How to solve 2d problems numerically.

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In summary, a 2d problem is a problem with two variables that requires finding values that satisfy given conditions or equations. Solving a 2d problem can be done either numerically or analytically, with the former using numerical methods and the latter using algebraic or calculus techniques. Some common numerical methods for solving 2d problems include the finite difference method, finite element method, and the boundary element method. The choice of numerical method depends on the specific problem and its characteristics, such as the type of equations, boundary conditions, and desired level of accuracy. Potential sources of error when solving 2d problems numerically include round-off error, truncation error, and errors in the initial or boundary conditions. It is important to
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I havn't had much classes on numerical methods in quantum mechanics and I'm wondering how one would solve a general problem involving 2d motion. With general, I mean a problem that cannot be separated. Consider for instance the hamiltonian

[itex]\hat{H} = \frac{\widehat{p}_{x}^{2}+\widehat{p}_{y}^{2}}{2m}+x^{2}y^{2}[/itex]

How does one find the eigenvalues and eigen functions numerically?
 
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1. Use a basis of e.g. harmonic oscillator eigenfunctions and a diagonalization routine for symmetric matrices (e.g. Lapack).
2. Use a grid of points and finite difference approximation for the derivatives. Then diagonalize the matrix like in 1.
 

Related to How to solve 2d problems numerically.

What is the definition of a 2d problem?

A 2d problem is a problem that involves two variables, typically represented on a two-dimensional graph, where the goal is to find the values of these variables that satisfy a given set of conditions or equations.

What is the difference between solving a 2d problem numerically and analytically?

Solving a 2d problem numerically involves using numerical methods, such as numerical integration or iterative methods, to approximate the solutions. Solving a 2d problem analytically involves finding the exact solutions using algebraic or calculus techniques.

What are some common numerical methods used to solve 2d problems?

Some common numerical methods used to solve 2d problems include the finite difference method, finite element method, and the boundary element method.

How do I choose the appropriate numerical method for a 2d problem?

The choice of numerical method depends on the specific problem and its characteristics. Factors to consider include the type of equations involved, the boundary conditions, and the desired level of accuracy.

What are some potential sources of error when solving 2d problems numerically?

Potential sources of error when solving 2d problems numerically include round-off error, truncation error, and errors in the initial or boundary conditions. It is important to use appropriate numerical techniques and carefully consider the accuracy and precision of the results.

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