# Applying the Schrödinger Equation in Three Dimensions

In quantum physics, you can apply the Schrödinger equation when you work on problems that have a central potential. These are problems where you're able to separate the wave function into a radial part (which depends on the form of the potential) and an angular part, which is a spherical harmonic.

Central potentials are spherically symmetrical potentials, of the kind where V(** r**) = V(

*r*). In other words, the potential is independent of the vector nature of the radius vector; the potential depends on only the magnitude of vector

**(which is**

*r**r*), not on the angle of

**.**

*r*The Schrödinger equation looks like this in three dimensions, where

is the Laplacian operator:

And the Laplacian operator looks like this in rectangular coordinates:

In spherical coordinates, it's a little messy, but you can simplify later. Check out the spherical Laplacian operator:

Here, **L**^{2} is the square of the orbital angular momentum:

So in spherical coordinates, the Schrödinger equation for a central potential looks like this when you substitute in the terms:

Take a look at the preceding equation. The first term actually corresponds to the *radial kinetic energy* — that is, the kinetic energy of the particle moving in the radial direction. The second term corresponds to the *rotational kinetic energy*. And the third term corresponds to the *potential energy*.

So what can you say about the solutions to this version of the Schrödinger equation? You can note that the first term depends only on *r*, as does the third, and that the second term depends only on angles. So you can break the wave function,

into two parts:

A radial part

A part that depends on the angles

This is a special property of problems with central potentials.