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The Ritz Method

Tuesday, September 22nd, 2009

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:

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

Tags: boundary conditions, differential equations, energy, Hamiltonian system, orthogonal, Physics, rayleigh-ritz method, ritz method, Science, trial wave function, Walter Ritz, wave
Posted in Differential Equations | No Comments »

Graphing: Cosine Function

Thursday, September 17th, 2009

How to Graph the Cosine Function

Description

A detailed tutorial on the solving of the graph of the cosine function. Step by step tutorial including several examples of how to solve the graph of the cosine function for reference.

Overview

Graphing the cosine function is not difficult, but there are a few steps you need to follow. The first is, you need to find all the different points on the graph. You do this by taking a unit circle and using radians and reference points to find all of your coordinates. Then plot your points on the graph, and “connect the dots”. The graph of the sine function should resemble a “wave” That simply goes down once in a big loop and comes back up again.

Tags: coordinates, cosine, function, graph, graphing, intercepts, Math, reference angle, trig, trigonometry, unit circle, wave, x, y
Posted in Trigonometry | No Comments »

Graphing: Sine Function

Thursday, September 17th, 2009

How to Graph the Sine Function

Description

A detailed tutorial on the solving of the graph of the sine function. Step by step tutorial including several examples of how to solve the graph of the sine function for reference.

Overview

Graphing the sine function is not difficult, but there are a few steps you need to follow. The first is, you need to find all the different points on the graph. You do this by taking a unit circle and using radians and reference points to find all of your coordinates. Then plot your points on the graph, and “connect the dots”. The graph of the sine function has points at (0, 0), (pi/2, 1), (pi, 0), (3pi/2, -1), and (2, 0). The x-coordinates are all the main points around the circle while the y-cooridnates are your reference points. The graph of the sine function should resemble a “wave” that starts at the origin and travels in curves going both up and down.

Tags: coordinates, function, graph, graphing, intercepts, Math, reference angle, sine, trig, trigonometry, unit circle, wave, x, y
Posted in Trigonometry | No Comments »