Outline

1. Wave Equations
2. Eigenfunctions
3. Potential Well
4. Radial Solution
5. Angular Solution
6. Visualization

Wave Equations

The motivation behind Schrodinger's view of quantum mechanics was that experiments showed that matter at some level seemed to interact as if it were a wave. The classic experiment in which this is seen is electron diffraction.

Schrodinger postulated that the equation that described elementary particles should be a wave equation.

Before studying Schrodinger's wave equation in detail, it is worth time reviewing the properties of wave equations.

Eigenfunctions

An important property of wave equations is that if they are subject to boundary conditions, it is often the case that only a few possible solutions can be obtained. These specific solutions are often called by their German name "eigensolutions". In linear algebra this is often seen as the solution to A x = b x, where you want to know what are the possible vectors which a matrix can be multiplied by with the result being proportional to the original vector. The vectors for which this is true are called the eigenvectors of the matrix, and the constant of proportionality b is called the eigenvalue.

In Physics, what one typically has is a differential equation for which a differential operator is applied to some function, and the result is proportional to the original function. What is typically desired are the solutions of that function that meet some boundary conditions, and those functions are called the eigenfunctions of the differential equation. The constants of proportionality are called the eigenvalues.

You should review eigenvalue problems in Physics.

Potential Well

Before attempting to apply Schrodinger's equation in 3 dimensions, it may be useful to consider a simple 1 dimensional case, the infinite potential well.

Radial Solution

Schrodinger's wave equation is an equation in 3 dimensions. Both analytically and numerically, problems in multiple dimensions are generally harder to solve than problems in 1 dimension.

A common method of approaching 3 dimensional problems is to attempt to break the problem up into 3 one dimensional problems. If the solution for each dimension does not depend on the other dimension, this can obviously be done. With Schrodinger's equation, this is not entirely true, but one can still consider the problem "separable" in that the problem in the azimuthal dimension (angle around the x-y plane) does not depend on any of the other dimensions, the problem in the polar (angle from the z-axis) dimension depends only on the azimuthal solution, and the radial problem can be solved if the polar solution is known. Each of the solutions solve their respective differential operator terms in Schrodinger's equations separately, and the final solution can be written as a product of the three independent solutions.

With some assumptions about the solution of the angular problem, you should solve the radial portion of Schrodinger's Equation.

Angular Solution

The solution of the angular term in the three dimensional wave equation comes up frequently in physics, and the solution has been well studied. Advanced students should attempt to solve the angular portion of Schrodinger's Equation

Visualization

The visualization of electron clouds for Hydrogen will combine the solutions of the radial and angular solutions to Shrodinger's Equation with advanced visualization tools.