# Numerical solution of Schrödinger equation

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• aaaa202
In summary, the conversation discusses different methods for solving the Schrödinger equation numerically, including discretization and applying a finite difference scheme. However, the speaker is now facing a problem where they need to impose a non-zero boundary condition on the interval, which cannot be easily handled with the previous method. The suggestion is to use the shooting method or a basis expansion to satisfy the boundary conditions, despite the challenge of having different conditions for each state.
aaaa202
Suppose I want to solve the Schrödinger equation numerically for some potential V(x). The easiest way to do so, is to discretize it on a grid of finite length, and apply a finite difference scheme to approximate the second order derivative. Doing so yields an eigenvalue equation on matrix form for the wavefunctions and their corresponding energies, which may then be found by diagonalization.

The above method was the way I always solved the Schrödinger equation, when numerical work was needed. However, now I am faced with a problem, where I need to impose a boundary condition on the boundary of my interval. In the method above you implicitly assume that the wavefunctions outside the domain you are looking at, but for my current problem this will no longer work. Is there a way to adapt the finite difference method above to handle the case with a non-zero boundary condition? Worse even, my boundary condition differs for the different eigenmodes. I.e. the groundstate has one value at the boundary, the first excited another and so on. Could this also be incorporated easily?

aaaa202 said:
Suppose I want to solve the Schrödinger equation numerically for some potential V(x). The easiest way to do so, is to discretize it on a grid of finite length, and apply a finite difference scheme to approximate the second order derivative. Doing so yields an eigenvalue equation on matrix form for the wavefunctions and their corresponding energies, which may then be found by diagonalization.

The above method was the way I always solved the Schrödinger equation, when numerical work was needed. However, now I am faced with a problem, where I need to impose a boundary condition on the boundary of my interval. In the method above you implicitly assume that the wavefunctions outside the domain you are looking at, but for my current problem this will no longer work. Is there a way to adapt the finite difference method above to handle the case with a non-zero boundary condition? Worse even, my boundary condition differs for the different eigenmodes. I.e. the groundstate has one value at the boundary, the first excited another and so on. Could this also be incorporated easily?

I would propose the shooting method, see e.g. https://en.wikipedia.org/wiki/Shooting_method, combined with Runge-Kutta or other method to first-order differential equation. Alternatively, you could try a basis expansion where the basis functions satisfy the boundary conditions. The fact that the different states have different boundary conditions could be a bit tricky, but probably be handled.

## 1. What is the Schrödinger equation?

The Schrödinger equation is a fundamental equation in quantum mechanics that describes the behavior of quantum particles, such as electrons, in a given system. It is a differential equation that relates the time-dependent wavefunction of a particle to its energy.

## 2. Why is numerical solution of the Schrödinger equation important?

Numerical solutions of the Schrödinger equation are important because they allow us to study and understand the behavior of quantum systems that cannot be solved analytically. This is especially useful in complex systems, such as atoms and molecules, where finding exact solutions is not possible.

## 3. How is the Schrödinger equation solved numerically?

The Schrödinger equation can be solved numerically using various methods, such as the finite difference method, the finite element method, and the variational method. These methods involve discretizing the wavefunction and solving the resulting matrix equations to obtain an approximate solution.

## 4. What are the limitations of numerical solutions of the Schrödinger equation?

Numerical solutions of the Schrödinger equation are subject to errors due to the discretization process and the choice of numerical method. This can lead to inaccuracies in the results, especially for complex systems with many interacting particles. In addition, numerical solutions are limited by the computational resources available.

## 5. How are numerical solutions of the Schrödinger equation used in practice?

Numerical solutions of the Schrödinger equation are used in many areas of physics and chemistry to study the behavior of quantum systems. They are particularly useful in predicting and analyzing the properties of materials and molecules, and in developing new technologies such as quantum computers and quantum sensors.

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