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Interactive apps for teaching

Each app is available in two versions. The .html version will run in a browser window, without the installation of the Geogebra software. All interactive features are available, but you cannot see or modify the source code. On some computers certain apps refuse to run for unknown reasons. The .ggb version is the original Geogebra file. To use it you must download the Geogebra software from the website. This method is more reliable (though still not infallible) and gives you full access to the underlying code.

If you try any of these, please send me an email with your reactions, comments, suggestions, or questions.

Math

This app is designed to help students explore and understand the ideas of average and local slope (derivative) of a function.

html version

ggb version

Top

This app is designed to help students explore and understand the relationship between a function and its derivative.

html version

ggb version

Top

This app is designed to help students explore and understand the relationship between a function and its integral.

html version

ggb version

Top

This app explores the various representations of a complex number, and its relationship to its complex conjugate.

html version

ggb version

Top

Waves and Quantum Mechanics

This app simulates interference between the waves from two in-phase sources. It can display both the wavefronts and the interaction of the waves at a particular point on the screen, and allows tracking of the interference pattern.

html version

ggb version

Top

This app uses the "wag the dog" method (see D.J. Griffiths, Introduction to Quantum Mechanics, ch. 2) to find approximate energy eigenstates and eigenvalues for various one-dimensional potentials.

html version

ggb version

Top

The analysis of the Kronig-Penney model of a one-dimensional solid leads to a rather opaque equation that implicitly gives the dispersion relation E(k), but can only be solved numerically (see R.L. Liboff, Introductory Quantum Mechanics, Ch. 8). This app provides some clarification of at least the mathematical problem, and allows you to trace out the E(k) curve for the three lowest bands.

html version

ggb version

Top

Shows the effect of a simple perturbation on the energies and wavefunctions of the two lowest states of a one-dimensional harmonic oscillator.

html version

ggb version

Top

Shows the effect of a linear perturbation on the energies and wavefunctions of the three lowest states of a one-dimensional harmonic oscillator, calculated in second-order perturbation theory.

html version

ggb version

Top

TopTopThis is a modified version of the "wag the dog" app above, intended to illustrate the variational approximation method. You can use the "wag the dog" method to find the energy and wavefunction for the first or second eigenstate of a linear or square potential, and then find the best variational approximation to the state using a trial wavefunction based either on a Gaussian or a cosine-squared function.

html version

ggb version

Top

This app illustrates the LCAO (linear combination of atomic orbitals) approximation method in a simple 1-D case. The potential is double harmonic well, and the wavefunction is the sum of two Gaussians centered at the well minima. The only variable is the well separation. As you drag the point at the bottom of one well, the expectation values of the kinetic energy <T>, potential energy <V> and total energy <H> are graphed; the presence of a minimum shows that there is a bound state. Because the program is doing a lot of integrating, the response time is slow, and you have to be patient.

html version

ggb version

Top

Each app is available in two versions. The .html version will run in a browser window, without the installation of the Geogebra software. All interactive features are available, but you cannot see or modify the source code. On some computers certain apps refuse to run for unknown reasons. The .ggb version is the original Geogebra file. To use it you must download the Geogebra software from the website. This method is more reliable (though still not infallible) and gives you full access to the underlying code.

If you try any of these, please send me an email with your reactions, comments, suggestions, or questions.

Math

This app is designed to help students explore and understand the ideas of average and local slope (derivative) of a function.

html version

ggb version

Top

This app is designed to help students explore and understand the relationship between a function and its derivative.

html version

ggb version

Top

This app is designed to help students explore and understand the relationship between a function and its integral.

html version

ggb version

Top

This app explores the various representations of a complex number, and its relationship to its complex conjugate.

html version

ggb version

Top

Waves and Quantum Mechanics

This app simulates interference between the waves from two in-phase sources. It can display both the wavefronts and the interaction of the waves at a particular point on the screen, and allows tracking of the interference pattern.

html version

ggb version

Top

This app uses the "wag the dog" method (see D.J. Griffiths, Introduction to Quantum Mechanics, ch. 2) to find approximate energy eigenstates and eigenvalues for various one-dimensional potentials.

html version

ggb version

Top

The analysis of the Kronig-Penney model of a one-dimensional solid leads to a rather opaque equation that implicitly gives the dispersion relation E(k), but can only be solved numerically (see R.L. Liboff, Introductory Quantum Mechanics, Ch. 8). This app provides some clarification of at least the mathematical problem, and allows you to trace out the E(k) curve for the three lowest bands.

html version

ggb version

Top

Shows the effect of a simple perturbation on the energies and wavefunctions of the two lowest states of a one-dimensional harmonic oscillator.

html version

ggb version

Top

Shows the effect of a linear perturbation on the energies and wavefunctions of the three lowest states of a one-dimensional harmonic oscillator, calculated in second-order perturbation theory.

html version

ggb version

Top

TopTopThis is a modified version of the "wag the dog" app above, intended to illustrate the variational approximation method. You can use the "wag the dog" method to find the energy and wavefunction for the first or second eigenstate of a linear or square potential, and then find the best variational approximation to the state using a trial wavefunction based either on a Gaussian or a cosine-squared function.

html version

ggb version

Top

This app illustrates the LCAO (linear combination of atomic orbitals) approximation method in a simple 1-D case. The potential is double harmonic well, and the wavefunction is the sum of two Gaussians centered at the well minima. The only variable is the well separation. As you drag the point at the bottom of one well, the expectation values of the kinetic energy <T>, potential energy <V> and total energy <H> are graphed; the presence of a minimum shows that there is a bound state. Because the program is doing a lot of integrating, the response time is slow, and you have to be patient.

html version

ggb version

Top