Your quantum physics instructor may ask you to solve for the wave function for a made-up particle of mass m in a hydrogen atom. To do this, you can begin by using a modified Schrödinger equation that solves for large and small r:
![image0.png](https://www.dummies.com/wp-content/uploads/396746.image0.png)
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:
![image1.png](https://www.dummies.com/wp-content/uploads/396747.image1.png)
Substituting the preceding equation into the one before it gives you the following:
![image2.png](https://www.dummies.com/wp-content/uploads/396748.image2.png)
And dividing this equation by
![image3.png](https://www.dummies.com/wp-content/uploads/396749.image3.png)
gives you
![image4.png](https://www.dummies.com/wp-content/uploads/396750.image4.png)
This equation has terms that depend on either
![image5.png](https://www.dummies.com/wp-content/uploads/396751.image5.png)
but not both. That means you can separate this equation into two equations, like this (where the total energy, E, equals ER + Er):
![image6.png](https://www.dummies.com/wp-content/uploads/396752.image6.png)
Multiplying
![image7.png](https://www.dummies.com/wp-content/uploads/396753.image7.png)
gives you
![image8.png](https://www.dummies.com/wp-content/uploads/396754.image8.png)
And multiplying
![image9.png](https://www.dummies.com/wp-content/uploads/396755.image9.png)
gives you
![image10.png](https://www.dummies.com/wp-content/uploads/396756.image10.png)
Now you can solve for r, both small and large.
Solving for small r
The Schrödinger equation for
![image11.png](https://www.dummies.com/wp-content/uploads/396757.image11.png)
is the wave function for a made-up particle of mass m (in practice,
![image12.png](https://www.dummies.com/wp-content/uploads/396758.image12.png)
is pretty close to
![image13.png](https://www.dummies.com/wp-content/uploads/396759.image13.png)
so the energy, Er, is pretty close to the electron’s energy). Here’s the Schrödinger equation for
![image14.png](https://www.dummies.com/wp-content/uploads/396760.image14.png)
You can break the solution,
![image15.png](https://www.dummies.com/wp-content/uploads/396761.image15.png)
into a radial part and an angular part:
![image16.png](https://www.dummies.com/wp-content/uploads/396762.image16.png)
The angular part of
![image17.png](https://www.dummies.com/wp-content/uploads/396763.image17.png)
is made up of spherical harmonics,
![image18.png](https://www.dummies.com/wp-content/uploads/396764.image18.png)
so that part’s okay. Now you have to solve for the radial part, Rnl(r). Here’s what the Schrödinger equation becomes for the radial part:
![image19.png](https://www.dummies.com/wp-content/uploads/396765.image19.png)
where
![image20.png](https://www.dummies.com/wp-content/uploads/396766.image20.png)
To solve this equation, you take a look at two cases — where r is very small and where r is very large. Putting them together gives you the rough form of the solution.
Solving for large r
For very large r,
![image21.png](https://www.dummies.com/wp-content/uploads/396767.image21.png)
Because the electron is in a bound state in the hydrogen atom, E < 0; thus, the solution to the preceding equation is proportional to
![image22.png](https://www.dummies.com/wp-content/uploads/396768.image22.png)
Note that
![image23.png](https://www.dummies.com/wp-content/uploads/396769.image23.png)
diverges as r goes to infinity because of the
![image24.png](https://www.dummies.com/wp-content/uploads/396770.image24.png)
term, so B must be equal to zero. That means that
![image25.png](https://www.dummies.com/wp-content/uploads/396771.image25.png)