Solving the Wave Function of R Using the Schrödinger Equation

If your quantum physics instructor asks you to solve for the wave function of the center of mass of the electron/proton system in a hydrogen atom, you can do so using a modified Schrödinger equation:

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What you will find is that you can actually ignore

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and go straight on to

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Here’s how it works.

Because the Schrödinger 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:

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

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And dividing this equation by

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gives you

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This equation has terms that depend on either

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

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Multiplying

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gives you

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And multiplying

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gives you

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

So, using

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you can now solve for

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which is the wave function of the center of mass of the electron/proton system. This is a straightforward differential equation, and the solution is

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Here, C is a constant and k is the wave vector, where

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In practice, however, ER is so small that people almost always just ignore

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— that is, they assume it to be 1. In other words, the real action is in

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is the wave function for the center of mass of the hydrogen atom, and

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is the wave function for a (fictitious) particle of mass m.

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