Quantum Physics For Dummies, Revised Edition
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When you apply the quantum mechanical Schrödinger equation for a hydrogen atom, the quantization condition for the wave function of r to remain finite as r goes to infinity is

image0.png

where

image1.png

Substituting

image2.png

into the quantization-condition equation gives you the following:

image3.png

Now solve for the energy, E. Squaring both sides of the preceding equation gives you

image4.png

So here’s the energy, E (Note: Because E depends on the principal quantum number, you rename it En):

image5.png

Physicists often write this result in terms of the Bohr radius — the orbital radius that Niels Bohr calculated for the electron in a hydrogen atom, r0. The Bohr radius is

image6.png

And in terms of r0, here’s what En equals:

image7.png

The ground state, where n = 1, works out to be about E = –13.6 eV.

Notice that this energy is negative because the electron is in a bound state — you’d have to add energy to the electron to free it from the hydrogen atom. Here are the first and second excited states:

  • First excited state, n = 2: E = –3.4 eV

  • Second excited state, n = 3: E = –1.5 eV

So you’ve now used the quantization condition, which is

image8.png

to determine the energy levels of the hydrogen atom.

About This Article

This article is from the book:

About the book author:

Steven Holzner is an award-winning author of technical and science books (like Physics For Dummies and Differential Equations For Dummies). He graduated from MIT and did his PhD in physics at Cornell University, where he was on the teaching faculty for 10 years. He’s also been on the faculty of MIT. Steve also teaches corporate groups around the country.

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