### The basics

- Select a colored off-diagonal matrix cell by clicking on it. Notice the highlighting of the two wavefunctions that correspond to the selected row and column.
- Rotate the angle dial (light blue disk) and see the effects on the rows and columns of the selected cell. Also watch the two selected wavefunctions mix with each other.
- Try to zero out the selected cell by rotating the dial until the cell is white (or black) and the displayed H
_{mn}value is close to zero. What does this do to the wavefunctions? - Continue zeroing out off-diagonal cells. What is happening to the matrix?
- If you are having a hard time distinguishing colored shades, adjust the contrast slider.
- Try playing around with the other settings or see below for explanations.

### Explanation

- The colored grid represents a Hamiltonian matrix for a quantum particle in one dimension, subject to the selected potential. The matrix is initially expressed in a basis of sinusoidal functions.
- Brighter grid cells represent larger matrix element magnitudes, while small magnitudes fade into a selected zero color (black or white).
- “Rotating a cell” corresponds to performing a rotation in the two-dimensional subspace spanned by the basis functions of the selected row and column; we’ll come back to this.
- As these rotations are performed the plotted wavefunctions are adjusted accordingly, at a vertical level corresponding to their average energies, with these energies also displayed numerically.
- The wavefunctions are drawn over a plot of the potential energy, so you can see which parts of the functions are in classically allowed regions. Note that the vertical scale for the wavefunctions is unrelated to the energy scale.
- The goal is to solve the time-independent Schrödinger equation (TISE) by diagonalizing the Hamiltonian matrix; this corresponds to making the grid white (or black) everywhere off the main diagonal. The plotted basis functions are then the energy eigenfunctions, and their energies are the corresponding eigenvalues.

### User interface

- The Rotate button is a short-cut to using the dial. It performs a rotation in the selected two-dimensional subspace to zero out the selected matrix element.
- You can use the Find max cell button to select the maximum (in magnitude) off-diagonal cell. Zeroing out this cell helps diagonalize the matrix more quickly.
- You can click on a diagonal matrix element to see its numerical energy value and highlight the corresponding wavefunction.
- Press the Reset button to reset the matrix to the original sine-wave basis functions.
- Use the n
_{max}slider to vary to size of the matrix. Higher settings give more accurate results, but the diagonalization process requires more steps. - Select a potential function to explore different one-dimensional quantum systems.
- Use the Draw a new potential option to draw a potential energy function of any shape, using the current potential function as a starting point.
- You can select colors to show positive and negative matrix elements, as well as a zero color.
- Adjust the contrast slider to change how the element magnitudes are related to brightness. A value of 1 sets the maximum brightness to the maximum energy on the grid.
- When you select a cell, its rows and columns are highlighted. To make this less pronounced you can uncheck “Cell highlighting”.