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:
![image0.png](https://www.dummies.com/wp-content/uploads/397882.image0.png)
What you will find is that you can actually ignore
![image1.png](https://www.dummies.com/wp-content/uploads/397883.image1.png)
and go straight on to
![image2.png](https://www.dummies.com/wp-content/uploads/397884.image2.png)
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:
![image3.png](https://www.dummies.com/wp-content/uploads/397885.image3.png)
Substituting the preceding equation into the one before it gives you the following:
![image4.png](https://www.dummies.com/wp-content/uploads/397886.image4.png)
And dividing this equation by
![image5.png](https://www.dummies.com/wp-content/uploads/397887.image5.png)
gives you
![image6.png](https://www.dummies.com/wp-content/uploads/397888.image6.png)
This equation has terms that depend on either
![image7.png](https://www.dummies.com/wp-content/uploads/397889.image7.png)
but not both. That means you can separate this equation into two equations, like this (where the total energy, E, equals ER + Er):
![image8.png](https://www.dummies.com/wp-content/uploads/397890.image8.png)
Multiplying
![image9.png](https://www.dummies.com/wp-content/uploads/397891.image9.png)
gives you
![image10.png](https://www.dummies.com/wp-content/uploads/397892.image10.png)
And multiplying
![image11.png](https://www.dummies.com/wp-content/uploads/397893.image11.png)
gives you
![image12.png](https://www.dummies.com/wp-content/uploads/397894.image12.png)
Now you have two Schrödinger equations, which you can solve independently.
So, using
![image13.png](https://www.dummies.com/wp-content/uploads/397895.image13.png)
you can now solve for
![image14.png](https://www.dummies.com/wp-content/uploads/397896.image14.png)
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
![image15.png](https://www.dummies.com/wp-content/uploads/397897.image15.png)
Here, C is a constant and k is the wave vector, where
![image16.png](https://www.dummies.com/wp-content/uploads/397898.image16.png)
In practice, however, ER is so small that people almost always just ignore
![image17.png](https://www.dummies.com/wp-content/uploads/397899.image17.png)
— that is, they assume it to be 1. In other words, the real action is in
![image18.png](https://www.dummies.com/wp-content/uploads/397900.image18.png)
is the wave function for the center of mass of the hydrogen atom, and
![image19.png](https://www.dummies.com/wp-content/uploads/397901.image19.png)
is the wave function for a (fictitious) particle of mass m.