The Ritz Method

Tags: boundary conditions, differential equations, energy, Hamiltonian system, orthogonal, Physics, rayleigh-ritz method, ritz method, Science, trial wave function, Walter Ritz, wave

September 22, 2009

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How to Use the Ritz Method

Description

A detailed tutorial on the solving of problems using the Ritz Method. Step by step tutorial including several examples of how to solve problems with the Ritz Method for reference.

Overview

The Ritz Method, often called the Rayleigh-Ritz Method, is the finite element method used to find the eigenvalues of a Hamiltonian system. The Ritz Method starts out with a trial wave function, that is expressed as $E_0 \le \int \Psi^* \hat{H} \Psi \, d\tau$

From there, we can substitute a value for the trial wave function, which is $\Psi = \sum_{i=1}^N c_i \Psi_i.$

The final expected value can be written as:

$\varepsilon = \frac{\left\langle \sum_{i=1}^N c_i\Psi_i \right| \hat{H} \left| \sum_{i=1}^Nc_i\Psi_i \right\rangle}{\left\langle \sum_{i=1}^N c_i\Psi_i \Big| \sum_{i=1}^Nc_i\Psi_i \right\rangle} = \frac{\sum_{i=1}^N\sum_{j=1}^Nc_i^*c_jH_{ij}}{\sum_{i=1}^N\sum_{j=1}^Nc_i^*c_jS_{ij}} \equiv \frac{A}{B}$

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The Ritz Method

How to Use the Ritz Method

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