How to Determine the Allowed Energies of a Hydrogen Atom

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

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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.

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