Calculate the Wave Function of a Hydrogen Atom Using the Schrödinger Equation

If your quantum physics instructor asks you to find the wave function of a hydrogen atom, you can start with the radial Schrödinger equation, Rnl(r), which tells you that

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The preceding equation comes from solving the radial Schrödinger equation:

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The solution is only good to a multiplicative constant, so you add such a constant, Anl (which turns out to depend on the principal quantum number n and the angular momentum quantum number l), like this:

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You find Anl by normalizing Rnl(r).

Now try to solve for Rnl(r) by just flat-out doing the math. For example, try to find R10(r). In this case, n = 1 and l = 0. Then, because N + l + 1 = n, you have N = nl – 1. So N = 0 here. That makes Rnl(r) look like this:

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And the summation in this equation is equal to

image4.png

And because l = 0, rl = 1, so

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Therefore, you can also write

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where r0 is the Bohr radius. To find A10 and a0, you normalize

image7.png

to 1, which means integrating

image8.png

over all space and setting the result to 1.

image9.png

and integrating the spherical harmonics, such as Y00, over a complete sphere,

image10.png

gives you 1. Therefore, you’re left with the radial part to normalize:

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Plugging

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into

image13.png

gives you

image14.png

You can solve this kind of integral with the following relation:

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With this relation, the equation

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becomes

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Therefore,

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This is a fairly simple result. Because A10 is just there to normalize the result, you can set A10 to 1 (this wouldn’t be the case if

image19.png

involved multiple terms). Therefore,

image20.png

That’s fine, and it makes R10(r), which is

image21.png

You know that

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And so

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becomes

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Whew. In general, here’s what the wave function

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looks like for hydrogen:

image26.png

where

image27.png

is a generalized Laguerre polynomial. Here are the first few generalized Laguerre polynomials:

image28.png
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