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How to Simplify and Split the Schrödinger Equation for Hydrogen

In quantum physics, you may need to simplify and split the Schrödinger equation for hydrogen. Here's the usual quantum mechanical Schrödinger equation for the hydrogen atom:

image0.png

The problem is that you're taking into account the distance the proton is from the center of mass of the atom, so the math is messy. If you were to assume that the proton is stationary and that rp = 0, this equation would break down to the following, which is much easier to solve:

image1.png

Unfortunately, that equation isn't exact because it ignores the movement of the proton, so you see the more-complete version of the equation in quantum mechanics texts.

To simplify the usual Schrödinger equation, you switch to center-of-mass coordinates. The center of mass of the proton/electron system is at this location:

image2.png

And the vector between the electron and proton is

r = rerp

Using vectors R and r instead of re and rp makes the Schrödinger equation easier to solve. The Laplacian for R is

image3.png

And the Laplacian for r is

image4.png

How can you relate

image5.png

to the usual equation's

image6.png

After the algebra settles, you get

image7.png

where M = me + mp is the total mass and

image8.png

is called the reduced mass. When you put together the equations for the center of mass, the vector between the proton and the electron, the total mass, and m, then the time-independent Schrödinger equation becomes the following:

image9.png

Then, given the vectors, R and r, the potential is given by,

image10.png

The Schrödinger equation then becomes

image11.png

This looks easier — the main improvement being that you now have |r| in the denominator of the potential energy term rather than |rerp|.

Because the equation contains terms involving either R or r but not both, the form of this equation indicates that it's a separable differential equation. And that means you can look for a solution of the following form:

image12.png

Substituting the preceding equation into the one before it gives you the following:

image13.png

And dividing this equation by

image14.png

gives you

image15.png

Well, well, well. This equation has terms that depend on either

image16.png

but not both. That means you can separate this equation into two equations, like this (where the total energy, E, equals ER + Er):

image17.png

Multiplying

image18.png

gives you

image19.png

And multiplying

image20.png

gives you

image21.png

Now you have two Schrödinger equations, which you can solve independently.

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