Boundary conditions of eigenfunctions with Yukawa potential

In summary, the conversation discusses solving the eigenfunctions and eigenvalues problem of the Schrödinger equation with Yukawa potential. The speaker asks if the boundary conditions for the eigenfunctions can be the same as those for the Coulomb potential. The other person responds that for the radial wave function, the boundary conditions at r=0 should be the same as for the Coulomb potential and for r → ∞, there is only one boundary condition for bound states, leading to quantized energy eigenvalues similar to those of the hydrogen atom.
  • #1
Riccardo Marinelli
5
1
TL;DR Summary
Where can I find information about the boundary condition of eigenfunctions of the schrödinger equation with Yukawa potential?
Hello, I was going to solve numerically the eigenfunctions and eigenvalues problem of the schrödinger equation with Yukawa Potential. I thought that the Boundary condition of the eigenfunctions could be the same as in the case of Coulomb potential. Am I wrong? In that case, do you know some articles/books where I can find information about that?
 
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  • #2
Which boundary conditions are you referring to? For the radial wave function the ones at ##r=0## should indeed be the same as for the Coulomb potential. For ##r \rightarrow \infty## you only need one for the bound states, which leads to the quantized energy eigenvalues as for the hydrogen atom.
 
  • #3
Hi, I was referring to the boundary conditions in r=0, thank you for your reply
 

1. What is the Yukawa potential?

The Yukawa potential is a mathematical function that describes the interaction between two particles, one of which carries a force known as the Yukawa force. It is commonly used in quantum mechanics to model the behavior of particles at very small distances.

2. How do boundary conditions affect eigenfunctions with Yukawa potential?

Boundary conditions are constraints on the behavior of a system at its boundaries. In the case of eigenfunctions with Yukawa potential, boundary conditions determine the behavior of the wave function at the boundaries of the potential. These conditions play a crucial role in determining the allowed energy levels and the shape of the eigenfunctions.

3. What are the boundary conditions for eigenfunctions with Yukawa potential?

The boundary conditions for eigenfunctions with Yukawa potential depend on the specific system being studied. However, in general, the wave function must be continuous and its derivative must also be continuous at the boundaries of the potential. Additionally, the wave function must approach zero as the distance from the potential approaches infinity.

4. How do boundary conditions affect the energy levels of eigenfunctions with Yukawa potential?

Boundary conditions play a crucial role in determining the energy levels of eigenfunctions with Yukawa potential. The allowed energy levels are determined by the boundary conditions, and different sets of boundary conditions can result in different energy levels. For example, a more restrictive set of boundary conditions may result in a larger energy gap between the allowed energy levels.

5. Can boundary conditions be used to control the behavior of eigenfunctions with Yukawa potential?

Yes, boundary conditions can be used to control the behavior of eigenfunctions with Yukawa potential. By choosing specific boundary conditions, one can influence the shape and behavior of the eigenfunctions, which in turn affects the energy levels and other properties of the system. This allows for a deeper understanding and manipulation of the behavior of particles at the quantum level.

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