Working with Three-Dimensional Rectangular Potentials
Applying the Schrödinger Equation in Three Dimensions
Calculate the Wave Function of a Hydrogen Atom Using the Schrödinger Equation

Finding the x, y, and z Equations for Three-Dimensional Free Particle Problems

At some point, your quantum physics instructor may want you to find the x, y, and z equations for three-dimensional free particle problems. Take a look at the x equation for the free particle,

image0.png

You can write its general solution as

image1.png

where Ay and Az are constants.

Because

image2.png

you get this for

image3.png

where A= Ax Ay Az.

The part in the parentheses in the exponent is the dot product of the vectors

image4.png

That is, if the vector a = (ax, ay, az) in terms of components and the vector b = (bx, by, bz), then the dot product of a and b is

image5.png

So here’s how you can rewrite the

image6.png
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Determining the Radial Part of a Wave Function
Translate the Schrödinger Equation to Three Dimensions
Applying the Radial Equation Inside the Square Well
Finding the Schrödinger Equation for the Hydrogen Atom
How to Add Time Dependence and Get a Physical Equation for Three-Dimensional Free Particle Problems
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