Finite square well potential numerical solution

In summary, the conversation discusses the use of an effective length approximation in the numerical solution for the eigenstates in a finite square well potential. The process involves finding the eigenstate for an infinite square well, defining the effective length as the nominal length plus 2/alpha, and using this length in subsequent iterations to find the eigenstate and define alpha. The equations converge when the energy of the nth iteration is equal to the energy of the previous iteration, resulting in the correct eigenstates. There is a conceptual doubt about whether the iteration is trying to fit the old solution in a new well, and the question of why using 4/alpha instead of 2/alpha does not work. The need for further explanation and clarification is also mentioned
  • #1
Braggplane
6
0
hi guys i need some help with the iteration made in a numerical solution for the eigenstates in a finite square well potential using the effective length approximation

First, i find the eigenstate for a infinite square well, then i define the related alpha and i use it to define an effective length as the nominal one plus 2/alpha, and so on.

the n iteration will use the nominal length plus 2/alpha(n-1) to find the n eigenstate and to define the n alpha (which depends on E)

In this way my equations converge when En=En-1 (and i find the rigth eigenstates).

I have a conceptual doubt: when i define the new length, is the iteration trying to fit the old solution in a new well? (nominal length plus 2/alpha contain almost all the wave function)

but why if i make the same thing with 4/alpha doesn't work?
 
Physics news on Phys.org
  • #2
Braggplane said:
hi guys i need some help with the iteration made in a numerical solution for the eigenstates in a finite square well potential using the effective length approximation

First, i find the eigenstate for a infinite square well, then i define the related alpha and i use it to define an effective length as the nominal one plus 2/alpha, and so on.

the n iteration will use the nominal length plus 2/alpha(n-1) to find the n eigenstate and to define the n alpha (which depends on E)

In this way my equations converge when En=En-1 (and i find the rigth eigenstates).

I have a conceptual doubt: when i define the new length, is the iteration trying to fit the old solution in a new well? (nominal length plus 2/alpha contain almost all the wave function)

but why if i make the same thing with 4/alpha doesn't work?

I think that we need to see a bit more explained. I am familiar with some methods (Cooley-Numerov) of numerically solving the 1-D Schrodinger equation.

What are you calculating as you iterate? What are you converging to?
 

1. What is a finite square well potential?

A finite square well potential is a type of potential energy function used in physics and engineering to describe the behavior of a particle in a confined space. It is characterized by a rectangular potential energy barrier, with a finite width and height, surrounded by regions of zero potential energy.

2. How is the finite square well potential solved numerically?

The finite square well potential can be solved numerically using several methods, such as the shooting method, the transfer matrix method, and the finite difference method. These methods involve discretizing the potential energy function and solving for the wave function of the particle within the well.

3. What are the boundary conditions for the finite square well potential?

The boundary conditions for the finite square well potential depend on the type of potential (attractive or repulsive) and the energy of the particle. In general, the wave function must be continuous at the boundaries of the well, and its first derivative must also be continuous unless there is an infinite potential at the boundaries.

4. How does the solution for the finite square well potential depend on the parameters of the well?

The solution for the finite square well potential depends on the width and height of the potential barrier, as well as the energy of the particle. As these parameters change, the shape and behavior of the wave function within the well will also change, leading to different solutions.

5. What are some applications of the finite square well potential?

The finite square well potential has various applications in physics, such as in quantum mechanics to model the behavior of particles in a confined space, and in solid-state physics to describe the behavior of electrons in a crystal lattice. It is also used in engineering to model the behavior of particles in potential wells, such as in electronic devices and particle accelerators.

Similar threads

  • Atomic and Condensed Matter
Replies
4
Views
1K
  • Atomic and Condensed Matter
Replies
0
Views
725
Replies
12
Views
243
  • Quantum Physics
Replies
10
Views
1K
Replies
7
Views
1K
  • Advanced Physics Homework Help
Replies
14
Views
740
  • Advanced Physics Homework Help
Replies
19
Views
327
  • Advanced Physics Homework Help
Replies
4
Views
2K
Replies
1
Views
1K
  • Linear and Abstract Algebra
Replies
2
Views
959
Back
Top