around the $z$-axis, it is possible to take advantage of this to reduce the number of independent variables. Specifically, you can make a substitution of the form $\phi = \alpha + \beta$ where $\alpha$ is an arbitrary constant and $\beta$ is the angle around the $z$-axis. This will reduce the equation to one in terms of just $\theta$ and $\alpha$, which can then be solved using separation of variables or other methods.I hope this helps!
 
  • #3
for sharing your work on solving this differential equation. It seems like you have made some progress in finding a solution using the substitution for the term containing i. However, it is important to note that the presence of complex eigenvalues does not necessarily mean that the solution is incorrect. In fact, many physical systems have complex eigenvalues and it is a common occurrence in quantum mechanics.

To address your first question, it may be helpful to review the properties of spherical harmonics. In particular, the phase factor in the definition of spherical harmonics can result in complex values for the coefficients. This is not a problem and does not affect the physical interpretation of the solution.

As for your second question, it is difficult to determine what may be going wrong without further information about your approach and calculations. It is possible that there may be an error in your algebraic manipulation or in the use of the Wigner 3-j symbols. It may be helpful to double check your calculations and equations to ensure accuracy. Additionally, it may be useful to consult with a colleague or mentor for guidance and support in solving this equation. Overall, it seems like you are on the right track and with some further analysis and refinement, you may be able to find a solution for this differential equation. Keep up the good work!
 

FAQ: Solving a differential equation similar to Legendre

Question 1: What is a differential equation?

A differential equation is a mathematical equation that contains one or more derivatives of an unknown function. It describes the relationship between the function and its derivatives.

Question 2: What is a Legendre differential equation?

A Legendre differential equation is a type of second-order ordinary differential equation that arises in many areas of physics and engineering. It is named after the French mathematician Adrien-Marie Legendre.

Question 3: How do you solve a differential equation similar to Legendre?

To solve a differential equation similar to Legendre, you can use various methods such as power series, Frobenius method, or the Laplace transform. These methods involve finding a general solution and then applying initial conditions to obtain a particular solution.

Question 4: What are the applications of solving a differential equation similar to Legendre?

Solving a differential equation similar to Legendre is important in many fields, including physics, engineering, and mathematical modeling. It can be used to describe the behavior of physical systems, such as the motion of a pendulum, or to model the growth of populations in biology.

Question 5: Is there any software available for solving differential equations similar to Legendre?

Yes, there are various software programs and packages available, such as MATLAB, Mathematica, and Maple, that can be used to solve differential equations similar to Legendre. These programs use numerical methods to find approximate solutions to the equations.

Similar threads

Back
Top