Working with Three-Dimensional Rectangular Potentials
How to Determine the Allowed Energies of a Hydrogen Atom
Applying the Schrödinger Equation in Three Dimensions

Applying the Radial Equation Outside the Square Well

In quantum physics, you can apply the radial equation outside a square well (where the radius is greater than a). In the region r > a, the particle is just like a free particle, so here's what the radial equation looks like:

image0.png

You solve this equation as follows:

image1.png

you substitute

image2.png

so that Rnl(r) becomes

image3.png

Using this substitution means that the radial equation takes the following form:

image4.png

The solution is a combination of spherical Bessel functions and spherical Neumann functions, where Bl is a constant:

image5.png

If the energy E < 0, you must have Cl = i Bl", so that the wave function decays exponentially at large distances r. So the radial solution outside the square well looks like this, where

image6.png

Given that the wave function inside the square well is

image7.png

So how do you find the constants Al and Bl? You find those constants through continuity constraints: At the inside/outside boundary, where r = a, the wave function and its first derivative must be continuous. So to determine Al and Bl, you have to solve these two equations:

image8.png
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Finding the Schrödinger Equation for the Hydrogen Atom
When to Use Spherical Coordinates Instead of Rectangular Coordinates
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Translate the Schrödinger Equation to Three Dimensions
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